# The Galois group of a Special Group

Kaique Matias de Andrade Roberto Hugo L. Mariano Institute of Mathematics and
Statistics, University of São Paulo, Brazil. Emails<EMAIL_ADDRESS><EMAIL_ADDRESS>authors want to express their gratitude to Coordenação
de Aperfeiçoamento de Pessoal de Nível Superior (Capes -Brazil) by the
financial support to develop this work: project Capes Math-Amsud number
88881.694471/2022-01.

###### Abstract

In this ongoing work, we extend to a class of well-behaved pre-special
hyperfields the work of J. Mináč and Spira ([14]) that describes a
(pro-2)-group of a field extension that encodes the quadratic form theory of a
given field $F$: in [1] it is shown that its associated cohomology ring
contains a copy of the cohomology ring of the field $F$. Our construction, a
contravariant functor into the category of "pointed" pro-2-groups, is
essentially given by generators and relations of profinite-2-groups. We prove
that such profinite groups $\mbox{Gal}(F)$ encode the space of orders of the
special group canonically associated to the hyperfield $F$ and provide a
criterion to detect when $F$ is formally real or not.

##  1 Introduction

The Igr’s functors $W_{*},k_{*}$ were extended by M. Dickmann and F.Miraglia
from the category of fields of characteristic $\neq 2$ to the category of
special groups (equivalently, the category of special hyperfields). Another
relevant Igr functor, the graded cohomology ring, $H^{*}(Gal(F^{s}|F),\\{\pm
1\\})$ remains defined only on the field setting. This chapter constitutes an
attempt to provide an Igr functor associated to a (Galois) cohomology theory
for special groups, based on the work of J. Minac and M. Spira [14]. We will
define - by "generator and relations", $\mbox{Gal}(G)$, the Galois Group of a
special group $G$, and provide some properties of this construction, as the
encoding of the orderings on $G$. However, since deeper results will depend of
a description of $\mbox{Gal}(G)$ "from below", and it still unavailable a
complete theory of algebraic extension of (super)hyperfields, we will not
pursue a more complete development of this cohomology theory in this work,
reserving it for a future research.

We will work in the category of pro-2-groups, and take, as usual, the
conventions: "subgroup" means "closed subgroup"; "subgroup generated by a
subset" means "the closure of the abstract group generated by the subset";
"morphism" means "continuous homomorphism".

The relationship between Galois groups of fields with orderings and quadratic
forms, established by the works of Artin-Schreier (1920’s) and Witt (late
1930’s) are reinforced by a seminal paper of John Milnor ([13], 1971) through
the definition of a (mod 2) k-theory graded ring that "interpolates" the
graded Witt ring and the cohomology ring of fields: the three graded rings
constructions determine functors from the category of fields where 2 is
invertible that, almost tree decades later, are proved to be naturally
isomorphic by the work of Voevodsky with co-authors.

Since the 1980’s, have appeared many abstract approaches to the algebraic
theory of quadratic forms over fields that are essentially equivalent (or
dually equivalent): between them we emphasize the theory of Special Groups
developed by Dickmann-Miraglia: a first-order axiomatization of the algebraic
theory of quadratic forms (see [8], [7], [9]), in this setting the quadratic
form behavior of a non 2 characteristic field $F$ is faithfully encoded by an
structure $G(F)$ consisting of an (multiplicative) exponent two group
$F^{\times}/F^{\times 2}$ with a distinguished element $-1/F^{\times 2}$ and a
quaternary relation that describes the isometry relation between two diagonal
quadratic forms of dimension 2. The notions of (graded) Witt rings and
k-theory are extended to the category of Special groups with remarkable pay-
offs on questions on quadratic forms over fields. In [7] they get a proof of
Marshall’s signature conjecture for Pythagorean by the relation between three
functors associated with the quadratic form theory of a field $F$ as described
by Milnor ([13]): the mod 2 k-theory functor of the field ($k_{\ast}(F)$) is a
source of two canonical arrows one for the graduated Witt Ring of the field
($(s_{n}(F):k_{n}(F)\rightarrow(I^{n}(F)/I^{n+1}(F)))_{n\in\mathbb{N}}$) and
the other one to the cohomology ring of the absolute Galois group of $F$ with
coefficients $\mathbb{Z}_{d}$ ($h_{n}(F):k_{n}(F)\rightarrow
H^{n}(gal(F^{s}|F),\mathbb{Z}_{d}))_{n\in\mathbb{N}}$). In [8] and [9] they
presented constructions that generalizes, to the wider context of special
groups, the first two constructions and obtained a similar natural connection
between them:
($(s_{n}(G):k_{n}(G)\rightarrow(I^{n}(F)/I^{n+1}(G)))_{n\in\mathbb{N}}$),
whenever $G$ is a special group.

In the work of Minac-Spira [14], for each field $F$ with $char(F)\neq 2$ they
prescribe an Galois extension $F^{(3)}|F$ that is a low subfield in the
genetic way to build the quadratic closure $F^{q}$ of $F$ 111Let
$F^{(2)}=F(\sqrt{a},a\in F)$ = compositum in $F^{q}$ of quadratic extensions
of $F$ then $F^{(3)}=F^{(2)}(\sqrt{y},$ such that $F^{(2)}(\sqrt{y}|F)$ is
Galois) encodes much of the quadratic forms information about $F$. They showed
that the Galois group of that extensions can be described by "generators and
relations" given by the dimension of the $\mathbb{Z}_{d}$-vector space
${F^{\times}}/{F^{\times}}^{2}$ : that is the starting point of the present
paper. In [1] it is proved that the cohomology ring of the Galois group of the
extension $F^{(3)}|F$ with coefficients $\mathbb{Z}_{d}$ contains the
cohomology ring of the absolute Galois group of $F$ with coefficients
$\mathbb{Z}_{d}$ and is the target of a canonical morphism of graduated rings
from the Milnor’s mod 2 reduced $K$-theory ring ($h_{n}(F):k_{n}(F)\rightarrow
H^{n}(gal(F^{s}|F),\mathbb{Z}_{d}))_{n\in\mathbb{N}}$).

A (commutative, unital) multiring is essentialy a (commutative, unital) ring
with multivalued sum and satisfying a weak distributivity law ([12]);
multirings that satisfy a full distributivity law are known as hyperrings. In
[6] and [2] are described encodings (i.e., equivalences of categories) between
the category of special groups (resp. pre-special groups; reduced special
groups) and the category of special hyperfields (resp. pre-special
hyperfields, real reduced hyperfields). Elements of a extension theory of
hyperfields are developed in [4], [roberto2022algebraic] and
[roberto2021hauptsatz].

In [3] is developed a k-theory for a category of hyperfields that contains the
category of pre-special hyperfields, this generalizes simultaneously Milnor’s
k-theory for fields ([13]) and Dickmann-Miraglia’s k-theory for special groups
([10]). In [5] is provided a detailed analysis of some categories of inductive
graded ring (IGR) - a concept introduced in [7] in order to provide a solution
of Marshall’s signature conjecture for Pythagorean fields; moreover the
functor of k-theory of a pre-special hyperfields is a (free) IGR.

In the present work we extend to a class of (well-behaved) pre-special
hyperfields the work of J. Mináč and Spira ([14]) that describes a
(pro-2)-group of a field extension that encodes the quadratic form theory of a
given field $F$: in [1] it is shown that its associated cohomology ring is
contains a copy of the cohomology ring of the field $F$. Our construction, a
contravariant functor into the category of "pointed" pro-2-groups, is
essentially given by generators and relations of profinite-2-groups. We prove
that such profinite groups $gal(F)$ encode the space of orders of the special
group canonically associated to the hyperfield $F$ and provide a criteria to
detect when $F$ is formally real or not. Moreover, we calculate the Galois
groups associated to some of the main constructions of special groups like
quotients, finite products, directed colimits and extensions.

A next step is to develop and understand the associated cohomology ring,
relating it with k-theory functor of the special groups, and apply such
machinery to questions on Special Groups theory. Our final goal is to develop
cohomological methods in SG-Theory, and apply such machinery to questions on
Special Groups theory. In particular, can we describe "cohomological
obstructions" for: (i) a reduced SG to satisfy Marshall’s signature
conjecture; (ii) a formally real SG to satisfy Lam’s conjecture? It should be
emphasized that is a SG does not satisfies any of the above properties, then
it cannot be isomorphic to a special group associated to a field, a major
question in the SG-theory.

##  2 The motivation: W-groups

The context that we will keep in mind is essentially that of the results
developed in Sections 1 and 2 of [14]. In this Section we will reproduce (and
expand the details of) part of these results.

Consider a field $F$. In [14], J. Minac and M. Spira define a special Galois
extension of the base field $F$, and determine its structure and its Galois
group through the behavior of quaternions algebras over $F$. As they developed
in [14], this extension contain essentially all the information need to
understand the behavior of quadratic forms over $F$.

Recall that the quadratic closure of $F$, denoted by $F_{q}$, is the smallest
extension of $F$ which is closed under taking of square roots (or more
explicit, the compositum of all 2-towers over $F$ inside a fixed algebraic
closure of $F$). The group $\mbox{Gal}_{F}(F_{q})$ will be denoted by
$G^{q}_{F}$.

Let $\\{a_{i}:i\in I\\}$ be a basis of $\dot{F}/\dot{F}^{2}$ (as Minac and
Spira did in their paper, we will assume that $1,2,..,n\in I$ with $1<2<...<n$
in order to easy our presentation). We define
$F^{(2)}=F(\sqrt{a}:a\in\dot{F})$ (note that $F^{(2)}=F(\sqrt{a_{i}}:i\in I$),
$\mathcal{E}=\\{y\in F^{(2)}:F^{(2)}(\sqrt{y})|F\mbox{ is Galois}\\}$ and
$F^{(3)}=F^{(2)}(\sqrt{y}:y\in\mathcal{E})$ if $F$; if $F$ is quadratically
closed we set $F^{(2)}=F^{(3)}=F$.

Minac and Spira built a strong connection between $F^{(3)}$ and the Witt ring
of $F$. They named $F^{(3)}$ as Witt closure of $F$. The group
$G_{F}:=\mbox{Gal}_{F}(F^{3}|F)$ is called the W-group of $F$. Our goal here
is to describe a way to factor $G_{F}$ as
$G_{F}\cong\mathcal{W}(I)/\mathcal{V}(I)$, with $\mathcal{W}(I)$ and
$\mathcal{V}(I)$ interesting profinite groups. This procedure will reveal how
to generalize $G_{F}$ in the context of abstract theories of quadratic forms,
and in particular, describe what would be a Galois group associated to a
special group. The first step is to describe $\mathcal{W}(I)$.

For an arbitrary group $G$, define $\hat{G}=G^{4}[G^{2},G]$, i.e, the (closed)
subgroup generated by fourth powers and by commutators of the form $[g^{2},h]$
for $g,h\in G$. Let $t^{4}[g^{2},h]\in\hat{G}$. Then, for each $z\in G$:

$\displaystyle z^{-1}(t^{4}[g^{2},h])z$
$\displaystyle=z^{-1}t^{4}[g^{2},h]z=(z^{-1}t^{4}z)(z^{-1}[g^{2},h]z)$

with $z^{-1}t^{4}z=(z^{-1}tz)^{4}\in G^{4}$ and
$z^{-1}[g^{2},h]z\in[G^{2},G]$, because

$\displaystyle z^{-1}[g^{2},h]z$ $\displaystyle=z^{-1}g^{-2}h^{-1}g^{2}hz$
$\displaystyle=(z^{-1}g^{-2}z)(z^{-1}h^{-1}z)(z^{-1}g^{2}z)(z^{-1}hz)$
$\displaystyle=(z^{-1}gz)^{-2}(z^{-1}hz)^{-1}(z^{-1}gz)^{2}(z^{-1}hz)$
$\displaystyle=[(z^{-1}gz)^{2},(z^{-1}hz)].$

Hence $\hat{G}$ is a normal subgroup of $G$, and we define
$\overline{G}=G/\hat{G}$. Let $\mathcal{C}$ denote the class of profinite
2-groups $G$ such that $\hat{G}=\\{1\\}$.

The main example (and the motivation to consider this full subcategory of
pro-2-groups) is the following fact: If $char(F)\neq 2$ then
$G=Gal(F^{(3)}|F)$ satisfies this condition $\hat{G}=\\{1\\}$, since, by
Proposition 2.1 in [14], $G\cong G_{q}/{G_{q}}^{4}.[{G_{q}}^{2},{G_{q}}]$,
where $G_{q}=Gal(F_{q}|F)$ and $F_{q}$ is a quadratic closure of $F$ .

A pro-$\mathcal{C}$-group will be called just $\mathcal{C}$-group, and
$\mathcal{C}$-group on $I$ if it has a minimal set of generators of
cardinality $|I|$.

Let $I$ be a well-ordered set. The next step is to describe a canonical way to
represent the elements of $\overline{S}$ where $S$ is the free pro-2-group on
a nonempty set $I$. Let

$\mathcal{W}(I):=\prod_{i\in
I}\mathbb{Z}_{2}\times\prod_{\begin{subarray}{c}i,j\in I\\\
i<j\end{subarray}}\mathbb{Z}_{2}\times\prod_{i\in I}\mathbb{Z}/2\mathbb{Z}.$

Here we are considering $\mathbb{Z}_{2}$ multiplicatively, i.e,
$\mathbb{Z}_{2}\cong\\{1,-1\\}$. A typical element of $\mathcal{W}_{I}$ will
be written as $(t_{i}^{\alpha_{i}})(t_{ij}^{\beta_{ij}})(x_{i}^{\gamma_{i}})$,
where $\alpha_{i},\beta_{ij},\gamma_{i}\in\\{0,1\\}$ and
$t_{i}=t_{ij}=x_{i}=-1$ for all $i,j\in I$.

Let $g,h\in\mathcal{W}(I)$

$\displaystyle g$
$\displaystyle=(t_{i}^{\alpha_{i}})(t_{ij}^{\beta_{ij}})(x_{i}^{\gamma_{i}})$
$\displaystyle h$
$\displaystyle=(t_{i}^{\alpha^{\prime}_{i}})(t_{ij}^{\beta^{\prime}_{ij}})(x_{i}^{\gamma^{\prime}_{i}}).$

We define

$gh=(t_{i}^{\alpha_{i}+\alpha^{\prime}_{i}+\gamma_{i}\gamma^{\prime}_{i}})(t_{ij}^{\beta_{ij}+\beta^{\prime}_{ij}+\gamma^{\prime}_{i}\gamma_{j}})(x_{i}^{\gamma_{i}+\gamma^{\prime}_{i}}),$

where the exponents are taken modulo 2.

######  Lemma 2.1.

With the notation described above, $\mathcal{W}(I)$ is a group.

###### Proof.

Let

$\displaystyle g$
$\displaystyle=(t_{i}^{\alpha_{i}})(t_{ij}^{\beta_{ij}})(x_{i}^{\gamma_{i}})$
$\displaystyle h$
$\displaystyle=(t_{i}^{\alpha^{\prime}_{i}})(t_{ij}^{\beta^{\prime}_{ij}})(x_{i}^{\gamma^{\prime}_{i}})$
$\displaystyle k$
$\displaystyle=(t_{i}^{\alpha^{\prime\prime}_{i}})(t_{ij}^{\beta^{\prime\prime}_{ij}})(x_{i}^{\gamma^{\prime\prime}_{i}})$
$\displaystyle 1$ $\displaystyle=(t_{i}^{0})(t_{ij}^{0})(x_{i}^{0}).$

First of all,

$\displaystyle g\cdot 1$ $\displaystyle=(t_{i}^{\alpha_{i}+0+\gamma_{i}\cdot
0})(t_{ij}^{\beta_{ij}+0+0\cdot\gamma_{j}})(x_{i}^{\gamma_{i}+0})=g,$
$\displaystyle 1\cdot g$
$\displaystyle=(t_{i}^{0+\alpha_{i}+0\cdot\gamma_{i}})(t_{ij}^{0+\beta_{ij}+\gamma_{i}\cdot
0})(x_{i}^{0+\gamma_{i}})=g,$

hence $1$ is in fact the neutral element of $\mathcal{W}(I)$. In order to find
$g^{-1}$, we need to solve a system of modulo 2 congruences.

$\displaystyle
gh:=1\Rightarrow\begin{cases}\alpha_{i}+\alpha^{\prime}_{i}+\gamma_{i}\gamma^{\prime}_{i}\equiv_{2}0\\\
\beta_{ij}+\beta^{\prime}_{ij}+\gamma^{\prime}_{i}\gamma_{j}\equiv_{2}0\\\
\gamma_{i}+\gamma^{\prime}_{i}\equiv_{2}0\end{cases}\Rightarrow\begin{cases}\gamma^{\prime}_{i}\equiv_{2}\gamma_{i}\\\
\beta^{\prime}_{ij}\equiv_{2}\beta_{ij}+\gamma_{i}\gamma_{j}\\\
\alpha^{\prime}_{i}\equiv_{2}\alpha_{i}+\gamma_{i}\end{cases}$

Then, taking

$g^{-1}=(t_{i}^{\alpha_{i}+\gamma_{i}})(t_{ij}^{\beta_{ij}+\gamma_{i}\gamma_{j}})(x_{i}^{\gamma_{i}})$

we obtain $g\cdot g^{-1}=g^{-1}\cdot g=1$.

Finally, for associativity, we have

$\displaystyle(g\cdot h)\cdot k$
$\displaystyle=[(t_{i}^{\alpha_{i}+\alpha^{\prime}_{i}+\gamma_{i}\gamma^{\prime}_{i}})(t_{ij}^{\beta_{ij}+\beta^{\prime}_{ij}+\gamma^{\prime}_{i}\gamma_{j}})(x_{i}^{\gamma_{i}+\gamma^{\prime}_{i}})]\cdot
k$
$\displaystyle=(t_{i}^{\alpha_{i}+\alpha^{\prime}_{i}+\alpha^{\prime\prime}_{i}+\gamma_{i}\gamma^{\prime}_{i}+(\gamma_{i}+\gamma^{\prime}_{i})\gamma^{\prime\prime}_{i}})(t_{ij}^{\beta_{ij}+\beta^{\prime}_{ij}+\beta^{\prime\prime}_{ij}+\gamma^{\prime}_{i}\gamma_{j}+\gamma^{\prime\prime}_{i}(\gamma_{j}+\gamma^{\prime}_{j})})(x_{i}^{\gamma_{i}+\gamma^{\prime}_{i}+\gamma^{\prime\prime}_{i}})$
$\displaystyle=(t_{i}^{\alpha_{i}+\alpha^{\prime}_{i}+\alpha^{\prime\prime}_{i}+\gamma_{i}\gamma^{\prime}_{i}+\gamma_{i}\gamma^{\prime\prime}_{i}+\gamma^{\prime}_{i}\gamma^{\prime\prime}_{i}})(t_{ij}^{\beta_{ij}+\beta^{\prime}_{ij}+\beta^{\prime\prime}_{ij}+\gamma^{\prime}_{i}\gamma_{j}+\gamma^{\prime\prime}_{i}\gamma_{j}+\gamma^{\prime\prime}_{i}\gamma^{\prime}_{j}})(x_{i}^{\gamma_{i}+\gamma^{\prime}_{i}+\gamma^{\prime\prime}_{i}})$

and

$\displaystyle g\cdot(h\cdot k)$
$\displaystyle=g\cdot[(t_{i}^{\alpha_{i}^{\prime}+\alpha^{\prime\prime}_{i}+\gamma^{\prime}_{i}\gamma^{\prime\prime}_{i}})(t_{ij}^{\beta^{\prime}_{ij}+\beta^{\prime\prime}_{ij}+\gamma^{\prime\prime}_{i}\gamma^{\prime}_{j}})(x_{i}^{\gamma^{\prime}_{i}+\gamma^{\prime\prime}_{i}})]$
$\displaystyle=(t_{i}^{\alpha_{i}+\alpha^{\prime}_{i}+\alpha^{\prime\prime}_{i}+\gamma^{\prime}_{i}\gamma^{\prime\prime}_{i}+\gamma_{i}(\gamma^{\prime}_{i}+\gamma^{\prime\prime}_{i})})(t_{ij}^{\beta_{ij}+\beta^{\prime}_{ij}+\beta^{\prime\prime}_{ij}+\gamma^{\prime\prime}_{i}\gamma^{\prime}_{j}+(\gamma^{\prime}_{i}+\gamma^{\prime\prime}_{i})\gamma_{j}})(x_{i}^{\gamma_{i}+\gamma^{\prime}_{i}+\gamma^{\prime\prime}_{i}})$
$\displaystyle=(t_{i}^{\alpha_{i}+\alpha^{\prime}_{i}+\alpha^{\prime\prime}_{i}+\gamma^{\prime}_{i}\gamma^{\prime\prime}_{i}+\gamma_{i}\gamma^{\prime}_{i}+\gamma_{i}\gamma^{\prime\prime}_{i}})(t_{ij}^{\beta_{ij}+\beta^{\prime}_{ij}+\beta^{\prime\prime}_{ij}+\gamma^{\prime\prime}_{i}\gamma^{\prime}_{j}+\gamma^{\prime}_{i}\gamma_{j}+\gamma^{\prime\prime}_{i}\gamma_{j}})(x_{i}^{\gamma_{i}+\gamma^{\prime}_{i}+\gamma^{\prime\prime}_{i}}).$

Thus $(g\cdot h)\cdot k=g\cdot(h\cdot k)$ completing the proof. ∎

######  Remark 2.2.

  1. a -

Note that, if $|I|=n$, then

$|\mathcal{W}(I)|=\left|\prod_{i\in
I}\mathbb{Z}_{2}\times\prod_{\begin{subarray}{c}i,j\in I\\\ i\leq
j\end{subarray}}\mathbb{Z}_{2}\times\prod_{i\in
I}\mathbb{Z}_{2}\right|=2^{(n^{2}+3n)/2}.$

  2. b -

The example above is the free $\mathcal{C}$ in $n$-generators. In fact: for
each $n\in\mathbb{N}$, let $F(n)$ be the free group in $n$-generators, then
$\mathcal{W}(n)\cong F(n)/\hat{F}(n)$.

  3. c -

It follows that the category of finite (discrete) groups $G$ with
$\hat{G}=\\{1\\}$ is a category of $\mathbb{Z}_{2}$-modules that is closed by
homomorphic images, subgroups and finite products. In particular,
$\mathbb{Z}_{2}=\mathbb{Z}/2\mathbb{Z},\mathbb{Z}_{4}=\mathbb{Z}/4\mathbb{Z}$
and $\mathbb{D}_{4}\cong\mathbb{Z}_{2}\ltimes\mathbb{Z}_{4}$ (the 8-element
dihedral group), are finite $\mathcal{C}$-groups.

Lets denote, for $k,l\in I$,

$\displaystyle t_{k}$ $\displaystyle:=(t_{i}^{\delta_{ik}})(1_{ij})(1_{i})$
$\displaystyle t_{kl}$
$\displaystyle:=(1_{i})(t_{ij}^{\delta_{(ij)(kl)}})(1_{i})$ $\displaystyle
x_{k}$ $\displaystyle:=(1_{i})(1_{ij})(x_{i}^{\delta_{ik}})$

where for all $i,j,k,l\in I$, $\delta_{ik}=1$ if $i=k$ and $\delta_{ik}=0$
otherwise; and $\delta_{(ij)(kl)}=1$ if $i=k$ and $j=l$, and
$\delta_{(ij)(kl)}=0$ otherwise. After some straightforward calculations we
obtain the following results.

######  Lemma 2.3.

Consider $t_{k},t_{kl},x_{k}$ as above. Then for all $g,h\in\mathcal{W}(I)$,
with $g=(t_{i}^{\alpha_{i}})(t_{ij}^{\beta_{ij}})(x_{i}^{\gamma_{i}})$,
$h=(t_{i}^{\alpha^{\prime}_{i}})(t_{ij}^{\beta^{\prime}_{ij}})(x_{i}^{\gamma^{\prime}_{i}})$
and
$z=(t_{i}^{\alpha^{\prime\prime}_{i}})(t_{ij}^{\beta^{\prime\prime}_{ij}})(x_{i}^{\gamma^{\prime\prime}_{i}})$,
we have the following:

  1. i -

$t_{k}\cdot t_{k}=1$.

  2. ii -

$x_{k}\cdot x_{k}=t_{k}$.

  3. iii -

If $k<l$ then $[x_{k},x_{l}]=t_{kl}$.

  4. iv -

$g^{-1}=(t_{i}^{\alpha_{i}+\gamma_{i}})(t_{ij}^{\beta_{ij}+\gamma_{i}\gamma_{j}})(x_{i}^{\gamma_{i}})$.

  5. v -

$g^{2}=(t_{i}^{\gamma_{i}})(t_{ij}^{\gamma_{i}\gamma_{j}})(1_{i})$.

  6. vi -

$h^{g}=ghg^{-1}=(t_{i}^{\alpha^{\prime}_{i}})(t_{ij}^{\beta^{\prime}_{ij}+\gamma_{i}\gamma^{\prime}_{j}+\gamma^{\prime}_{i}\gamma_{j}})(x_{i}^{\gamma^{\prime}_{i}})$.

  7. vii -

$[g,h]=(1_{i})(t_{ij}^{\gamma_{i}\gamma^{\prime}_{j}+\gamma^{\prime}_{i}\gamma_{j}})(1_{j})$.

  8. viii -

$g^{4}=[g^{2},h]=1$.

  9. ix -

$[[z,w],h]=1$.

######  Remark 2.4.

In [14], they simply denote

$g=(x_{i}^{2\alpha_{i}})([x_{i},x_{j}]^{\beta_{ij}})(x_{i}^{\gamma_{i}}).$

But we are not going to use this simplification here.

Now, for each $i,j\in I$, consider the following three sets:

$\displaystyle M_{i}$
$\displaystyle:=\left\\{g=(t_{i}^{\alpha_{i}})(t_{ij}^{\beta_{ij}})(x_{i}^{\gamma_{i}})\in\mathcal{W}(I):\gamma_{i}=0\right\\},$
$\displaystyle S_{i}$
$\displaystyle:=\left\\{g=(t_{i}^{\alpha_{i}})(t_{ij}^{\beta_{ij}})(x_{i}^{\gamma_{i}})\in\mathcal{W}(I):\alpha_{i}=\gamma_{i}=0\right\\},$
$\displaystyle D_{ij}$
$\displaystyle:=\left\\{g=(t_{i}^{\alpha_{i}})(t_{ij}^{\beta_{ij}})(x_{i}^{\gamma_{i}})\in\mathcal{W}(I):\beta_{ij}=\gamma_{i}=\gamma_{j}=0\right\\}.$

Now, consider the following families

$\displaystyle M(I)$ $\displaystyle:=\\{M_{i}:i\in I\\},\,S(I):=\\{S_{i}:i\in
I\\},\,D(I):=\\{D_{ij}:i,j\in I,i\leq j\\},$ $\displaystyle V$
$\displaystyle:=M(I)\cup S(I)\cup D(I).$

######  Proposition 2.5.

Let $i,j\in I$.

  1. a -

$M_{i}$ is a maximal normal subgroup of $\mathcal{W}(I)$ such that
$\mathcal{W}(I)/M_{i}\cong\mathbb{Z}_{2}$.

  2. b -

$S_{i}$ is a normal subgroup of $\mathcal{W}(I)$ such that $S_{i}\subseteq
M_{i}$ and $\mathcal{W}(I)/S_{i}\cong\mathbb{Z}_{4}$.

  3. c -

$D_{ij}$ is a normal subgroup of $\mathcal{W}(I)$ such that $D_{ij}\subseteq
M_{i}\cap M_{j}$ and $\mathcal{W}(I)/D_{ij}\cong\mathbb{D}_{4}$.

  4. d -

$\bigcap V=\\{1\\}$.

###### Proof.

We establish the following notation: let $g\in\mathcal{W}(I)$. Then
$g=(t_{i}^{\alpha_{i}})(t_{ij}^{\beta_{ij}})(x_{i}^{\gamma_{i}})$ for suitable
$\alpha_{i},\beta_{ij},\gamma_{i}\in\\{0,1\\}$. We denote

$\displaystyle\alpha_{i}(g):=\alpha_{i},\,\beta_{ij}(g):=\beta_{ij}\mbox{ and
}\gamma_{i}(g):=\gamma_{i}.$

In this sense, if $g,h\in\mathcal{W}(I)$ then

$gh=(t_{i}^{\alpha_{i}(g)+\alpha_{i}(h)+\gamma_{i}(g)\gamma_{i}(h)})(t_{ij}^{\beta_{ij}(g)+\beta_{ij}(h)+\gamma_{i}(h)\gamma_{j}(g)})(x_{i}^{\gamma_{i}(g)+\gamma_{i}(h)}).$

Moreover, using the formulas in Lemma 2.3, we obtain that for all $i,j\in I$,
$M_{i}$, $S_{i}$ and $D_{ij}$ are proper normal subgroups of $\mathcal{W}(I)$.

  1. a -

Let $\tau,\theta\in\mathcal{W}(I)\setminus M_{i}$. Then
$\gamma_{i}(\tau)=\gamma_{i}(\theta)=1$ and

$\gamma_{i}(\theta^{-1}\tau)=\gamma_{i}(\theta)+\gamma_{i}(\tau)=0.$

Therefore $\theta^{-1}\tau\in M_{i}$ which imply

$\mathcal{W}(I)/M_{i}=\\{\overline{1},\overline{\tau}\\}\cong\mathbb{Z}_{2}.$

  2. b -

Note that $S_{i}\subseteq M_{i}$. Now, let $\tau,\theta\in M_{i}\setminus
S_{i}$. Then $\alpha_{i}(\tau)=\alpha_{i}(\theta)=1$ and

$\alpha_{i}(\theta^{-1}\tau)=[\alpha_{i}(\theta)+\gamma_{i}(\theta)]+\alpha_{i}(\tau)+\gamma_{i}(\theta)\gamma_{i}(\tau)=0.$

Hence $\theta^{-1}\tau\in S_{i}$, and $M_{i}/S_{i}\cong\mathbb{Z}_{2}$. So we
have an exact sequence

$1\rightarrow
M_{i}/S_{i}\xrightarrow{\iota}\mathcal{W}(I)/S_{i}\xrightarrow{\pi}\mathcal{W}(I)/M_{i}\rightarrow
1,$

where $\iota$ and $\pi$ are respectively the canonical inclusion and canonical
projection. Moreover

$\mathcal{W}(I)/S_{i}\cong M_{i}/S_{i}\rtimes\mathcal{W}(I)/M_{i}\mbox{ or
}\mathcal{W}(I)/S_{i}\cong\mathcal{W}(I)/M_{i}\ltimes M_{i}/S_{i}.$

In both cases,

$|\mathcal{W}(I)/S_{i}|=|M_{i}/S_{i}\times\mathcal{W}(I)/M_{i}|=|\mathbb{Z}_{2}\times\mathbb{Z}_{2}|=4.$

Now let $\sigma\in\mathcal{M}\setminus M_{i}$. We have $\sigma^{4}=1\in S_{i}$
with

$\displaystyle\gamma_{i}(\sigma^{3})$ $\displaystyle:=\gamma_{i}(\sigma)=1.$

Then $\overline{\sigma}^{3}\neq\overline{1}$ in $\mathcal{W}(I)/S_{i}$, which
proves that $\mathcal{W}(I)/S_{i}$ has an element of order 4. Then

$\mathcal{W}(I)/S_{i}\cong\mathbb{Z}_{4}.$

  3. c -

Remember that

$\mathbb{D}_{4}:=\langle r,s:r^{4}=s^{2}=(sr)^{2}=1\rangle.$

Using the same argument of item (b), we get $(M_{i}\cap
M_{j})/D_{ij}\cong\mathbb{Z}_{2}$ and $|\mathcal{W}(I)/(M_{i}\cap M_{j})|=4$
with

$|\mathcal{W}(I)/D_{ij}|=|(M_{i}\cap
M_{j})/D_{ij}\times\mathcal{W}(I)/(M_{i}\cap M_{j})|=8.$

More specifically, if we get $\tau_{1}\in(M_{i}\cap M_{j})\setminus D_{ij}$,
$\tau_{2},\theta_{2}\in M_{i}\setminus M_{j}$, $\tau_{3},\theta_{3}\in
M_{j}\setminus M_{i}$ and
$\tau_{4},\theta_{4}\in\mathcal{W}(I)\setminus(M_{i}\cup M_{j})$, with

$\displaystyle\beta_{ij}(\tau_{2})=\beta_{ij}(\tau_{3})=\beta_{ij}(\tau_{4})=1$
$\displaystyle\beta_{ij}(\theta_{2})=\theta_{ij}(\theta_{3})=\theta_{ij}(\tau_{4})=0$

then the following equations hold in $\mathcal{W}(I)/D_{ij}$

$\displaystyle\overline{\tau_{1}}^{2}=\overline{1}$
$\displaystyle\overline{\tau^{-1}_{4}\theta_{4}}=\overline{\tau^{-1}_{3}\theta_{3}}=\overline{\tau^{-1}_{2}\theta_{2}}=\overline{\tau_{1}}$

Then

$\mathcal{W}(I)/D_{ij}=\\{\overline{1},\overline{\tau_{1}},\overline{\tau_{2}},\overline{\tau_{3}},\overline{\tau_{4}},\overline{\tau_{1}\tau_{2}},\overline{\tau_{1}\tau_{3}},\overline{\tau_{1}\tau_{4}}\\},$

with the following table of multiplication:

$\cdot$ | $\overline{\tau_{1}}$ | $\overline{\tau_{2}}$ | $\overline{\tau_{3}}$ | $\overline{\tau_{4}}$ | $\overline{\tau_{1}\tau_{2}}$ | $\overline{\tau_{1}\tau_{3}}$ | $\overline{\tau_{1}\tau_{4}}$
---|---|---|---|---|---|---|---
$\overline{\tau_{1}}$ | $\overline{1}$ | $\overline{\tau_{1}\tau_{2}}$ | $\overline{\tau_{1}\tau_{3}}$ | $\overline{\tau_{1}\tau_{4}}$ | $\overline{\tau_{2}}$ | $\overline{\tau_{3}}$ | $\overline{\tau_{4}}$
$\overline{\tau_{2}}$ | $\overline{\tau_{1}\tau_{2}}$ | $\overline{1}$ | $\overline{\tau_{1}\tau_{4}}$ | $\overline{\tau_{3}}$ | $\overline{\tau_{1}}$ | $\overline{\tau_{4}}$ | $\overline{\tau_{1}\tau_{3}}$
$\overline{\tau_{3}}$ | $\overline{\tau_{1}\tau_{3}}$ | $\overline{\tau_{4}}$ | $\overline{1}$ | $\overline{\tau_{1}\tau_{2}}$ | $\overline{\tau_{1}\tau_{4}}$ | $\overline{\tau_{1}}$ | $\overline{\tau_{2}}$
$\overline{\tau_{4}}$ | $\overline{\tau_{1}\tau_{4}}$ | $\overline{\tau_{1}\tau_{3}}$ | $\overline{\tau_{2}}$ | $\overline{\tau_{1}}$ | $\overline{\tau_{3}}$ | $\overline{\tau_{1}\tau_{2}}$ | $\overline{1}$
$\overline{\tau_{1}\tau_{2}}$ | $\overline{\tau_{2}}$ | $\overline{\tau_{1}}$ | $\overline{\tau_{4}}$ | $\overline{\tau_{1}\tau_{3}}$ | $\overline{1}$ | $\overline{\tau_{1}\tau_{4}}$ | $\overline{\tau_{3}}$
$\overline{\tau_{1}\tau_{3}}$ | $\overline{\tau_{3}}$ | $\overline{\tau_{1}\tau_{4}}$ | $\overline{\tau_{1}}$ | $\overline{\tau_{2}}$ | $\overline{\tau_{4}}$ | $\overline{1}$ | $\overline{\tau_{1}\tau_{2}}$
$\overline{\tau_{1}\tau_{4}}$ | $\overline{\tau_{4}}$ | $\overline{\tau_{3}}$ | $\overline{\tau_{1}\tau_{2}}$ | $\overline{1}$ | $\overline{\tau_{1}\tau_{3}}$ | $\overline{\tau_{2}}$ | $\overline{\tau_{1}}$

Then denoting $r=\tau_{4}$ and $s=\tau_{2}$ we get $r^{4}=s^{2}=(sr)^{2}=1$
and

$\displaystyle\overline{1}$ $\displaystyle=s^{2}=r^{4},$
$\displaystyle\overline{\tau_{1}}$ $\displaystyle=r^{2},$
$\displaystyle\overline{\tau_{2}}$ $\displaystyle=s,$
$\displaystyle\overline{\tau_{3}}$ $\displaystyle=sr,$
$\displaystyle\overline{\tau_{4}}$ $\displaystyle=r,$
$\displaystyle\overline{\tau_{1}\tau_{2}}$ $\displaystyle=r^{2}s,$
$\displaystyle\overline{\tau_{1}\tau_{3}}$ $\displaystyle=r^{2}sr=rs,$
$\displaystyle\overline{\tau_{1}\tau_{4}}$ $\displaystyle=r^{3};$

witnessing the desired isomorphism.

  4. d -

By the very definition

$\displaystyle\bigcap
V=\left\\{g=(t_{i}^{\alpha_{i}})(t_{ij}^{\beta_{ij}})(x_{i}^{\gamma_{i}})\in\mathcal{W}(I):\beta_{ij}=\alpha_{i}=\gamma_{i}=\gamma_{j}=0\mbox{
for all }i,j\in I\right\\}=\\{1\\}.$

∎

Let $V$ as in Proposition 2.5 and let $P_{fin}(V)$ be the set of finite
subsets of $V$ and for $A\in P_{fin}(V)$, denote

$X_{A}=W_{I}/\bigcap A.$

Note that $P_{fin}(V)$ is a directed poset with the partial ordering induced
by inclusion. If $B\subseteq A$, denote by $\pi_{AB}:X_{A}\rightarrow X_{B}$
the canonical projection. Then $(X_{A},\pi_{AB},P_{fin}(V))$ is a projective
system, in the sense that $\pi_{AA}=id_{X_{A}}$ and, if $E\subseteq B\subseteq
A$, then $\pi_{AE}=\pi_{BE}\circ\pi_{AB}$.

######  Proposition 2.6.

The canonical “diagonal” function

$\mathcal{W}(I)\rightarrow\varprojlim_{A\in P_{fin}(V)}X_{A},\mbox{ which is
given by the rule }g\mapsto\left(g/X_{A}\right)_{A\in P_{fin}(V)}$

is an abstract group isomorphism, so, by transport, $\mathcal{W}(I)$ is a
(topological) profinite 2-group with $P_{fin}(V)$ as fundamental system of
clopen neighborhoods of $\\{1\\}$.

###### Proof.

Let $X=\prod_{A\in P_{fin}(V)}X_{A}$ and $\pi_{A}:X\rightarrow X_{A}$ be the
canonical projection. Denote $\Delta:\mathcal{W}(I)\rightarrow X$ the morphism
given by the rule $\Delta(g):=\left(g/X_{A}\right)_{A\in P_{fin}(V)}$. This
morphism $\Delta$ is injective since

$\mbox{Ker}(\Delta)=\bigcap_{A\in P_{fin}(V)}=\\{1\\}.$

Now, let $\overline{g}=\left(g/X_{A}\right)_{A\in
P_{fin}(V)}\in\mbox{Im}(\Delta)$. If $B\subseteq A$, we get

$\overline{g}_{B}=g/X_{B}=(g/X_{A})/X_{B}=(\pi_{AB}(g))/X_{B}.$

Moreover $\mbox{Im}(\Delta)\subseteq\varprojlim_{A\in P_{fin}(V)}X_{A}$. To
prove the surjectivity of $\Delta$, consider the morphism
$\pi_{A}:\mathcal{W}(I)\rightarrow X_{A}$ given by the canonical projection.
Then $(\mathcal{W}(I),\pi_{A})$ is a compatible system of surjective morphisms
where $\Delta$ is the exact morphism induced by $(\mathcal{W}(I),\pi_{A})$.
Then $\mbox{Im}(\Delta)$ is a dense subset of $\varprojlim_{A\in
P_{fin}(V)}X_{A}$ (for instance, see Lemma 1.1.7 of [15]) which is also
closed. If $\varphi_{A}:\varprojlim_{A\in P_{fin}(V)}X_{A}\rightarrow X_{A}$
denote the projection, we have a new projective system
$(\varphi_{A}(\mbox{Im}(\Delta),\pi_{AB}|_{\varphi_{A}(\mbox{Im}(\Delta)})$.
Then (using Corollary 1.1.8 of [15]) we get

$\mbox{Im}(\Delta)=\varprojlim_{A\in
P_{fin}(V)}\varphi_{A}(\mbox{Im}(\Delta)=\overline{\mbox{Im}(\Delta)}=\varprojlim_{A\in
P_{fin}(V)}X_{A}.$

Moreover

$\\{\mbox{Ker}(\pi_{A}):\mathcal{W}(I)\rightarrow X_{A}\\}_{A\in
P_{fin}(V)}=P_{fin}(V)$

is a fundamental system of neighborhoods of $\\{1\\}$. ∎

Lets invoke some terminology from the theory of profinite groups:

######  Definition 2.7.

Let $G$ be a profinite group.

  1. i -

We say that $X$ generates $G$ as a profinite group if $G=\overline{\langle
G\rangle}$. In that case, we call $X$ a set of topological generators of $G$.

  2. ii -

We say that $X\subseteq G$ converges to 1 if every open subgroup $U$ of $G$
contains all but a finite number of the elements in $X$.

  3. iii -

Let $G$ be a profinite group. The Frattini subgroup of $G$, notation
$\Phi(G)$, is the intersection of all its maximal open subgroups.

######  Fact 2.8.

Let $G$ be a profinite group.

  1. i -

$G$ is compact, Hausdorff and totally disconnected (has a topological basis of
clopens).

  2. ii -

A subgroup $U$ is open if and only if is closed of finite index (Lemma 2.1.2
of [15]).

  3. iii -

A closed subgroup $H$ of a profinite group $G$ is the intersection of all open
subgroups of $G$ containing $H$. If $H$ is normal, then $H$ is the
intersection of all open normal subgroups of $G$ containing $H$ (Proposition
2.1.4 of [15]).

  4. iv -

A maximal closed subgroup is necessarily open.

  5. v -

$\Phi(G)$ is a characteristic subgroup of $G$: for every automorphism
$\psi:G\rightarrow G$ of $G$ we have $\psi[\Phi(G)]=\Phi(G)$.

  6. vi -

If $h:H\rightarrow G$ is a continuous homomorphism of pro-2-groups then
$h[\Phi(H)]\subseteq\Phi[G]$.

######  Lemma 2.9 (Lemma 2.8.7 of [15]).

Let $p$ be a prime number and let $G$ be a pro-$p$ group.

  1. a -

Every maximal closed subgroup $M$ of $G$ has index $p$.

  2. b -

The Frattini quotient $G/\Phi(G)$ is a $p$-elementary abelian profinite group
and hence a vector space of the field $\mathbb{F}_{p}$ with $p$ elements.

  3. c -

$\Phi(G)=\overline{G^{p}[G,G]}$, where $G^{p}=\\{x^{p}:x\in G\\}$ and $[G,G]$
denotes the commutator subgroup of $G$.

######  Lemma 2.10.

Let ${\mathcal{G}}_{i}$, $i=0,1$ be projectives profinite groups and
${\mathcal{V}}_{i}\subseteq{\mathcal{G}}_{i}$ be normal closed subgroups such
that ${\mathcal{V}}_{i}\subseteq\Phi({\mathcal{G}}_{i})$. If
$f:{\mathcal{G}}_{0}/{\mathcal{V}}_{0}\ \to\
{\mathcal{G}}_{1}/{\mathcal{V}}_{1}$ is an epimorphism (respectively an
isomorphism) then there is some continuous homomorphism
$\tilde{f}:{\mathcal{G}}_{0}\ \to\ {\mathcal{G}}_{1}$ such that
$q_{1}\circ\tilde{f}=f\circ q_{0}$ where the $q_{i}$ are the projections on
quotient; besides any such lifting $\tilde{f}$ is an epimorphism (resp. an
isomorphism).

$\textstyle{\mathcal{G}_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{f}}$$\scriptstyle{q_{0}}$$\textstyle{\mathcal{G}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{1}}$$\textstyle{\mathcal{G}_{0}/\mathcal{V}_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\mathcal{G}_{1}/\mathcal{V}_{1}}$

In [14] is established the main categorical property of $\mathcal{W}(I)$:

######  Theorem 2.11 (Universal Property of $\mathcal{W}(I)$, Theorem 1.1 of
[14]).

The group $\mathcal{W}(I)$ is the $\mathcal{C}$-free group on $I$-generators.
In other words, $I\subseteq\mathcal{W}(I)$ is a generator set converging to 1
and if $f:I\rightarrow G$ is any function to a $\mathcal{C}$-group $G$ such
that $f[I]\subseteq G$ converges to 1 then there is an (unique) continuous
homomorphism $\tilde{f}:\mathcal{W}(I)\rightarrow G$ such that
$\tilde{f}|_{I}=f$. Moreover, if $H$ is any $\mathcal{C}$-group then $H$ has a
generator set converging to 1 of cardinality $|I|$ if and only if there is an
epimorphism $\mathcal{W}(I)\twoheadrightarrow H$ with kernel $V$ contained in
$\Phi(I)$, the Frattini subgroup of $\mathcal{W}(I)$.

######  Corollary 2.12.

Let $X=\\{x_{i}:i\in I\\}\subseteq\mathcal{W}(I)$. Then $X$ is a set of
generators of $\mathcal{W}(I)$ converging to 1.

######  Proposition 2.13.

We have $\Phi(I)=\mathcal{W}^{2}_{I}$, and $\Phi(I)$ has
$\\{x_{i},t_{ij}:i<j\in I\\}$ as a minimal set of generators converging to 1.
Moreover

$\displaystyle\Phi(I)$ $\displaystyle=\mathcal{W}^{2}_{I}=\bigcap M(I)$
$\displaystyle=\left\\{g=(t_{i}^{\alpha_{i}})(t_{ij}^{\beta_{ij}})(x_{i}^{\gamma_{i}})\in\mathcal{W}(I):\gamma_{i}=0\mbox{
for all }i\in I\right\\}$ $\displaystyle=\\{g\in\mathcal{W}(I):\mbox{there
exists }g_{1},g_{2},g_{3}\in\mathcal{W}(I)\mbox{ such that
}g=g_{1}^{2}g_{2}^{2}g_{3}^{2}\\}.$

We have some kind of duality theorems for $\Phi(I)$ and $\mathcal{W}(I)$.

######  Proposition 2.14.

Consider the $\mathbb{Z}_{2}$-module of homogeneous quadratic polynomials in
$I$ variables $\\{z_{i}\\}_{i\in I}$

$P_{2}(I)=\\{q\in\mathbb{Z}_{2}[I]:q=\sum_{i\in I}a_{i}z_{i}^{2}+\sum_{i<j\in
I}b_{ij}z_{i}z_{j}\\}\cong\bigoplus_{i\in
I}\mathbb{Z}_{2}\oplus\bigoplus_{i<j\in I}\mathbb{Z}_{2}.$

Then we have a topological group isomorphism

$\Phi(I)\cong\mbox{Hom}(P_{2}(I),\mathbb{Z}_{2})\cong\prod_{i\in
I}\mathbb{Z}_{2}\times\prod_{i<j\in I}\mathbb{Z}_{2},$

with the associated “perfect pairing”

$\langle\\_,\\_\rangle:\Phi(I)\times P_{2}(I)\rightarrow\mathbb{Z}_{2}.$

###### Proof.

First of all, note that $P_{2}(I)$ is generated (as $\mathbb{Z}_{2}$-vector
space) by the set of monomials $B=\\{z_{i}^{2},z_{i}z_{j}\\}_{i<j\in I}$. In
fact, this is a $\mathbb{Z}_{2}$-basis of $P_{2}(I)$. Let $B^{*}$ be the dual
basis

$B^{*}:=\\{\varphi_{ij}\\}_{i\leq j\in I},$

where $\varphi_{ij}:P_{2}(I)\rightarrow\mathbb{Z}_{2}$ is given by

$\varphi_{ij}(z_{k}z_{l})=\begin{cases}1\mbox{ if }i=k\mbox{ and }j=l\\\
0\mbox{ otherwise}\end{cases}$

Now we define a function $\lambda:\\{t_{i},t_{ij}:i<j\in I\\}\rightarrow
B^{*}$ by the rules $t_{i}\mapsto\varphi_{ii}$ and
$t_{ij}\mapsto\varphi_{ij}$. Since $\\{t_{i},t_{ij}:i<j\in I\\}$ is a set of
generators of $\Phi(I)$, this function $\lambda$ induces a continuous group
homomorphism
$\tilde{\lambda}:\Phi(I)\rightarrow\mbox{Hom}(P_{2}(I),\mathbb{Z}_{2})$ by the
following: let $g=(t_{i}^{\alpha_{i}})(t_{ij}^{\beta_{ij}})(1_{i})\in\Phi(I)$
(see Proposition 2.13). Define
$\tilde{\lambda}(g):P_{2}(I)\rightarrow\mathbb{Z}_{2}$ for $q=\sum_{i\in
I}a_{i}z_{i}^{2}+\sum_{i<j\in I}b_{ij}z_{i}z_{j}\in P_{2}(I)$ by

$\tilde{\lambda}(g)(q):=\sum_{i\in I}a_{i}\alpha_{i}+\sum_{i<j\in
I}b_{ij}\beta_{ij}.$

We immediately have that $\tilde{\lambda}$ is a continuous injective group
homomorphism. Since $\lambda$ is bijective and $B^{*}$ is a
$\mathbb{Z}_{2}$-basis of $\mbox{Hom}(P_{2}(I),\mathbb{Z}_{2})$, we have that
$\tilde{\lambda}$ is an isomorphism. ∎

######  Proposition 2.15.

Consider the $\mathbb{Z}_{2}$-module of homogeneous linear polynomials in $I$
variables $\\{z_{i}\\}_{i\in I}$

$P_{1}(I)=\\{q\in\mathbb{Z}_{2}[I]:q=\sum_{i\in
I}c_{i}z_{i}\\}\cong\bigoplus_{i\in I}\mathbb{Z}_{2}.$

Then we have a topological group isomorphism

$\mathcal{W}(I)/\Phi(I)\cong\mbox{Hom}(P_{1}(I),\mathbb{Z}_{2})\cong\prod_{i\in
I}\mathbb{Z}_{2},$

with the associated “perfect pairing”

$\langle\\_,\\_\rangle:\mathcal{W}(I)/\Phi(I)\times
P_{1}(I)\rightarrow\mathbb{Z}_{2}.$

###### Proof.

First of all, note that $P_{1}(I)$ is generated (as $\mathbb{Z}_{2}$-vector
space) by the set of monomials $B=\\{z_{i}\\}_{i\in I}$. In fact, this is a
$\mathbb{Z}_{2}$-basis of $P_{1}(I)$. Let $B^{*}$ be the dual basis

$B^{*}:=\\{\varphi_{i}\\}_{i\in I},$

where $\varphi_{i}:P_{1}(I)\rightarrow\mathbb{Z}_{2}$ is given by

$\varphi_{i}(z_{j})=\begin{cases}1\mbox{ if }i=j\\\ 0\mbox{
otherwise}\end{cases}$

Now we define
$\theta:\mathcal{W}(I)\rightarrow\mbox{Hom}(P_{1}(I),\mathbb{Z}_{2})$ by the
following: for
$g=(t_{i}^{\alpha_{i}})(t_{ij}^{\beta_{ij}})(x_{i}^{\gamma_{i}})\in\mathcal{W}(I)$,
let $\theta(g):P_{1}(I)\rightarrow\mathbb{Z}_{2}$ be the morphism defined by
the rule

$\mbox{For }q=\sum_{i\in I}c_{i}z_{i}\in P_{1}(I),\,\theta(g)(q)=\sum_{i\in
I}c_{i}\gamma_{i}.$

Then $\theta$ is a surjective morphism (because
$B^{*}\subseteq\mbox{Im}(\theta)$) and by Proposition 2.13
$\mbox{Ker}(\theta)=\Phi(I)$. Hence

$\mathcal{W}(I)/\Phi(I)=\mathcal{W}(I)/\mbox{Ker}(\theta)\cong\mbox{Hom}(P_{1}(I),\mathbb{Z}_{2}).$

∎

Now, is time to return to our first goal: present the description of
$G_{F}:=\mbox{Gal}_{F}(F^{(3)})$ by $G_{F}\cong\mathcal{W}(I)/\mathcal{V}(I)$.

######  Proposition 2.16 (2.1 in [14]).

$G_{F}\cong\overline{G^{q}_{F}}$.

Then $G_{F}$ is a $C$-group on $B$, where $B=\\{a_{i}:i\in I\\}$ is an well-
ordered basis of $\dot{F}/\dot{F}^{2}$, so, using Theorem 2.11, there exists
an epimorphism $\pi_{B}:\mathcal{W}(B)\rightarrow G_{F}$. Then we simply take
$\mathcal{V}(B):=\mbox{Ker}(\pi_{B})$.

Moreover, J. Minac and M. Spira (again, in [14]) gave a nice explicit
description to $\mathcal{V}(B)$. Let $\mbox{Quat}(F)$ be the subgroup of
$Br(F)$, the Brauer group of $F$, generated by the quaternion algebras of $F$.
By Merkurjev’s Theorem ([16]), we have

$\mbox{Quat}(F)\cong\mbox{Br}_{2}(F)\cong k_{2}(F),$

where $Br_{2}(F)$ is the subgroup of $Br(F)$ generated by elements of order 2
and $k_{*}(F)$ is the graded ring of Milnor’s mod 2 reduced K-theory.

Consider $\varphi_{B}:P_{2}(B)\twoheadrightarrow k_{2}(F)$ as the epimorphism
defined by the rule

$\left(\sum_{i\in I}\alpha_{i}z_{i}^{2}+\sum_{i<j\in
I}\beta_{ij}z_{i}z_{j}\right)\mapsto\left(\sum_{i\in
I}\alpha_{i}l(a_{i})l(a_{i})+\sum_{i<j\in
I}\beta_{ij}l(a_{i})l(a_{j})\right).$

Let $Q_{B}:=\mbox{Ker}(\varphi_{B})$.

######  Fact 2.17 (Essentially 2.20 in [14]).

$\mathcal{V}(B)=Q_{B}^{\perp}$, where $Q_{B}^{\perp}=\\{v\in\Phi(B):\langle
v,Q_{B}\rangle=0\\}$ and $\langle\ ,\ \rangle$ is the perfect pairing
described in Proposition 2.14.

Let a $F$ be a field with $char(F)\neq 2$. By “Pontryaguin duality”, let
$M_{F}$ denotes the unique maximal clopen subgroup of
$Gal_{F}=Gal_{F}(F^{(3)})$ corresponding to $-1\in SG(F)=\dot{F}/\dot{F}^{2}$
(Proposition 2.15).

######  Fact 2.18 (Essentially 3.3, 3.5, 3.6, 3.7 in [14]).

Let $F,L\in Field_{2}$. Then are equivalent:

  1. a -

$(W(F),\dot{F}/\dot{F}^{2})\cong(W(L),\dot{L}/\dot{L}^{2})$ as abstract Witt
rings.

  2. b -

$SG(F)\cong SG(L)$ as special groups.

  3. c -

$(G_{F},M_{F})\cong(G_{L},M_{L})$ as pointed profinite-${\mathcal{C}}$-groups.

Our next step, is use all these facts to obtain a group associated to a
(pre)-special group.

##  3 The Galois Group of a Pre Special Group

Lets deal first, with a special group $G$. Let $B=\\{a_{i}:i\in I\\}$ be a
well ordered $\mathbb{Z}_{2}$-basis of $G$ and consider the $\mathcal{C}$-free
group in $B$-generators $\mathcal{W}(B)$. Define an epimorphism
$\pi_{B}:P_{2}(B)\rightarrow k_{2}(G)$ by the rule

$\left(\sum_{i\in I}\alpha_{i}z_{i}^{2}+\sum_{i<j\in
I}\beta_{ij}z_{i}z_{j}\right)\mapsto\left(\sum_{i\in
I}\alpha_{i}l(a_{i})l(a_{i})+\sum_{i<j\in I}\beta_{ij}l(a_{i})l(a_{j})\right)$

with kernel $Q(B)$. Take
$\mathcal{V}(B):=Q(B)^{\perp}\subseteq\Phi(B)\subseteq\mathcal{W}(B)$. Since
$\Phi(B)$ is the center of $\mathcal{W}(B)$ then
$\mathcal{V}(B)\subseteq\mathcal{W}(B)$ is a (closed) normal subgroup of
$\mathcal{W}(B)$ and we can consider the $\mathcal{C}$-group
$\mathcal{W}(B)/\mathcal{V}(B)$.

######  Definition 3.1 (Galois Group - base dependent version).

Let $G$ be a special group and $B,\mathcal{W}(B)$ and $\mathcal{V}(B)$ as
above. We define the Galois group of $G$ with respect to $B$ by

$\mbox{Gal}(G,B):=\mathcal{W}(B)/\mathcal{V}(B)$

The most essential information of our Galois group is encoded by
$Q(B)=\mbox{Ker}(\pi_{B})$. We have a useful description by generators that
generalizes the one described by J. Minac and M. Spira:

######  Proposition 3.2.

Let $G$ be a special group and $B,\mathcal{W}(B)$ and $\mathcal{V}(B)$ as
above. Consider a finite subset $B^{\prime}\subseteq B$,
$B^{\prime}=\\{a_{i_{0}},...,a_{i_{n-1}}\\}$ ($i_{0}<...<i_{n-1}$), and $a,b$
in the linear span of $B^{\prime}$, say

$a=\prod_{k<n}a_{i_{k}}^{\alpha_{i_{k}}}\mbox{ and
}b=\prod_{k<n}a_{i_{k}}^{\beta_{i_{k}}},\,\alpha_{i_{k}},\beta_{i_{k}}\in\\{0,1\\}.$

Consider the polynomial $q^{B}_{a,b}\in P_{2}(B)$ given by

$q^{B}_{a,b}=\sum_{k<n}\alpha_{i_{k}}\beta_{i_{k}}z_{i_{k}}^{2}+\sum_{k<l<n}(\alpha_{i_{k}}\beta_{i_{l}}+\alpha_{i_{l}}\beta_{i_{k}})z_{i_{k}}z_{i_{l}}.$

Note that $q^{B}_{a,(b_{0}\cdot...\cdot b_{n-1})}=\sum_{k<n}q^{B}_{a,b_{k}}$.
Moreover we have the following properties.

  1. i -

$\pi_{B}(q^{B}_{a,b})=l(a)l(b)\in k_{2}(G)$.

  2. ii -

$q^{B}_{a,b}$ does not depend on the particular choice of the finite subset
$B^{\prime}\subseteq B$.

  3. iii -

$Q(B)$ is generated by $\\{q^{B}_{a,b}:l(a)l(b)=0\\}$.

###### Proof.

  1. i -

Note that

$l(a)=\sum_{k<n}\alpha_{i_{k}}l(a_{i_{k}})\mbox{ and
}l(b)=\sum_{k<n}\beta_{i_{k}}l(a_{i_{k}}).$

Then

$\displaystyle l(a)l(b)$
$\displaystyle=\left(\sum_{k<n}\alpha_{i_{k}}l(a_{i_{k}})\right)\left(\sum_{k<n}\beta_{i_{k}}l(a_{i_{k}})\right)=\sum_{k<n}\sum_{p<n}\alpha_{i_{k}}\beta_{i_{p}}l(a_{i_{k}})l(a_{i_{p}})$
$\displaystyle=\sum_{k<n}\alpha_{i_{k}}\beta_{i_{k}}l(a_{i_{k}})l(a_{i_{k}})+\sum_{k<n}\sum_{k<p<n}\alpha_{i_{k}}\beta_{i_{p}}l(a_{i_{k}})l(a_{i_{p}})+\sum_{k<n}\sum_{p<k<n}\alpha_{i_{k}}\beta_{i_{p}}l(a_{i_{k}})l(a_{i_{p}})$
$\displaystyle=\sum_{k<n}\alpha_{i_{k}}\beta_{i_{k}}l(a_{i_{k}})l(a_{i_{k}})+\sum_{k<n}\sum_{k<p<n}(\alpha_{i_{k}}\beta_{i_{p}}+\alpha_{i_{p}}\beta_{i_{k}})l(a_{i_{k}})l(a_{i_{p}})$

On the other hand, by definition of $\pi_{B}$ we get

$\displaystyle\pi_{B}(q^{B}_{a,b})$
$\displaystyle=\sum_{k<n}\alpha_{i_{k}}\beta_{i_{k}}l(a_{i_{k}})l(a_{i_{k}})+\sum_{k<p<n}(\alpha_{i_{k}}\beta_{i_{p}}+\alpha_{i_{p}}\beta_{i_{k}})l(a_{i_{k}})l(a_{i_{p}}),$

completing the proof.

  2. ii -

It is an immediate consequence of previous item: if $B_{1},B_{2}$ are finite
subsets of $B$ and $a,b$ are elements in the linear span of $B_{1}$ and in the
linear span of $B_{2}$, then

$q^{B_{1}}_{a,b}=q^{B_{2}}_{a,b}.$

  3. iii -

Of course, $q^{B}_{a,b}\in Q(B)$ if and only if $l(a)l(b)=0$ in $k_{2}(G)$ and
hence

$[\\{q^{B}_{a,b}:l(a)l(b)=0\\}]\subseteq Q(B).$

To get the reverse inclusion, let
$q=\sum_{k<n}\alpha_{i_{k}}z^{2}_{i_{k}}+\sum_{k,p<n}\beta_{i_{k}i_{p}}z_{i_{k}}z_{i_{p}}\in
Q$. Then

$\sum_{k<n}\alpha_{i_{k}}l(a_{i_{k}})l(a_{i_{k}})+\sum_{k,p<n}\beta_{i_{k}i_{p}}l(a_{i_{k}})l(a_{i_{p}})=0\mbox{
in }k_{2}(G).$

Now, for each $k<n$ let

$b_{k}:=a_{i_{k}}^{\alpha_{i_{k}}}\prod_{k<p}a_{i_{p}}^{\beta_{i_{k}i_{p}}}.$

Then $q=\sum_{k<n}q^{B}_{a_{i_{k}},b_{i_{k}}}$ and

$\sum_{k<n}l(a_{i_{k}})l(b_{i_{k}})=0\mbox{ in }k_{2}(G).$

We are under the hypothesis of Lemma 3.3. Thus, according Theorem
LABEL:fixsg3-ktheory, there exists subsets
$\\{c_{0},...,c_{m-1}\\},\\{d_{0},...,d_{n-1}\\}$ of $G$ with $m\geq n$ such
that

    1. (a)

$\\{c_{0},...,c_{m-1}\\}$ is linearly independent and $c_{k}=a_{i_{k}}$ for
all $k<n$;

    2. (b)

$d_{k}=b_{i_{k}}$ for all $k<n$ and $d_{k}=1$ for $k=n,...,m-1$.

    3. (c)

For all $x\in[c_{0},...,c_{m-1}]$, there is some $r_{x}\in D_{G}(1,-x)$ such
that for each $k<m$

$d_{k}=\prod_{x\in C_{k}}r_{x}$

where

$C_{k}=\left\\{\prod_{p<m}c_{p}^{\varepsilon_{p}}:\varepsilon_{p}\in\\{0,1\\}\mbox{
and }\varepsilon_{k}=1\right\\}.$

It follows that

$\displaystyle q$
$\displaystyle=\sum_{k<n}q^{B}_{a_{i_{k}},b_{i_{k}}}=\sum_{k<m}q^{B}_{c_{k},d_{k}}=\sum_{k<m}q^{B}_{c_{k},\prod_{x\in
C_{k}}r_{x}}$ $\displaystyle=\sum_{k<m}\sum_{x\in C_{k}}q^{B}_{c_{k},r_{x}}.$

Denoting $C:=[c_{0},...,c_{m-1}]$, we have $C=C_{0}\cup...\cup C_{m-1}$. Then

$\displaystyle q=\sum_{k<m}\sum_{x\in C_{k}}q^{B}_{c_{k},r_{x}}=\sum_{x\in
C}q^{B}_{x,r_{x}}.$

Since $r_{x}\in D_{G}(1,-x)$, we have $l(x)l(r_{x})=0$ in $k_{2}(G)$. Then

$q=\sum_{x\in C}q^{B}_{x,r_{x}}\in[\\{q^{B}_{a,b}:l(a)l(b)=0\\}].$

∎

Now, we will generalize the Galois group for pre-special groups. The K-theory
developed by M. Dickmann and F. Miraglia in [10] is available for pre-special
groups. Then we can take the same $B,\mathcal{W}(B)$ and $\mathcal{V}(B)$ we
are considering until now.

Let $G$ be a special group and $B=\\{v_{i}\\}_{i\in I}$, $C=\\{w_{i}\\}_{i\in
I}$, $D=\\{z_{i}\\}_{i\in I}$ be ordered $\mathbb{Z}_{2}$-basis of $G$. Then,
for all $i\in I$ we have an expression

$\displaystyle w_{i}=\prod_{k\in I}v_{k}^{m_{ik}},\,m_{ik}\in\\{0,1\\}\mbox{
for all }i,k\in I,$ (1)

where the above product has finite support (i.e, $|\\{i,k\in I:m_{ik}\neq
0\\}|<\infty$). In other words, for all $i\in I$, there exist unique sequence
in $I$ $i_{0}<i_{1}<\cdots<i_{n}$ such that

$\displaystyle w_{i}=v_{i_{0}}\cdot v_{i_{1}}\cdot...\cdot v_{i_{n(i)}}.$ (2)

By abuse of notation, let $C=\\{x_{i}:i\in I\\}\subseteq\mathcal{W}(C)$. We
define a function $\mu_{BC}:C\rightarrow\mathcal{W}(B)$ by the rule

$\displaystyle x_{i}\mapsto x_{i_{0}}\cdot x_{i_{1}}\cdot...\cdot
x_{i_{n}}\mbox{ if }w_{i}=v_{i_{0}}\cdot v_{i_{1}}\cdot...\cdot v_{i_{n}}.$
(3)

This function is well-defined because both $B$ and $C$ are basis, so the
expression 2 is unique.

######  Lemma 3.3.

Let $G$ be a pre-special group and $B,C,\mu_{BC}$ as above. There is a unique
continuous homomorphism $\mu_{BC}:\mathcal{W}(C)\rightarrow\mathcal{W}(B)$
that extends $\mu_{BC}$. Also $\mu_{BC}[\Phi(C)]\subseteq\Phi(B)$.

###### Proof.

By abuse of notation, let $B=\\{x_{i}:i\in I\\}\subseteq\mathcal{W}(B)$ and
$C=\\{x_{i}:i\in I\\}\subseteq\mathcal{W}(C)$. We have $B$ and $C$ as a set of
generators converging to 1 in $\mathcal{W}(B)$ and $\mathcal{W}(C)$
respectively.

Let $X=\mu_{BC}[C]\subseteq\mathcal{W}(B)$. Since $\langle X\rangle=\langle
B\rangle$ and $\mathcal{W}(B)=\overline{\langle B\rangle}$, we have that $X$
is a set of generators of $\mathcal{W}(B)$.

Let $U\subseteq\mathcal{W}(B)$ be an open subgroup. Since $B$ is a set of
generators converging to 1, there is a finite subset
$Y=\\{x_{i_{1}},...,x_{j_{m}}\\}\subseteq B$ with $U\cap Y=\emptyset$. Since
the set of $\mathbb{F}_{2}$-linear combinations of a finite set is finite,
there is a finite quantity of elements in $\mu_{BC}[C]$ not belonging to $U$.

The existence and continuity of $\mu_{BC}$ is an immediate consequence of the
Universal Property of $\mathcal{W}(I)$ (Theorem 2.11). An explicit formula for
$\mu_{BC}$ is given by the following rule: for
$g=(t_{i}^{\alpha_{i}})(t_{ij}^{\beta_{ij}})(x_{i}^{\gamma_{i}})$ we set

$\mu_{BC}(g):=\left(t_{i}^{\left(\alpha_{i}\sum_{k\in
I}m_{ik}\right)}\right)\left(t_{ij}^{\left(\beta_{ij}\sum_{r,s\in
I}m_{ir}m_{js}\right)}\right)\left(x_{i}^{\left(\gamma_{i}\sum_{k\in
I}m_{ik}\right)}\right).$

Since $\Phi(\mathcal{W}(C))=\mathcal{W}(C)^{2}$ and
$\Phi(\mathcal{W}(B))=\mathcal{W}(B)^{2}$, we get
$\mu_{BC}[\Phi(C)]\subseteq\Phi(B)$. ∎

######  Remark 3.4.

A direct calculation show for all $a,b\in G$ that

$m^{2}_{B,B^{\prime}}(q^{B}_{ab})=q^{B^{\prime}}_{ab}.$

Denote $\mu^{(2)}_{B,B^{\prime}}:\Phi(B^{\prime})\to\Phi(B)$ the restriction
of ${\mu}_{B,B^{\prime}}$ to the Frattini’s subgroups and
$\mu^{(1)}_{B,B^{\prime}}:{\mathcal{W}}(B^{\prime})/\Phi(B^{\prime})\to{\mathcal{W}}(B)/\Phi(B)$
the quotient of $\mu_{B,B^{\prime}}$. Then, from the isomorphism in
Proposition 2.14 $\Phi(B)\cong Hom(P_{2}(B),{\mathbb{Z}}_{2})$, we have:

$\Phi(B^{\prime})\times
P_{2}(B)\to{\mathbb{Z}}_{2}:<\mu^{1}_{B,B^{\prime}}-,->_{B}=<-,m^{1}_{B,B^{\prime}}->_{B^{\prime}}$

so , for all $\sigma^{\prime}\in\Phi(B^{\prime})$

$<\mu^{2}_{B,B^{\prime}}(\sigma^{\prime}),q^{B}_{ab}>_{B}=<\sigma^{\prime},m^{2}_{B,B^{\prime}}(q^{B}_{ab})>_{B^{\prime}}=<\sigma^{\prime},q^{B^{\prime}}_{ab}>_{B^{\prime}}$

and then
$\mu^{2}_{B,B^{\prime}}[(q^{B^{\prime}}_{a,b})^{\perp}]=(q^{B}_{ab})^{\perp}$

######  Lemma 3.5.

The morphism $\mu_{BC}$ is an isomorphism, $\mu_{BB}=id_{\mathcal{W}(B)}$,
$\mu_{BC}^{-1}=\mu_{CB}$ and

$\mu_{BD}=\mu_{BC}\circ\mu_{CD}.$

###### Proof.

The fact that $\mu_{BB}=id_{\mathcal{W}(B)}$ and $\mu_{BC}^{-1}=\mu_{CB}$ is
an immediate consequence of Lemma 3.3. For the other part, let
$B=\\{v_{i}\\}_{i\in I}$, $C=\\{w_{i}\\}_{i\in I}$ and $D=\\{z_{i}\\}_{i\in
I}$ be $\mathbb{F}_{2}$-basis of $G$. Then for all $i\in I$,

$\displaystyle w_{i}=\prod_{k\in I}v_{k}^{m_{ik}},\,z_{i}=\prod_{k\in
I}w_{k}^{n_{ik}},\,z_{i}=\prod_{k\in I}v_{k}^{p_{ik}},$

such that all these products has finite support. Then

$\displaystyle z_{i}$ $\displaystyle=\prod_{k\in I}w_{k}^{n_{ik}}=\prod_{k\in
I}\left(\prod_{r\in I}v_{r}^{m_{kr}}\right)^{n_{ik}}=\prod_{k\in I}\prod_{r\in
I}v_{r}^{n_{ik}m_{kr}}=\prod_{r\in I}\prod_{k\in
I}v_{r}^{n_{ik}m_{kr}}=\prod_{r\in I}v_{r}^{p_{ir}}.$

Moreover

$\sum_{k\in I}\sum_{r\in I}n_{ik}m_{kr}=\sum_{r\in I}\sum_{k\in
I}n_{ik}m_{kr}=\sum_{r\in I}p_{ir}.$

Then for all
$g=(t_{i}^{\alpha_{i}})(t_{ij}^{\beta_{ij}})(x_{i}^{\gamma_{i}})\in\mathcal{W}(D)$,

$\displaystyle\mu_{BC}\circ\mu_{CD}(g)=\mu_{BC}(\mu_{CD}(g))=$
$\displaystyle\mu_{BC}\left(\left(t_{i}^{\alpha_{i}\left(\sum_{k\in
I}n_{ik}\right)}\right)\left(t_{ij}^{\left(\beta_{ij}\sum_{r,s\in
I}n_{ir}n_{js}\right)}\right)\left(x_{i}^{\left(\gamma_{i}\sum_{k\in
I}n_{ik}\right)}\right)\right)=$
$\displaystyle\left(t_{i}^{\alpha_{i}\left(\sum_{k\in
I}n_{ik}\right)\left(\sum_{r\in
I}m_{kr}\right)}\right)\left(t_{ij}^{\beta_{ij}\left(\sum_{r,s\in
I}n_{ir}n_{js}\right)\left(\sum_{e,f\in
I}m_{re}m_{sf}\right)}\right)\left(x_{i}^{\gamma_{i}\left(\sum_{k\in
I}n_{ik}\right)\left(\sum_{r\in I}m_{kr}\right)}\right)=$
$\displaystyle\left(t_{i}^{\alpha_{i}\left(\sum_{r\in I}\sum_{k\in
I}n_{ik}m_{kr}\right)}\right)\left(t_{ij}^{\beta_{ij}\left(\sum_{r,s\in
I}\sum_{e,f\in
I}(n_{ir}m_{re})(n_{js}m_{sf})\right)}\right)\left(x_{i}^{\gamma_{i}\left(\sum_{r\in
I}\sum_{k\in I}n_{ik}m_{kr}\right)}\right)=$
$\displaystyle\left(t_{i}^{\alpha_{i}\left(\sum_{r\in
I}p_{ir}\right)}\right)\left(t_{ij}^{\beta_{ij}\left(\sum_{e,f\in
I}p_{ie}p_{jf}\right)}\right)\left(x_{i}^{\gamma_{i}\left(\sum_{r\in
I}p_{ir}\right)}\right)=\mu_{BD}(g).$

Then we get $\mu_{BD}=\mu_{BC}\circ\mu_{CD}$. ∎

Now, consider the Equation 1. This expression induce isomorphisms
$m^{1}_{BC}:P_{1}(B)\rightarrow P_{1}(C)$ and $m^{2}_{BC}:P_{2}(B)\rightarrow
P_{2}(C)$ given by the rules

$\displaystyle m^{1}_{BC}\left(\sum_{i\in I}c_{i}z_{i}\right)$
$\displaystyle:=\sum_{i\in I}c_{i}\left(\sum_{k\in I}m_{ik}\right)z_{i}$
$\displaystyle m^{2}_{BC}\left(\sum_{i\in I}a_{i}z_{i}^{2}+\sum_{i<j\in
I}b_{ij}z_{i}z_{j}\right)$ $\displaystyle:=\sum_{i\in I}a_{i}\left(\sum_{k\in
I}m_{ik}\right)z^{2}_{i}+\sum_{i<j\in I}b_{ij}\left[\left(\sum_{r\in
I}m_{ir}\right)\left(\sum_{s\in I}m_{js}\right)\right]z_{i}z_{j}.$

where all these sums has finite support.

######  Lemma 3.6.

We have

$m^{1}_{BD}=m^{1}_{BC}\circ m^{1}_{CD}\mbox{ and }m^{2}_{BD}=m^{2}_{BC}\circ
m^{2}_{CD}.$

###### Proof.

Lets recover the calculations in the proof of Lemma 3.5: let
$B=\\{v_{i}\\}_{i\in I}$, $C=\\{w_{i}\\}_{i\in I}$ and $D=\\{z_{i}\\}_{i\in
I}$ be $\mathbb{F}_{2}$-basis of $G$. Then for all $i\in I$,

$\displaystyle w_{i}=\prod_{k\in I}v_{k}^{m_{ik}},\,z_{i}=\prod_{k\in
I}w_{k}^{n_{ik}},\,z_{i}=\prod_{k\in I}v_{k}^{p_{ik}},$

such that all these products has finite support. Then

$\displaystyle z_{i}$ $\displaystyle=\prod_{k\in I}w_{k}^{n_{ik}}=\prod_{k\in
I}\left(\prod_{r\in I}v_{r}^{m_{kr}}\right)^{n_{ik}}=\prod_{k\in I}\prod_{r\in
I}v_{r}^{n_{ik}m_{kr}}=\prod_{r\in I}\prod_{k\in
I}v_{r}^{n_{ik}m_{kr}}=\prod_{r\in I}v_{r}^{p_{ir}}.$

Moreover

$\sum_{k\in I}\sum_{r\in I}n_{ik}m_{kr}=\sum_{r\in I}\sum_{k\in
I}n_{ik}m_{kr}=\sum_{r\in I}p_{ir}.$

Then

$\displaystyle m^{1}_{BC}\left(m^{1}_{CD}\left(\sum_{k\in
I}c_{k}z_{i}\right)\right)$ $\displaystyle=m^{1}_{BC}\left(\sum_{i\in
I}c_{i}\left(\sum_{k\in I}n_{ik}\right)z_{i}\right)=\sum_{i\in
I}c_{i}\left(\sum_{k\in I}n_{ik}\right)\left(\sum_{r\in I}m_{kr}\right)z_{i}$
$\displaystyle=\left(\sum_{r\in
I}p_{ir}\right)z_{i}=m^{1}_{BD}\left(\sum_{k\in I}c_{k}z_{i}\right)$

and hence $m^{1}_{BD}=m^{1}_{BC}\circ m^{1}_{CD}$. In the same reasoning,

$\displaystyle m^{2}_{BC}\left(m^{2}_{CD}\left(\sum_{i\in
I}a_{i}z_{i}^{2}+\sum_{i<j\in I}b_{ij}z_{i}z_{j}\right)\right)=$
$\displaystyle m^{2}_{BC}\left(\sum_{i\in I}a_{i}\left(\sum_{k\in
I}n_{ik}\right)z^{2}_{i}+\sum_{i<j\in I}b_{ij}\left[\left(\sum_{r\in
I}n_{ir}\right)\left(\sum_{s\in I}n_{js}\right)\right]z_{i}z_{j}\right)=$
$\displaystyle\sum_{i\in I}a_{i}\left(m^{2}_{BC}\left(\left(\sum_{k\in
I}n_{ik}\right)z^{2}_{i}\right)\right)+m^{2}_{BC}\left(\sum_{i<j\in
I}b_{ij}\left[\left(\sum_{r\in I}n_{ir}\right)\left(\sum_{s\in
I}n_{js}\right)\right]z_{i}z_{j}\right)=$ $\displaystyle\sum_{i\in
I}a_{i}\left(\sum_{k\in I}n_{ik}\left(\sum_{r\in
I}m_{kr}\right)z^{2}_{i}\right)+\sum_{i<j\in I}b_{ij}\left[\left(\sum_{r\in
I}n_{ir}\right)\left(\sum_{s\in I}n_{js}\right)\left(\sum_{e\in
I}m_{re}\right)\left(\sum_{f\in I}m_{sf}\right)\right]z_{i}z_{j}=$
$\displaystyle\sum_{i\in I}a_{i}\left(\left(\sum_{r\in I}\sum_{k\in
I}n_{ik}m_{kr}\right)z^{2}_{i}\right)+\sum_{i<j\in
I}b_{ij}\left[\left(\sum_{e\in I}\sum_{r\in
I}n_{ir}m_{re}\right)\left(\sum_{f\in I}\sum_{s\in
I}n_{js}m_{sf}\right)\right]z_{i}z_{j}=$ $\displaystyle\sum_{i\in
I}a_{i}\left(\sum_{r\in I}p_{ir}\right)z^{2}_{i}+\sum_{i<j\in
I}b_{ij}\left[\left(\sum_{e\in I}p_{ie}\right)\left(\sum_{f\in
I}p_{jf}\right)\right]z_{i}z_{j}=$ $\displaystyle m^{2}_{BD}\left(\sum_{i\in
I}a_{i}z_{i}^{2}+\sum_{i<j\in I}b_{ij}z_{i}z_{j}\right)$

provide that $m^{2}_{BD}=m^{2}_{BC}\circ m^{2}_{CD}$. ∎

Note that $m^{1}_{BC},m^{2}_{BC}$ induces respectively the isomorphisms

$\displaystyle\mathfrak{m}^{1}_{BC}$
$\displaystyle:\mbox{Hom}(P_{1}(C),\mathbb{Z}_{2})\rightarrow\mbox{Hom}(P_{1}(B),\mathbb{Z}_{2})$
$\displaystyle\mathfrak{m}^{2}_{BC}$
$\displaystyle:\mbox{Hom}(P_{2}(C),\mathbb{Z}_{2})\rightarrow\mbox{Hom}(P_{2}(B),\mathbb{Z}_{2}).$

given by the respective rules: if $f:P_{1}(C)\rightarrow\mathbb{Z}_{2}$ and
$q=\sum_{i\in I}c_{i}z_{i}\in P_{1}(B)$ then

$\displaystyle\mathfrak{m}^{1}_{BC}(f)(q)$
$\displaystyle:=f(m^{1}_{BC}(q))=f\left(\sum_{i\in I}c_{i}\left(\sum_{k\in
I}m_{ik}\right)z_{i}\right).$

In the same reasoning, if $f:P_{2}(C)\rightarrow\mathbb{Z}_{2}$ and
$q=\sum_{i\in I}a_{i}z_{i}^{2}+\sum_{i<j\in I}b_{ij}z_{i}z_{j}\in P_{2}(B)$
then

$\displaystyle\mathfrak{m}^{2}_{BC}(f)(q)$
$\displaystyle:=f(m^{2}_{BC}(q))=f\left(\sum_{i\in I}a_{i}\left(\sum_{k\in
I}m_{ik}\right)z^{2}_{i}+\sum_{i<j\in I}b_{ij}\left[\left(\sum_{r\in
I}m_{ir}\right)\left(\sum_{s\in I}m_{js}\right)\right]z_{i}z_{j}\right)$

Now denote

$\mu^{1}_{BC}:\mathcal{W}(C)/\Phi(C)\rightarrow\mathcal{W}(B)/\Phi(B)$

the quotient of $\mu_{BC}$ and

$\mu^{2}_{BC}:\Phi(C)\rightarrow\Phi(B)$

the restriction of $\mu_{BC}$ to the Frattini’s subgroups. Also consider the
isomorphisms

$\displaystyle\tilde{\theta}:\mathcal{W}(I)/\Phi(I)$
$\displaystyle\xrightarrow{\cong}\mbox{Hom}(P_{1}(I),\mathbb{Z}_{2})$
$\displaystyle\tilde{\lambda}:\Phi(I)$
$\displaystyle\xrightarrow{\cong}\mbox{Hom}(P_{2}(I),\mathbb{Z}_{2})$

of Propositions 2.14, and 2.15.

######  Lemma 3.7.

Denote $\pi_{B}:\mathcal{W}(B)\rightarrow\mathcal{W}(B)/\Phi(B)$ the canonical
projection, with the same for $\pi_{C}$. Then we have a commutative diagram

$\textstyle{\mathcal{W}(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{BC}}$$\scriptstyle{\theta_{C}}$$\textstyle{\mbox{Hom}(P_{1}(C),\mathbb{Z}_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathfrak{m}^{1}_{BC}}$$\textstyle{\mathcal{W}(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta_{B}}$$\textstyle{\mbox{Hom}(P_{1}(B),\mathbb{Z}_{2})}$

which induces a commutative diagram

$\textstyle{\mathcal{W}(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{C}}$$\scriptstyle{\theta_{C}}$$\textstyle{\mathcal{W}(C)/\Phi(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\theta}_{C}}$$\scriptstyle{\mu^{1}_{BC}}$$\textstyle{\mbox{Hom}(P_{1}(C),\mathbb{Z}_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathfrak{m}^{1}_{BC}}$$\textstyle{\mathcal{W}(B)/\Phi(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\theta}_{B}}$$\textstyle{\mbox{Hom}(P_{1}(B),\mathbb{Z}_{2})}$$\textstyle{\mathcal{W}(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{B}}$$\scriptstyle{\theta_{B}}$

###### Proof.

Let
$g=(t_{i}^{\alpha_{i}})(t_{ij}^{\beta_{ij}})(x_{i}^{\gamma_{i}})\in\mathcal{W}(C)$
and $q=\sum_{i\in I}c_{i}z_{i}\in P_{1}(B)$. Then

$\displaystyle(\mathfrak{m}^{1}_{BC}\circ\theta)(g)(q)$
$\displaystyle=\mathfrak{m}^{1}_{BC}(\theta(g)(q))=\theta(g)(m^{1}_{BC}(q))$
$\displaystyle=\theta(g)\left(\sum_{i\in I}c_{i}\left(\sum_{k\in
I}m_{ik}\right)z_{i}\right)=\sum_{i\in I}c_{i}\left(\sum_{k\in
I}m_{ik}\right)\gamma_{i}$ $\displaystyle=\sum_{i\in I}\sum_{k\in
I}c_{i}m_{ik}\gamma_{i}.$

On the other hand,

$\displaystyle(\theta\circ\mu_{BC})(g)(q)$
$\displaystyle=\theta(\mu_{BC}(g))(q)$
$\displaystyle=\theta\left(\left(t_{i}^{\alpha_{i}\left(\sum_{k\in
I}m_{ik}\right)}\right)\left(t_{ij}^{\beta_{ij}\left(\sum_{r,s\in
I}m_{ir}m_{js}\right)}\right)\left(x_{i}^{\gamma_{i}\left(\sum_{k\in
I}m_{ik}\right)}\right)\right)\left(\sum_{i\in I}c_{i}z_{i}\right)$
$\displaystyle=\sum_{i\in I}c_{i}\left(\gamma_{i}\left(\sum_{k\in
I}m_{ik}\right)\right)=\sum_{i\in I}\sum_{k\in
I}c_{i}m_{ik}\gamma_{i}=(\mathfrak{m}^{1}_{BC}\circ\theta)(g)(q).$

Then $\mathfrak{m}^{1}_{BC}\circ\theta=\theta\circ\mu_{BC}$. Since
$\theta_{B}=\tilde{\theta}_{B}\circ\pi_{B}$ and
$\theta_{C}=\tilde{\theta}_{C}\circ\pi_{C}$ we have the desired commutative
diagram. ∎

######  Lemma 3.8.

We have a commutative diagram

$\textstyle{\Phi(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\lambda}_{C}}$$\scriptstyle{\mu^{2}_{BC}}$$\textstyle{\mbox{Hom}(P_{2}(C),\mathbb{Z}_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathfrak{m}^{2}_{BC}}$$\textstyle{\Phi(B)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\lambda}_{B}}$$\textstyle{\mbox{Hom}(P_{2}(B),\mathbb{Z}_{2})}$

###### Proof.

Let $g=(t_{i}^{\alpha_{i}})(t_{ij}^{\beta_{ij}})(1_{i})\in\Phi(C)$ and
$\sum_{i\in I}a_{i}z_{i}^{2}+\sum_{i<j\in I}b_{ij}z_{i}z_{j}\in P_{2}(B)$. We
have

$\displaystyle(\mathfrak{m}^{2}_{BC}\circ\tilde{\lambda}_{C})(g)(q)=\tilde{\lambda}_{C}(g)(m^{2}_{BC}(q))=$
$\displaystyle\tilde{\lambda}_{C}(g)\left(\sum_{i\in I}a_{i}\left(\sum_{k\in
I}m_{ik}\right)z^{2}_{i}+\sum_{i<j\in I}b_{ij}\left[\left(\sum_{r\in
I}m_{ir}\right)\left(\sum_{s\in I}m_{js}\right)\right]z_{i}z_{j}\right)=$
$\displaystyle\sum_{i\in I}a_{i}\left(\sum_{k\in
I}m_{ik}\right)\alpha_{i}+\sum_{i<j\in I}b_{ij}\left[\left(\sum_{r\in
I}m_{ir}\right)\left(\sum_{s\in I}m_{js}\right)\right]\beta_{ij}$

On the other hand,

$\displaystyle(\tilde{\lambda}_{B}\circ\mu^{2}_{BC})(g)(q)=\tilde{\lambda}(\mu^{2}_{BC}(g)(q))$
$\displaystyle=\tilde{\lambda}_{B}\left(\left(t_{i}^{\alpha_{i}\left(\sum_{k\in
I}m_{ik}\right)}\right)\left(t_{ij}^{\beta_{ij}\left(\sum_{r,s\in
I}m_{ir}m_{js}\right)}\right)(1_{i})\right)\left(\sum_{i\in
I}a_{i}z_{i}^{2}+\sum_{i<j\in I}b_{ij}z_{i}z_{j}\right)=$
$\displaystyle\sum_{i\in I}a_{i}\left(\alpha_{i}\left(\sum_{k\in
I}m_{ik}\right)\right)+\sum_{i<j\in I}b_{ij}\left(\beta_{ij}\left(\sum_{r,s\in
I}m_{ir}m_{js}\right)\right)$

proving that
$\mathfrak{m}^{2}_{BC}\circ\tilde{\lambda}_{C}=\tilde{\lambda}_{B}\circ\mu^{2}_{BC}$.
∎

######  Lemma 3.9.

With the notations of Lemmas 3.2-3.7 we have the following.

  1. i -

The arrows $\mu^{1}_{BC}$ and $\mu^{2}_{BC}$ are isomorphisms. Moreover, for
all well-ordered basis $B,C,D$ we have $\mu^{1}_{BB}=id$, $\mu^{2}_{BB}=id$,
$\mu^{1}_{BD}=\mu^{1}_{BC}\circ\mu^{1}_{CD}$ and
$\mu^{2}_{BD}=\mu^{2}_{BC}\circ\mu^{2}_{CD}$.

  2. ii -

The isomorphism $\mu_{BC}:\mathcal{W}(C)\rightarrow\mathcal{W}(B)$ restricts
to an isomorphism $\mathcal{V}(C)\rightarrow\mathcal{V}(B)$ so we get quotient
isomorphism

$\tilde{\mu}_{BC}:\mathcal{W}(C)/\mathcal{V}(C)\rightarrow\mathcal{W}(B)/\mathcal{V}(B).$

  3. iii -

If $B,C,D$ are well-ordered base of $G$, then $\tilde{\mu}_{CC}=id$ and
$\tilde{\mu}_{BD}=\tilde{\mu}_{BC}\circ\tilde{\mu}_{CD}$.

###### Proof.

  1. i -

Just use the same calculations made in Lemma 3.5.

  2. ii -

By Proposition 3.2 we have

$\mathfrak{m}^{2}_{BC}(Q(B))=Q(C).$

Since $\mathcal{V}(B)=Q(B)^{\perp}$, we have an induced isomorphism
$\mu_{BC}|_{\mathcal{V}(B)}:\mathcal{V}(B)\xrightarrow{\cong}\mathcal{V}(C)$,
legitimating the quotient isomorphism

$\tilde{\mu}_{BC}:\mathcal{W}(C)/\mathcal{V}(C)\rightarrow\mathcal{W}(B)/\mathcal{V}(B).$

  3. iii -

It is an immediate consequence of item (i).

∎

######  Definition 3.10 (Galois Group - base independent version).

Let $G$ be a pre-special group. Take

$E_{G}=\\{B:B\mbox{ is a well-ordered }\mathbb{F}_{2}\mbox{-basis of }G\\}.$

Consider the set $E_{G}$ endowed with the trivial groupoid operation of
concatenation of pairs (i.e., the arrows are $E_{G}\times E_{G}$) and take the
functor $\mbox{Gal}:E_{G}\rightarrow\mathcal{C}$, (where $\mathcal{C}$ is the
category of $\mathcal{C}$-groups) given by the following rules: for an object
$B\in E_{G}$, $\mbox{Gal}(B):=\mathcal{W}(B)/\mathcal{V}(B)$ and for an arrow
$(B,C)\in E^{2}_{G}$,

$\mbox{Gal}(B,C)=\mu_{BC}:\mathcal{W}(C)/\mathcal{V}(C)\rightarrow\mathcal{W}(B)/\mathcal{V}(B).$

We define the Galois group  of $G$, notation $\mbox{Gal}(G)$ by

$\mbox{Gal}(G):=\varprojlim_{B\in E_{G}}\mathcal{W}(B)/\mathcal{V}(B).$

######  Remark 3.11.

Keeping the notation above, note that

$\pi_{B}:\mbox{Gal}(G):=\varprojlim_{B\in
E_{G}}\mathcal{W}(B)/\mathcal{V}(B)\to\mathcal{W}(B)/\mathcal{V}(B)$

is an isomorphism, for each $B\in E_{G}$. This holds because

$\mbox{Gal}(B,C)=\mu_{BC}:\mathcal{W}(C)/\mathcal{V}(C)\rightarrow\mathcal{W}(B)/\mathcal{V}(B)$

is an isomorphism for each arrow $(B,C)\in E^{2}_{G}$.

##  4 On the structure of Galois Groups of Pre Special Groups

As occurs with fields, the Galois group of a pre-special groups (in a certain
subclass) is able to encode many relevant quadratic information.

All pre-special groups occurring is this section will be assumed $k$-stable.

We start this Section, by providing more details on the structure of
$\mathcal{C}$-groups.

Let $\mathcal{G}$ be a $\mathcal{C}$-group on $I$-minimal generators. By
Theorem 2.11, There is an epimorphism
$\lambda:\mathcal{W}(I)\rightarrow\mathcal{G}$ with kernel
$\mathcal{V}\subseteq\Phi(I)$, and then, we have an isomorphism
$\tilde{\lambda}:\mathcal{W}(I)/\mathcal{V}\rightarrow\mathcal{G}$.

######  Proposition 4.1.

With the above notation, we have the following.

  1. i -

We have a natural bijection

$\displaystyle\\{M\subseteq\mathcal{G}:M\mbox{ is a maximal open
subgroup}\\}\cong$ $\displaystyle\\{M\subseteq\mathcal{W}(I):M\mbox{ is a
maximal open subgroup}\\}.$

Then $\Phi(\mathcal{G})\cong\Phi(I)/\mathcal{V}$ and
$\mathcal{G}/\Phi(\mathcal{G})\cong\mathcal{W}(I)/\Phi(I)$.

  2. ii -

The maximal closed subgroups of $\mathcal{W}(I)$ are precisely the clopen
(normal) subgroups with quotient $\mathbb{Z}_{2}$. Moreover, we have a natural
bijection

$\\{M\subseteq\mathcal{G}:M\mbox{ is a maximal open subgroup}\\}\cong
P_{fin}(I)\setminus\\{\emptyset\\}.$

and that extends to a natural bijection

$\\{N\subseteq\mathcal{G}:N\mbox{ is an open subgroup with index}\leq
2\\}\cong P_{fin}(I).$

  3. iii -

We have a natural isomorphism of $\mathbb{Z}_{2}$-modules

$Homcont(\mathcal{G},\mathbb{Z}_{2})\cong fsFunc(I,\mathbb{Z}_{2})$

where $fsFunc(I,\mathbb{Z}_{2})$ is the set of all function
$f:I\to\mathbb{Z}_{2}$ with finite support.

###### Proof.

  1. i -

The desired bijection follows by the bijection

$\displaystyle\\{M\subseteq\mathcal{W}(I)/\mathcal{V}:M\mbox{ is a maximal
open subgroup}\\}\cong$ $\displaystyle\\{M\subseteq\mathcal{W}(I):M\mbox{ is a
maximal open subgroup}\\}.$

given by the following rule: lets
$q:\mathcal{W}(I)\rightarrow\mathcal{W}(I)/\mathcal{V}$ denote the canonical
projection. We have a function
$\overline{q}:\mathcal{P}(\mathcal{W}(I)/\mathcal{V})\rightarrow\mathcal{P}(\mathcal{W}(I))$
given by the rule

$\overline{q}(X):=q^{-1}[X]\mbox{ (the inverse image)}.$

This function $\overline{q}$ induces the desired bijection. Since the Frattini
subgroup of $\mathcal{G}$ is the intersection of all open normal subgroups we
have (via $\tilde{\lambda}$ and the bijection)
$\Phi(\mathcal{G})\cong\Phi(I)/\mathcal{V}$. Then

$\mathcal{G}/\phi(\mathcal{G})\cong(\mathcal{W}(I)/\mathcal{V})/(\Phi(I)/\mathcal{V})\cong\mathcal{W}(I)/\Phi(I).$

  2. ii -

For $\\{i_{0},...,i_{n}\\}\subseteq I$ denote

$\zeta_{I}(i_{0},...,i_{n}):=\\{\sigma\in\mathcal{W}(I):\gamma_{i_{0}}(\sigma)+...+\gamma_{i_{n}}(\sigma)=0\\}.$

We have that $\zeta_{I}(i_{0},...,i_{n})$ is a subgroup of $\mathcal{W}(I)$.
Now let $\tau,\theta\in\mathcal{W}(I)\setminus\zeta_{I}(i_{0},...,i_{n})$.
Then for all $i\in I$,

$\gamma_{i}(\theta^{-1}\tau)=\gamma_{i}(\theta)+\gamma_{i}(\tau).$

Therefore

$\sum^{n}_{p=1}\gamma_{i_{p}}(\theta^{-1}\tau)=\sum^{n}_{p=1}[\gamma_{i_{p}}(\theta)+\gamma_{i_{p}}(\tau)]=\sum^{n}_{p=1}\gamma_{i_{p}}(\theta)+\sum^{n}_{p=1}\gamma_{i_{p}}(\tau)=1+1=0.$

Then $\theta^{-1}\tau\in\zeta_{I}(i_{0},...,i_{n})$ which imply

$\mathcal{W}(I)/\zeta_{I}(i_{0},...,i_{n})=\\{\overline{1},\overline{\tau}\\}\cong\mathbb{Z}_{2}.$

To verify that $\zeta_{I}(i_{0},...,i_{n})$ is clopen, note that

$\gamma_{i_{0}}(\sigma)+...+\gamma_{i_{n}}(\sigma)=0\mbox{ iff
}\begin{cases}\gamma_{i_{0}}(\sigma)+...+\gamma_{i_{n-1}}(\sigma)=0\mbox{ and
}\gamma_{i_{n}}(\sigma)=0\mbox{ or }\\\
\gamma_{i_{0}}(\sigma)+...+\gamma_{i_{n-1}}(\sigma)=1\mbox{ and
}\gamma_{i_{n}}(\sigma)=1.\end{cases}$

In other words,

$\zeta_{I}(i_{0},...,i_{n})=[\zeta_{I}(i_{0},...,i_{n-1})\cap
M_{i_{n}}]\cup[\zeta_{I}(i_{0},...,i_{n-1})^{c}\cap M_{i_{n}}^{c}].$

So, in order to verify that $\zeta_{I}(i_{0},...,i_{n})$ is clopen is enough
to deal with the case $n=0$. But $\zeta_{I}(i_{0})=M_{i_{0}}$ is in fact a
clopen, which provide (by induction) that $\zeta_{I}(i_{0},...,i_{n})$ is
clopen for all $i_{0},...,i_{n}\in I$. Then $\zeta_{I}(i_{0},...,i_{n})$ is a
maximal clopen subgroup of $\mathcal{W}(I)$, and we have a well-defined
injective function

$\zeta_{I}:\mathcal{P}_{fin}(I)\setminus\\{\emptyset\\}\rightarrow\\{M\subseteq\mathcal{W}(I):M\mbox{
is a maximal open subgroup}\\}$

given by the rule $\\{i_{0},...,i_{n}\\}\mapsto\zeta_{I}(i_{0},...,i_{n})$.

For surjectivity, let $M$ be a maximal open subgroup. Then $M$ is closed of
finite index and by Lemma 2.9(a), $M$ has index 2 in $G$. Using Propositions
2.5 and 2.6 and the compacity of $\mathcal{W}(I)$, there exists
$i_{1},...,i_{n},j_{1},...,j_{m},k_{1},...,k_{p}\in I$ with

$M_{i_{1}}\cap...\cap M_{i_{n}}\cap S_{j_{1}}\cap...\cap S_{j_{m}}\cap
D_{k_{1}}\cap...\cap D_{k_{p}}\subseteq M.$

Note that we have

$M_{i_{1}}\cap...\cap M_{i_{n}}\cap S_{j_{1}}\cap...\cap S_{j_{m}}\cap
D_{k_{1}}\cap...\cap
D_{k_{p}}\subseteq\zeta_{I}(i_{1},...,i_{n},j_{1},...,j_{m},k_{1},...,k_{p}).$

Lets denote
$\zeta_{I}(i_{1},...,i_{n},j_{1},...,j_{m},k_{1},...,k_{p}):=\zeta_{I}(\vec{i},\vec{j},\vec{k})$
and $H:=M\cap\zeta_{I}(\vec{i},\vec{j},\vec{k})$. Suppose $H\neq M$ and let
$\tau,\theta\in M\setminus H$. The same calculations made for injectivity
shows that $\theta^{-1}\tau\in\zeta_{I}(\vec{i},\vec{j},\vec{k})$ which imply

$M/H=\\{\overline{1},\overline{\tau}\\}\cong\mathbb{Z}_{2}.$

Moreover, using the same calculations made in Proposition 2.5(b) we have that
$H$ has index $\mathbb{Z}_{4}$ in $\mathcal{W}(I)$. Since $H\subseteq M$ and
$H\subseteq\zeta_{I}(\vec{i},\vec{j},\vec{k})$ with both maximal clopen
subgroups, by Lemma 4.7(i) we have $M=\zeta_{I}(\vec{i},\vec{j},\vec{k})$,
contradicting the assumption $H\neq M$. Then $H=M$ and we have
$M=\zeta_{I}(\vec{i},\vec{j},\vec{k})$. Therefore we have bijections

$\displaystyle\mathcal{P}_{fin}(I)\setminus\\{\emptyset\\}$
$\displaystyle\cong\\{M\subseteq\mathcal{W}(I):M\mbox{ is a maximal open
subgroup}\\}$ $\displaystyle\cong\\{M\subseteq\mathcal{G}:M\mbox{ is a maximal
open subgroup}\\}.$

Therefore

$\\{N\subseteq\mathcal{G}:N\mbox{ is an open subgroup with index}\leq
2\\}\cong P_{fin}(I).$

  3. iii -

Since $\mathcal{G}=\mathcal{W}(I)/{\mathcal{V}}$ and

${\mathcal{V}}\subseteq\Phi(I)=\bigcap\\{ker(\varphi):\varphi\in
Homcont(\mathcal{W}(I),\mathbb{Z}_{2})\\},$

then the natural epimorphism
$\mathcal{W}(I)\twoheadrightarrow\mathcal{W}(I)/{\mathcal{V}}$ induces the
isomorphism

$Homcont(\mathcal{G},\mathbb{Z}_{2})\cong
Homcont(\mathcal{W}(I),\mathbb{Z}_{2}).$

Since $\mathbb{Z}_{2}$ is a finite/discrete $\mathcal{C}$-group, then the
universal property of $\mathcal{W}(I)$ (Theorem 2.11) gives a natural
isomorphism $Homcont(\mathcal{W}(I),\mathbb{Z}_{2})\cong
fsFunc(I,\mathbb{Z}_{2})$.

Alternatively, the result follows also from the item (ii) above and the
(obvious) natural bijections

$Homcont(\mathcal{G},\mathbb{Z}_{2})\cong\\{N\subseteq\mathcal{G}:N\mbox{ is
an open subgroup with index}\leq 2\\}$ $P_{fin}(I)\cong
fsFunc(I,\mathbb{Z}_{2}).$

∎

######  Proposition 4.2 (Prontryagin duality).

Let $\mathcal{G}=\mbox{Gal}(G)$ for some pre-special group $G$. Then

  1. i -

There is a canonical bijection

$\mathbb{M}:G\overset{\cong}{\to}\\{M\subseteq\mathcal{G}:M\mbox{ is a
(closed, normal) subgroup of index less or equal to }2\\}$

$a\mapsto M_{a}$ such that it induces a canonical bijection

$G\setminus\\{1\\}\cong\\{M\subseteq\mathcal{G}:M\mbox{ is a maximal
subgroup}\\}.$

  2. ii -

There is a canonical isomorphism of $\mathbb{Z}_{2}$-modules
$\psi_{G}:G\overset{\cong}{\to}Homcont(\mathcal{G},\mathbb{Z}_{2})$,
$a\mapsto\mu_{a}$, where $\mu_{a}:\mathcal{G}\to\mathbb{Z}_{2}$ is the unique
continuous homomorphism such that $ker(\mu_{a})=M_{a}$.

  3. iii -

There is a canonical isomorphism of pro-2-groups
$\phi_{G}:\mathcal{G}/\Phi(\mathcal{G})\rightarrow\mbox{Hom}(G,\mathbb{Z}_{2})$.

###### Proof.

  1. i -

This follows directly from Proposition 4.1, since
$\pi_{B}:Gal(G)\overset{\cong}{\to}{\mathcal{W}}(B)/{\mathcal{V}}(B)$ and
${\mathcal{V}}(B)\subseteq\Phi(B)$, for every well ordered basis $B$ of $G$.

  2. ii -

By Proposition 4.1(iii), for each well ordered basis $B$ in $G$, we have a
natural isomorphism of $\mathbb{Z}_{2}$-modules.

$Homcont(\mathcal{W}(B)/\mathcal{V}(B),\mathbb{Z}_{2})\cong
fsFunc(B,\mathbb{Z}_{2})$

This is, in fact, an isomorphism of $\mathbb{Z}_{2}$-modules. Taking into
account the isomorphisms of "change of base", we glue the above isomorphisms
to obtain the natural isomorphism

$Homcont(Gal(G),\mathbb{Z}_{2})=Homcont(\varprojlim_{B\in
E_{G}}\mathcal{W}(B)/\mathcal{V}(B),\mathbb{Z}_{2})\cong\varprojlim_{B\in
E_{G}}fSFunc(B,\mathbb{Z}_{2})\cong G$

  3. iii -

Note that
$\pi_{\mathcal{G}}:{\mathcal{G}}\twoheadrightarrow{\mathcal{G}}/\Phi({\mathcal{G}})$
induces a $\mathbb{Z}_{2}$-isomorphism

$\pi^{*}_{\mathcal{G}}:Homcont(\mathcal{G}/\Phi({\mathcal{G}}),\mathbb{Z}_{2})\overset{\cong}{\to}Homcont(\mathcal{G},\mathbb{Z}_{2}).$

By Lemma 3.7, the isomorphisms described in Proposition 2.15, namely

$\mathcal{W}(B)/\Phi(B)\cong\mbox{Hom}(P_{1}(B),\mathbb{Z}_{2}),$

are natural. Thus we obtain a natural isomorphism

$\mathcal{G}/\Phi({\mathcal{G}})\cong Hom(G,\mathbb{Z}_{2})$

∎

######  Remark 4.3.

Note that combining items (iii), (ii) of the Proposition above, we obtain the
Pontryagin duality:

$\mathcal{G}/\Phi({\mathcal{G}})\cong Hom(G,\mathbb{Z}_{2})\cong
Hom(Homcont(\mathcal{G}/\Phi({\mathcal{G}}),\mathbb{Z}_{2}),\mathbb{Z}_{2})$

$G\cong Homcont(\mathcal{G}/\Phi({\mathcal{G}}),\mathbb{Z}_{2})\cong
Homcont(Hom(G,\mathbb{Z}_{2}),\mathbb{Z}_{2})$

This induces a canonical duality between the pointed $\mathbb{Z}_{2}$-module
$(G,-1)$ and the "pointed" pro-2-group $(Gal(G),M)$, where $M\subseteq Gal(G)$
is an open subgroup of index $\leq 2$.

Let $G$ be a k-stable special group. Write $\mathcal{G}=\mbox{Gal}(G)$.

The isomorphism of pro-2-groups
$\phi_{G}:\mathcal{G}/\Phi(\mathcal{G})\rightarrow\mbox{Hom}(G,\mathbb{Z}_{2})$,
determines a "perfect pairing"
$\hat{\phi}_{G}:\mathcal{G}/\Phi(\mathcal{G})\times
G\rightarrow\mathbb{Z}_{2}$ given by the rule

$\hat{\phi}_{G}(\overline{\alpha},g):=\langle\overline{\alpha},g\rangle:=\phi_{G}(\overline{\alpha})(g).$

We will denote $(\ )^{\perp}$, generically, both the correspondences between
subsets of $\mathcal{G}/\Phi(\mathcal{G})$ and subsets of $G$.

######  Proposition 4.4.

The perfect pairing $\hat{\phi}_{G}:\mathcal{G}/\Phi(\mathcal{G})\times
G\rightarrow\mathbb{Z}_{2}$ gives an anti-isomorphism of complete lattices
between the poset

$\\{R\subseteq\mathcal{G}/\Phi(\mathcal{G}):R\mbox{ is a closed subgroup of
}\mathcal{G}/\Phi(\mathcal{G})\\}=$
$\\{\pi(T)\subseteq\mathcal{G}/\Phi(\mathcal{G}):T\mbox{ is a closed subgroup
of }\mathcal{G}\\},$

with $\pi:\mathcal{G}\rightarrow\mathcal{G}/\Phi(\mathcal{G})$ being the
canonical projection, and the poset

$\\{\Delta\subseteq G:\Delta\mbox{ is a subgroup of }G\\}.$

These anti-isomorphisms are given by the rules

$\displaystyle\pi(T)$ $\displaystyle\mapsto\pi(T)^{\perp}:=\\{a\in
G:\hat{\phi}_{G}(\sigma/\Phi(G),a)=0\mbox{for all }\sigma\in T\\}$
$\displaystyle H$ $\displaystyle\mapsto
H^{\perp}:=\\{\sigma/\Phi(G):\hat{\phi}_{G}(\sigma/\Phi(G),a)=0\mbox{ for all
}a\in H\\}.$

From this we get:

  1. i -

An anti-isomorphism of complete lattices between the posets

$\\{\Delta\subseteq G:\Delta\mbox{ is a subgroup of }G\\}$

and

$\\{T\subseteq\mathcal{G}:T\mbox{ is a closed subgroup and
}\Phi(\mathcal{G})\subseteq T\\}.$

  2. ii -

A bijection between the sets

$\\{\Delta\subseteq G:\Delta\mbox{ is a maximal subgroup of }G\\}$

and

$\\{\pi(T)\subseteq\mathcal{G}/\Phi(\mathcal{G}):T\mbox{ is a discrete
subgroup with order }\ 2\\}=$
$\\{\pi(T)\subseteq\mathcal{G}/\Phi(\mathcal{G}):T\mbox{ is a closed subgroup
with }\pi(T)=\\{id,\sigma/\Phi\\}\mbox{, for some
}\sigma\in{\mathcal{G}}\setminus\Phi({\mathcal{G}})\\}.$

###### Proof.

All items are immediate consequences of the isomorphism.

Since ${\mathcal{G}}$ is a compact Hausdorff group and
$\Phi({\mathcal{G}})\subseteq{\mathcal{G}}$ is a closed normal subgroup, note
that then the quotient map
${\mathcal{G}}\twoheadrightarrow{\mathcal{G}}/{\Phi(\mathcal{G})}$ gives an
isomorphism of complete lattices between the poset of closed subgroups of
${\mathcal{G}}$ which contains $\Phi({\mathcal{G}})$ and the poset of closed
subgroups of ${\mathcal{G}}/{\Phi(\mathcal{G})}$. ∎

######  Remark 4.5.

Let $\sigma\in{\mathcal{G}}\setminus\Phi({\mathcal{G}})$. It follows from the
definition of pairing that:

  * •

For any $x\in G-\\{1\\}$: $\sigma\in M_{x}$ iff
$<\sigma/\Phi({\mathcal{G}}),x>=0$ iff
$x\in\\{\Phi({\mathcal{G}}),\sigma.\Phi({\mathcal{G}})\\}^{\perp}$.

  * •

If there is an involution in $\sigma.\Phi(\mathcal{G})$ then of all element in
$\sigma.\Phi({\mathcal{G}})$ are involutions.

To obtain quadratic information from the Galois groups, we will need develop
deeper group theoretic results.

######  Lemma 4.6.

Let $B$ an well ordered basis of $G$ and consider
$\eta_{B}=\pi_{B}^{-1}:\mathcal{W}(B)/\mathcal{V}(B)\ \overset{\cong}{\to}\
Gal(G)$.

  1. i-

Let $a\neq 1$, choose $B=\\{a_{i}:i\in I\\}$ an well ordered basis of $G$ such
that $a\in B$, say $a=a_{i}$. Then
$\eta_{B}[M^{\prime}_{i}/\mathcal{V}(B)]=M_{a}$.

  2. ii-

Let $a,b\neq 1$ , $a\neq b$ so $\\{a,b\\}$ is a $\mathbb{Z}_{2}$-l.i. subset
of $G$, choose $B=\\{a_{i}:i\in I\\}$ an well ordered basis of $G$ such that
$a,b\in B$, say $a=a_{i}$, $b=a_{j}$ , $i<j\in I$. Let
$\\{M^{\prime}_{i},M^{\prime}_{j},M^{\prime}\\}$ be the three maximal
subgroups of $\mathcal{W}(B)$ above $M^{\prime}_{i}\cap M^{\prime}_{j}$. Then
$\eta_{B}[M^{\prime}_{i}/{\mathcal{V}}(B)]=M_{a}$,
$\eta_{B}[M^{\prime}_{j}/\mathcal{V}(B)]=M_{b}$ and
$\eta_{B}[M^{\prime}/\mathcal{V}(B)]=M_{ab}$.

  3. iii-

Let $\\{M_{1},M_{2},M_{3}\\}\subseteq Gal(G)$ maximal subgroups that are
pairwise distinct. Then are equivalent:

$\bullet$ $\\{M_{1},M_{2},M_{3}\\}$ are independent, which means that for each
of 3 enumerations $\\{u,v,w\\}$ of $\\{1,2,3\\}$, $M_{u}\cap M_{v}\nsubseteq
M_{w}$.

$\bullet$ There is some enumeration $\\{u,v,w\\}$ of $\\{1,2,3\\}$ with
$M_{u}\cap M_{v}\nsubseteq M_{w}$.

$\bullet$ $Gal(G)/(M_{1}\cap M_{2}\cap
M_{3})\cong\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}$

###### Proof.

  1. i-

Recall that $\mathcal{V}(B)\subseteq\Phi(B)\subseteq M^{\prime}_{k},\forall
k\in I$, $M^{\prime}_{i}=\\{\sigma\in\mathcal{W}(B):\gamma_{i}(\sigma)=0\\}$.
The "isomorphic" perfect pairings,

$\displaystyle<,>_{B}$ $\displaystyle:\mathcal{W}(B)/\Phi(B)\times
P_{1}(B)\to{\mathbb{Z}}_{2}$ $\displaystyle<,>$
$\displaystyle:Gal(G)/\Phi(Gal(G))\times G\to\mathbb{Z}_{2}$

provides $M^{\prime}_{i}/\Phi(B)=\\{z_{i}\\}^{\perp}$ and
$M_{a_{i}}/\Phi(Gal(G))=\\{a_{i}\\}^{\perp}$. Since the pairings are
“compatible”, i.e., the dual of the isomorphism
$P_{1}(B)\overset{\cong}{\to}G$ : $z_{k}\mapsto a_{k}$, $k\in I$ corresponds
to the isomorphism

$\mathcal{W}(B)/\Phi(B)\underset{can}{\overset{\cong}{\to}}(\mathcal{W}(B)/\mathcal{V}(B))/\Phi((\mathcal{W}(B)/{\mathcal{V}}(B)))\underset{\bar{\eta}_{B}}{\overset{\cong}{\to}}Gal(G)/\Phi(Gal(G)),$

we have $\bar{\eta}_{B}\circ
can[M^{\prime}_{i}/\Phi(B)]=M_{a_{i}}/\Phi(Gal(G))$. Therefore, as the
Frattini subgroups are contained in all maximal open subgroups, we get
$\eta_{B}[M^{\prime}_{i}/{\mathcal{V}}(B)]=M_{a_{i}}$.

  2. ii-

$M^{\prime}=\\{\sigma\in\mathcal{W}(B):\gamma_{i}(\sigma)+\gamma_{j}(\sigma)=0\\}$.
The "isomorphic" perfect pairings,

$\mathcal{W}(B)/\Phi(B)\times P_{1}(B)\to{\mathbb{Z}}_{2}\mbox{ and
}{Gal(G)/\Phi(Gal(G))}\times G\to{\mathbb{Z}}_{2}$

provides $M_{a}/\Phi(Gal(G))=\\{a\\}^{\perp}$ and
$M_{b}/\Phi(Gal(G))=\\{b\\}^{\perp}$, so

$\displaystyle M_{ab}/\Phi(Gal(G))$
$\displaystyle=\\{ab\\}^{\perp}=\\{\theta/\Phi(Gal(G)):<\theta/\Phi(Gal(G)),ab>=0\\}$
$\displaystyle\subseteq(M_{a})/\Phi(Gal(G))\cap
M_{b})/\Phi(Gal(G)))=(M_{a}\cap M_{b})/\Phi(Gal(G))$

Therefore $\\{M_{a},M_{b},M_{ab}\\}$ are the three maximal subgroups of
$Gal(G)$ above $M_{a}\cap M_{b}$. Since
$\\{M^{\prime}_{i},M^{\prime}_{j},M^{\prime}\\}$ are the three maximal
subgroups of $\mathcal{W}(B)$ above $M^{\prime}_{i}\cap M^{\prime}_{j}$ and

$\eta_{B}[M^{\prime}_{i}/{\mathcal{V}}(B)]=M_{a}\mbox{ and
}\eta_{B}[M^{\prime}_{j}/{\mathcal{V}}(B)]=M_{b},$

we must have $\eta_{B}[M^{\prime}/{\mathcal{V}}(B)]=M_{ab}$

  3. iii-

We have uniquely determined $\\{a,b,c\\}\subseteq G\setminus\\{1\\}$, with
$M_{1}=M_{a},M_{2}=M_{b},M_{3}=M_{c}$ and, from the hypothesis, $a,b,c$ are
pairwise distinct so the result follows from (ii) since if $\\{x,y\\}$ is a
${\mathbb{Z}}_{2}$-li set then $\\{x,xy\\}$ and $\\{y,xy\\}$ are
${\mathbb{Z}}_{2}$-li sets and those 3 sets are ${\mathbb{Z}}_{2}$-basis of
the group $\\{1,x,y,xy\\}$.

∎

######  Lemma 4.7.

Let $G$ be a $k$-stable pre-special group and denote $\mathcal{G}:=Gal(G)$.

  1. i -

Let $S\subseteq\mathcal{G}$ a normal closed of with
$\mathcal{G}/S\cong\mathbb{Z}_{4}$ then there is a unique maximal subgroup
$H\subseteq\mathcal{G}$ such that $S\subseteq H$.

  2. ii -

Let $D\subseteq\mathcal{G}$ a normal closed of with
$\mathcal{G}/S\cong\mathbb{D}_{4}$ then there is a unique set
$\\{H_{1},H_{2}\\}$ with $H_{1}\neq H_{2}$, $H_{i}\subseteq\mathcal{G}$
maximal subgroups such that $D\subseteq H_{1}\cap H_{2}$ and if
$\\{H,H_{1},H_{2}\\}$ are the three maximal subgroups above $H_{1}\cap H_{2}$
then $H/D\cong\mathbb{Z}_{4}$.

###### Proof.

  1. i -

For the existential part take $r\in\mathcal{G}\setminus S$ such that
$r^{2}\notin S$ and $\mathcal{G}/S=\\{1.S,r.S,r^{2}.S,r^{3}.S\\}$
$\cong\mathbb{Z}_{4}$ and take the maximal subgroup $H=1S\cup r^{2}S$. Then
$S\subseteq H$ and the canonical epimorphism
$\mathcal{G}/S\twoheadrightarrow\mathcal{G}/H$ corresponds to the epimorphism
$\mathbb{Z}_{4}\twoheadrightarrow\mathbb{Z}_{2}$. This maximal $H\supseteq S$
is unique because if $S\subseteq H_{1},H_{2}$ with $H_{1}\neq H_{2}$, then
$S\subseteq H_{1}\cap H_{2}$ and there will be an epimorphism
$\mathbb{Z}_{4}\cong\mathcal{G}/S\twoheadrightarrow\mathcal{G}/H_{1}\cap
H_{2}\cong\mathbb{Z}_{2}\times\mathbb{Z}_{2}$, however there is no element in
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ of order 4.

  2. ii -

For the existential part let $r,s\in\mathcal{G}\setminus D$ be such that
$r^{2}\notin D,s^{2}\in D$ and then

$\displaystyle\mathbb{D}_{4}$
$\displaystyle\cong\mathcal{G}/D=\\{1.D,r.D,r^{2}.D,r^{3}.D,s.D,sr.D,sr^{2}.D,sr^{3}.D\\}$

Then each one of the maximal subgroups above $D$ is a reunion of four classes
so they must contain $1D,r^{2}D$. they are three:

$\displaystyle 1D\cup r^{2}D\cup rD\cup r^{3}D$ $\displaystyle 1D\cup
r^{2}D\cup sD\cup sr^{2}D$ $\displaystyle 1D\cup r^{2}D\cup srD\cup sr^{3}D$

Then take

$\displaystyle\\{H_{1},H_{2}\\}$ $\displaystyle=\\{1D\cup r^{2}D\cup sD\cup
sr^{2}D,1D\cup r^{2}D\cup srD\cup sr^{3}D\\}\mbox{ and }$ $\displaystyle H$
$\displaystyle=1D\cup r^{2}D\cup rD\cup r^{3}D.$

Then $D\subseteq H_{1}\cap H_{2}=1D\cup r^{2}D\subseteq H,H_{1},H_{2}$. Then
the canonical epimorphism
$\mathcal{G}/D\twoheadrightarrow\mathcal{G}/H_{1}\cap H_{2}$ corresponds to
the epimorphism
$\mathbb{D}_{4}\twoheadrightarrow\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ and
$H/D\cong\mathbb{Z}_{4}$ . This pair of maximals $H_{1},H_{2}\supseteq D$ is
unique because if $H_{3}$ is a maximal $S\subseteq H_{3}$ with $H_{3}\neq
H_{1},H_{2}$, then $D\subseteq H_{1}\cap H_{2}\cap H_{3}$ and we have two to
consider:
$\bullet$ $H_{1}\cap H_{2}\subseteq H_{3}$: in this case $H_{3}=H=1D\cup
r^{2}D\cup rD\cup r^{3}D$, then

$1D=\cup r^{2}D=H_{1}\cap H_{2}=H\cap H_{1}=H\cap H_{2}$

so $D\subseteq H\cap H_{1}$, but we see directly that
$H_{2}/D\ncong{\mathbb{Z}}_{4}$; similarly $D\subseteq H\cap H_{2}$ but also
$H_{1}/D\ncong{\mathbb{Z}}_{4}$;
$\bullet$ $H_{1}\cap H_{2}\nsubseteq H_{3}$ then $H_{3}\neq H,H_{1},H_{2}$,
also $H_{1}\cap H_{3}\nsubseteq H_{2}$ (because if $H_{1}\cap H_{3}\subseteq
H_{2}$, then $H_{1}\cap H_{2}\nsubseteq H_{3}$, see the previous Lemma) and
similarly $H_{2}\cap H_{3}\nsubseteq H_{1}$, then $\\{H_{1},H_{2},H_{3}\\}$
are independent, so in this case, the epimorphism
$\mathcal{G}/D\twoheadrightarrow\mathcal{G}/H_{1}\cap H_{2}\cap H_{3}$
corresponds to an epimorphism
${\mathbb{D}}_{4}\twoheadrightarrow{\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$
but there is no element in
${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$ of order 4.

∎

######  Theorem 4.8.

  1. i -

Let $a\in G\setminus\\{1\\}$.

$l(a).l(a)=0\in k_{2}(G)\ \Rightarrow$

There is $S\subseteq Gal(G)$, a normal clopen subgroup such that
$Gal(G)/S\cong{\mathbb{Z}}_{4}$ and $S\subseteq M_{a}$.

  2. ii -

Let $a,b\in G\setminus\\{1\\}$ such that $a\neq b$

$l(a).l(b)=0\in k_{2}(G)\ \Rightarrow$

There is $D\subseteq Gal(G)$, a normal clopen subgroup such that
$Gal(G)/D\cong{\mathbb{D}}_{4}$ and $D\subseteq M_{a}\cap M_{b}$,
$M_{ab}/D\cong{\mathbb{Z}}_{4}$.

###### Proof.

  1. i -

Since $\\{a\\}\subseteq G$ is a l.i. subset, take an well ordered basis
$B=\\{a_{i}:i\in I\\}\subseteq G$ such that $a\in B$, say $a=a_{i}$. Then
$\eta_{B}:{\mathcal{W}}(B)/{\mathcal{V}}(B)\overset{\cong}{\to}\ Gal(G)$ with
${\mathcal{V}}(B)\subseteq\Phi(B)$ and ${\mathcal{V}}(B)=Q(B)^{\perp}$ where
$Q(B)=ker(P_{2}(B)\twoheadrightarrow k_{2}(G))$, then, by Proposition 3.2,
$Q(B)=[\\{q^{B}_{xy}:l(x)l(y)=0\\}]$ so

${\mathcal{V}}(B)=\bigcap\\{(q^{B}_{xy})^{\perp}:l(x)l(y)=0\\}=\bigcap\\{(q^{B}_{xy})^{\perp}:x,y\neq
1,l(x)l(y)=0\\}.$

Denote $M^{\prime}_{i}=\\{\sigma\in{\mathcal{W}}(B):\gamma_{i}(\sigma)=0\\}$
and
$S^{\prime}_{i}=\\{\sigma\in{\mathcal{W}}(B):\alpha_{i}(\sigma)=\gamma_{i}(\sigma)=0\\}$
then, by Proposition 2.5(i), $S^{\prime}_{i}\subseteq
M^{\prime}_{i}\subseteq{\mathcal{W}}(B)$ are clopen normal subgroups with
${\mathcal{W}}(B)/M^{\prime}_{i}\cong{\mathbb{Z}}_{2}$,
${\mathcal{W}}(B)/S^{\prime}_{i}\cong{\mathbb{Z}}_{4}$. We have
${\mathcal{V}}(B)\subseteq\Phi(B)\subseteq M^{\prime}_{i}$, and we state the

Claim: ${\mathcal{V}}(B)\subseteq S^{\prime}_{i}$.

This entails that $M_{a}=\eta_{B}[M^{\prime}_{i}/{\mathcal{V}}(B)]\subseteq
Gal(G)$, $Gal(G)/M_{a}\cong{\mathbb{Z}}_{2}$ and

$S:=\eta_{B}[S^{\prime}_{i}/{\mathcal{V}}(B)]\subseteq Gal(G)$

is a clopen normal subgroup of $Gal(G)$ with $Gal(G)/S\cong{\mathbb{Z}}_{4}$
and $S\subseteq M_{a}$, as we need.

Proof of the Claim: We will see that
$S^{\prime}_{i}\cap\Phi(B)=(q^{B}_{a_{i}a_{i}})^{\perp}$ then as $a=a_{i}$ and
$l(a)l(a)=0$ we get ${\mathcal{V}}(B)\subseteq(q^{B}_{a_{i}a_{i}})^{\perp}$ so
${\mathcal{V}}(B)\subseteq S^{\prime}_{i}\cap\Phi(B)$. Since $1\neq a=a_{i}$
it follows that $q^{B}_{a_{i}a_{i}}=z^{2}_{i}\in P_{2}(B)$ is such that
$(q^{B}_{a_{i}a_{i}})^{\perp}\subseteq\Phi(B)$ has index 2 and we will proof
that $S^{\prime}_{i}\cap\Phi(B)\subseteq\Phi(B)$ has also index 2 and
$S^{\prime}_{i}\cap\Phi(B)\subseteq(q^{B}_{a_{i}a_{i}})^{\perp}$ so we get
$S^{\prime}_{i}\cap\Phi(B)=(q^{B}_{a_{i}a_{i}})^{\perp}$. Firstly we show that
$\Phi(B)/S^{\prime}_{i}\cap\Phi(B)\cong{\mathbb{Z}}_{2}$: as
$\Phi(B)\hookrightarrow M^{\prime}_{i}$ then
$\Phi(B)/S^{\prime}_{i}\cap\Phi(B)\rightarrowtail
M^{\prime}_{i}/S^{\prime}_{i}$ and
$M^{\prime}_{i}/S^{\prime}_{i}\cong{\mathbb{Z}}_{2}$ so
$\Phi(B)/S^{\prime}_{i}\cap\Phi(B)$ has 1 or 2 elements. However, it cannot
has 1 element: if $S^{\prime}_{i}\cap\Phi(B)=\Phi(B)$ then, by Proposition
2.13,

$\bigcap\\{M^{\prime}_{j}:j\in I\\}=\Phi(B)\subseteq S^{\prime}_{i}$

but ${\mathcal{W}}(B)$ is a compact space and
$S^{\prime}_{i}\subseteq{\mathcal{W}}(B)$ is open subset ,
$M^{\prime}_{j}\subseteq{\mathcal{W}}(B)$ is a closed subset $j\in I$ so there
is a finite subset $\\{j_{0},\ldots,j_{n}\\}\subseteq I$ such that
$M^{\prime}_{j_{0}}\cap\ldots\cap M^{\prime}_{j_{n}}\subseteq S^{\prime}_{i}$,
choose $n\in{\mathbb{N}}$ minimum with this property so for each $m\leq n$ ,
$\bigcap\\{M^{\prime}_{j_{l}}:l\neq m\\}\nsubseteq M^{\prime}_{j_{m}}$ then we
have an isomorphism

$Gal(G)/\bigcap\\{M^{\prime}_{j_{l}}:l\leq
n\\}\overset{\cong}{\to}\prod_{l\leq n}Gal(G)/M^{\prime}_{j_{l}}$

so the epimorphism $Gal(G)/\bigcap\\{M^{\prime}_{j_{l}}:l\leq
n\\}\twoheadrightarrow Gal(G)/S^{\prime}_{i}$ corresponds to an epimorphism
$\prod_{l\leq n}{\mathbb{Z}}_{2}\twoheadrightarrow{\mathbb{Z}}_{4}$, but the
two elements of order 4 in ${\mathbb{Z}}_{4}$ cannot be in the image of the
homomorphism. Now we prove that
$S^{\prime}_{i}\cap\Phi(B)\subseteq(q^{B}_{a_{i}a_{i}})^{\perp}$: we have
$(q^{B}_{a_{i}a_{i}})^{\perp}=\\{z^{2}_{i}\\}^{\perp}$ and

$S^{\prime}_{i}\cap\Phi(B)=\\{\sigma\in{\mathcal{W}}(I):\alpha_{i}(\sigma)=0\mbox{
and }\gamma_{j}(\sigma)=0\mbox{ for each }j\in I\\}$

and it follows from of the group operation and the definition of the pairing
$<,>:\Phi(B)\times P_{2}(B)\to{\mathbb{Z}}_{2}$ that

$\\{x_{k}x_{l}:k<l\in I\\}\cup\\{x^{2}_{j}:i\neq j\in
I\\}\subseteq\left(S^{\prime}_{i}\cap\Phi(B)\right)\cap\\{z^{2}_{i}\\}^{\perp}$

Since $S^{\prime}_{i}\cap\Phi(B),\\{z^{2}_{i}\\}^{\perp}\subseteq\Phi(B)$ are
closed subgroups,

$closure([\\{x_{k}x_{l}:k<l\in I\\}\cup\\{x^{2}_{j}:i\neq j\in
I\\}])\subseteq\left(S^{\prime}_{i}\cap\Phi(B)\right)\cap\\{z^{2}_{i}\\}^{\perp}.$

Now we will prove that $S^{\prime}_{i}\cap\Phi(B)\subseteq
closure([\\{x_{k}x_{l}:k<l\in I\\}\cup\\{x^{2}_{j}:i\neq j\in I\\}])$; it is
enough find for each $\sigma\in S^{\prime}_{i}\cap\Phi(B)$ and each basic
neighborhood $T$ of $1\in{\mathcal{W}}(B)$ two finite sets $\\{j_{1},\ldots
j_{n}\\}\subseteq I-\\{i\\}$ and
$\\{(k_{1},l_{1}),\ldots(k_{m},l_{m}):k_{u}<l_{u}\in I,1\leq u\leq m\\}$ such
that
$(x^{2}_{j_{1}}.\ldots.x^{2}_{j_{n}}.x_{k_{1}}x_{l_{1}}.\ldots.x_{k_{m}}.x_{l_{m}})\in\sigma.T$
: let $T=\bigcap U$ where

$\displaystyle U\subseteq_{fin}V$ $\displaystyle=\\{M^{\prime}_{j}:j\in
I\\}\cup\\{S^{\prime}_{j}:j\in I\\}\cup\\{D^{\prime}_{kl}:k<l\in I\\}$

and take

$\displaystyle\\{j_{1},\ldots j_{n}\\}$ $\displaystyle=\\{j\in
I:S^{\prime}_{j}\in U,\alpha_{j}(\sigma)=1\\}\subseteq I\setminus\\{i\\}\mbox{
and }$ $\displaystyle\\{k_{1}<l_{1},\ldots k_{m}<l_{m}\\}$
$\displaystyle=\\{k<l\in I:D^{\prime}_{kl}\in U,\beta_{kl}(\sigma)=1\\}$

$\\{j_{1},\ldots j_{n}\\}=\\{j\in I:S^{\prime}_{j}\in
U,\alpha_{j}(\sigma)=1\\}\subseteq I\setminus\\{i\\}$ and
$\\{k_{1}<l_{1},\ldots k_{m}<l_{m}\\}=\\{k<l\in I:D^{\prime}_{kl}\in
U,\beta_{kl}(\sigma)=1\\}$ then, since $\gamma_{j}(\sigma)=0$, for all $j\in
I$, we get

$(x^{2}_{j_{1}}.\ldots.x^{2}_{j_{n}}.x_{k_{1}}x_{l_{1}}.\ldots.x_{k_{m}}.x_{l_{m}})\in\sigma.T$

  2. ii -

Since $\\{a,b\\}\subseteq G$ is a l.i. subset, take an well ordered basis
$B=\\{a_{i}:i\in I\\}\subseteq G$ such that $a,b\in B$, say
$a=a_{i},b=a_{j},i<j\in I$. We denote

$\displaystyle M^{\prime}_{i}$
$\displaystyle=\\{\sigma\in{\mathcal{W}}(B):\gamma_{i}(\sigma)=0\\}$
$\displaystyle M^{\prime}_{j}$
$\displaystyle=\\{\sigma\in{\mathcal{W}}(B):\gamma_{j}(\sigma)=0\\}$
$\displaystyle M^{\prime}$
$\displaystyle=\\{\sigma\in{\mathcal{W}}(B):\gamma_{i}(\sigma)+\gamma_{j}(\sigma)=0\\}$
$\displaystyle D^{\prime}_{ij}$
$\displaystyle=\\{\sigma\in{\mathcal{W}}(B):\beta_{ij}(\sigma)=\gamma_{i}(\sigma)=\gamma_{j}(\sigma)=0\\}$

By Proposition 2.5(ii), $D^{\prime}_{ij}\subseteq
M^{\prime}_{i},M^{\prime}_{j}\subseteq{\mathcal{W}}(B)$ are clopen normal
subgroups with ${\mathcal{W}}(B)/M^{\prime}_{i}\cap
M^{\prime}_{j}\cong{\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$,
${\mathcal{W}}(B)/D^{\prime}_{ij}\cong{\mathbb{D}}_{4}$. Besides
${\mathcal{V}}(B)\subseteq\Phi(B)\subseteq M^{\prime}_{i}\cap M^{\prime}_{j}$
and we stat the

Claim: ${\mathcal{V}}(B)\subseteq D^{\prime}_{ij}$.

This entails that $M_{a}=\eta_{B}[M^{\prime}_{i}/{\mathcal{V}}(B)]\subseteq
Gal(G)$, $M_{b}=\eta_{B}[M^{\prime}_{j}/{\mathcal{V}}(B)]$,
$M_{ab}=\eta_{B}[M^{\prime}/{\mathcal{V}}(B)]$ ,
$Gal(G)/M_{a}\cong{\mathbb{Z}}_{2}\cong Gal(G)/M_{b}$ and
$D\doteq\eta_{B}[S^{\prime}_{i}/{\mathcal{V}}(B)]\subseteq Gal(G)$ is a clopen
normal subgroup of $Gal(G)$ with $Gal(G)/D\cong{\mathbb{D}}_{4}$ such that
$D\subseteq M_{a}\cap M_{b}$ and $M_{ab}/D\cong{\mathbb{Z}}_{4}$, as we need.

Proof of the Claim: We will see that
$D^{\prime}_{ij}\cap\Phi(B)=(q^{B}_{a_{i}a_{j}})^{\perp}$ then as
$a=a_{i},b=a_{j}$ and $l(a)l(b)=0$ we get
${\mathcal{V}}(B)\subseteq(q^{B}_{a_{i}a_{j}})^{\perp}$ so
${\mathcal{V}}(B)\subseteq D^{\prime}_{ij}\cap\Phi(B)$. As $1\neq a=a_{i}$ and
$1\neq b=a_{j}$ with $i<j\in I$, it follows that
$q^{B}_{a_{i}a_{j}}=z_{i}z_{j}\in P_{2}(B)$ is such that
$(q^{B}_{a_{i}a_{j}})^{\perp}\subseteq\Phi(B)$ has index 2 and we will proof
that $D^{\prime}_{ij}\cap\Phi(B)\subseteq\Phi(B)$ has also index 2 and
$D^{\prime}_{ij}\cap\Phi(B)\subseteq(q^{B}_{a_{i}a_{j}})^{\perp}$ so we get
$D^{\prime}_{ij}\cap\Phi(B)=(q^{B}_{a_{i}a_{j}})^{\perp}$.

Firstly, we will prove that
$\Phi(B)/D^{\prime}_{ij}\cap\Phi(B)\cong{\mathbb{Z}}_{2}$: since
$\Phi(B)\hookrightarrow M^{\prime}_{i}\cap M^{\prime}_{j}$ then

$\Phi(B)/D^{\prime}_{ij}\cap\Phi(B)\rightarrowtail M^{\prime}_{i}\cap
M^{\prime}_{j}/D^{\prime}_{ij}\mbox{ and }M^{\prime}_{i}\cap
M^{\prime}_{j}/D^{\prime}_{ij}\cong{\mathbb{Z}}_{2}$

so $\Phi(B)/D^{\prime}_{ij}\cap\Phi(B)$ has 1 or 2 elements. However it cannot
has 1 element: if $D^{\prime}_{ij}\cap\Phi(B)=\Phi(B)$ then
$\bigcap\\{M^{\prime}_{k}:k\in I\\}=\Phi(B)\subseteq D^{\prime}_{ij}$ but
${\mathcal{W}}(B)$ is a compact space and
$D^{\prime}_{ij}\subseteq{\mathcal{W}}(B)$ is open subset ,
$M^{\prime}_{k}\subseteq{\mathcal{W}}(B)$ is a closed subset $k\in I$ so there
is a finite subset $\\{k_{0},\ldots,k_{n}\\}\subseteq I$ such that
$M^{\prime}_{k_{0}}\cap\ldots\cap M^{\prime}_{k_{n}}\subseteq
D^{\prime}_{ij}$, choose $n\in{\mathbb{N}}$ minimum with this property so for
each $m\leq n$ , $\bigcap\\{M^{\prime}_{k_{l}}:l\neq m\\}\nsubseteq
M^{\prime}_{k_{m}}$ then we have an isomorphism

$Gal(G)/\bigcap\\{M^{\prime}_{k_{l}}:l\leq
n\\}\overset{\cong}{\to}\prod_{l\leq n}Gal(G)/M^{\prime}_{k_{l}}$

so the epimorphism $Gal(G)/\bigcap\\{M^{\prime}_{k_{l}}:l\leq
n\\}\twoheadrightarrow Gal(G)/D^{\prime}_{ij}$ corresponds to an epimorphism
$\prod_{l\leq n}{\mathbb{Z}}_{2}\twoheadrightarrow{\mathbb{D}}_{4}$, but the
two elements of order 4 in ${\mathbb{D}}_{4}$ cannot be in the image of the
homomorphism.

Now we prove that
$D^{\prime}_{ij}\cap\Phi(B)\subseteq(q^{B}_{a_{i}a_{j}})^{\perp}$: we have
$(q^{B}_{a_{i}a_{j}})^{\perp}=\\{z_{i}z_{j}\\}^{\perp}$ and

$D^{\prime}_{ij}\cap\Phi(B)=\\{\sigma\in{\mathcal{W}}(I):\beta_{ij}(\sigma)=0\mbox{
and }\gamma_{k}(\sigma)=0\mbox{ for each }k\in I\\}$

and it follows from of the group operation and the definition of the pairing

$<,>:\Phi(B)\times P_{2}(B)\to{\mathbb{Z}}_{2}$

that

$\\{x_{k}x_{l}:k<l\in I,(k,l)\neq(i,j)\\}\cup\\{x^{2}_{k}:k\in
I\\}\subseteq\left(D^{\prime}_{ij}\cap\Phi(B)\right)\cap\\{z_{i}z_{j}\\}^{\perp}$

then, since
$D^{\prime}_{ij}\cap\Phi(B),\\{z_{i}z_{j}\\}^{\perp}\subseteq\Phi(B)$ are
closed subgroups,

$closure([\\{x_{k}x_{l}:k<l\in I,(k,l)\neq(i,j)\\}\cup\\{x^{2}_{k}:k\in
I\\}])\subseteq\left(D^{\prime}_{ij}\cap\Phi(B)\right)\cap\\{z_{i}z_{j}\\}^{\perp}.$

Now we will prove that $D^{\prime}_{ij}\cap\Phi(B)\subseteq
closure([\\{x_{k}x_{l}:k<l\in I,(k,l)\neq(i,j)\\}\cup\\{x^{2}_{k}:k\in
I\\}])$; it is enough find for each $\sigma\in D^{\prime}_{ij}\cap\Phi(B)$ and
each basic neighborhood $T$ of $1\in{\mathcal{W}}(B)$ two finite sets
$\\{j_{1},\ldots,j_{n}\\}\subseteq I$ and
$\\{(k_{1},l_{1}),\ldots(k_{m},l_{m}):k_{u}<l_{u}\in
I,(k_{u},l_{u})\neq(i,j),1\leq u\leq m\\}$ such that
$(x^{2}_{j_{1}}.\ldots.x^{2}_{j_{n}}.x_{k_{1}}x_{l_{1}}.\ldots.x_{k_{m}}.x_{l_{m}})\in\sigma.T$
: let $T=\bigcap U$ where

$\displaystyle U\subseteq_{fin}V$ $\displaystyle=\\{M^{\prime}_{j}:j\in
I\\}\cup\\{S^{\prime}_{j}:j\in I\\}\cup\\{D^{\prime}_{kl}:k<l\in I\\}$

and take

$\displaystyle\\{j_{1},\ldots j_{n}\\}$ $\displaystyle=\\{j\in
I:S^{\prime}_{j}\in U,\alpha_{j}(\sigma)=1\\}\subseteq I\mbox{ and }$
$\displaystyle\\{k_{1}<l_{1},\ldots k_{m}<l_{m}\\}$ $\displaystyle=\\{k<l\in
I:D^{\prime}_{kl}\in U,\beta_{kl}(\sigma)=1\\}\subseteq I\times
I\setminus\\{(i,j)\\}$

then, since $\gamma_{k}(\sigma)=0$, for all $k\in I$, we get

$(x^{2}_{j_{1}}.\ldots.x^{2}_{j_{n}}.x_{k_{1}}x_{l_{1}}.\ldots.x_{k_{m}}.x_{l_{m}})\in\sigma.T.$

∎

The above proposition suggests the following:

######  Definition 4.9.

A pre-special group $G$ is said to be standard if it is a $k$-stable pre-
special group and holds both the reverse implications in the Theorem 4.8
above.

######  Remark 4.10.

Lemma 4.7 determines (injective) maps

$\displaystyle j_{1}$ $\displaystyle:\\{S\subseteq\mathcal{G}:S\mbox{ is a
normal subgroup of index }\mathbb{Z}_{4}\\}\rightarrow$
$\displaystyle\\{M\subseteq\mathcal{G}:M\mbox{ is a maximal subgroup}\\};$
$\displaystyle j_{2}$ $\displaystyle:\\{D\subseteq\mathcal{G}:D\mbox{ is a
normal subgroup of index }\mathbb{D}_{4}\\}\rightarrow$
$\displaystyle\qquad\\{\\{M_{1},M_{2}\\}:M_{1},M_{2}\subseteq\mathcal{G},\,M_{1}\neq
M_{2}\mbox{ are maximal subgroups}\\}.$

By the canonical bijection $\mathbb{M}:G\setminus\\{1\\}\
\overset{\cong}{\to}\ \\{M\subseteq Gal(G):M$ is a maximal clopen subgroup
$\\}$, it is natural to ask:
(1) Which subset of $\\{a\in G:a\neq 1\\}$ corresponds bijectively with
$image(j_{1})$?
(2) Which subset of $\\{\\{a,b\\}\subseteq G:a,b\neq 1,a\neq b\\}$ corresponds
bijectively with $image(j_{2})$?

The concept of standard  pre-special group provides a full answer to these
questions:
(1) The set $\\{\\{a\\}\subseteq G:\\{a\\}l.i.,l(a)l(a)=0\in k_{2}(G)\\}$
corresponds bijectively with $image(j_{1})$.
(2) The set $\\{\\{a,b\\}\subseteq G:\\{a,b\\}l.i.,l(a)l(b)=0\in k_{2}(G)\\}$
corresponds bijectively with $image(j_{2})$.

It follows from Propositions 2.3 and 2.4 in [14] that $SG(F)$ is a standard
special group, for every field $F$ with $char(F)\neq 2$.

We have already established that every special group $G$ is $k$-stable (see
Proposition LABEL:k-stable-aph(iii)).

These suggest the following:

######  Question 4.11.

Is every special group $G$ standard?222We are unable to solve this question
with the methods so far developed. We believe that to address this question,
we will have to develop the theory of quadratic extensions of pre-special
hyperfields.

In the sequel, we will see the relevance of the subclass of standard pre-
special groups. We invite the reader to recall Proposition 4.4.

######  Proposition 4.12.

Let $G$ be a $k$-stable pre special group and denote ${\mathcal{G}}:=Gal(G)$.

  1. i-

Let $\sigma\in{\mathcal{G}}\setminus\Phi({\mathcal{G}})$ be such that
$\sigma^{2}=id$. Then
$\\{\Phi({\mathcal{G}}),\sigma.\Phi({\mathcal{G}})\\}^{\perp}\subseteq G$ is a
maximal saturated subgroup of $G$.

  2. ii-

Suppose that $G$ is a standard pre-special group. Let
$\sigma\in{\mathcal{G}}\setminus\Phi(\mathcal{G})$ be such that
$\sigma^{2}\neq id$ (so $\sigma^{4}=id$). Then
$\\{\Phi({\mathcal{G}}),\sigma.\Phi({\mathcal{G}})\\}^{\perp}\subseteq G$ is
not a saturated subgroup of $G$.

  3. iii-

Suppose that $G$ is a standard special group. The set of all classes of
conjugacy of involutions $\sigma\in{\mathcal{G}}\setminus\Phi(G)$ corresponds
to the set of all orderings (= maximal saturated subgroups) of $G$.

  4. iv-

Suppose that $G$ is a standard pre-special group. Then we have an anti-
isomorphism of complete lattices between the posets

$\\{\Delta\subseteq G:\Delta\mbox{ is a saturated subgroups of}\ G\\}$

and

$\\{T\subseteq{\mathcal{G}}:T\mbox{ is a closed subgroup of}\
{\mathcal{G}}\mbox{(topologically) generated by involutions such that}\
\Phi({\mathcal{G}})\subseteq T\\}$

###### Proof.

  1. i-

Let $\bar{T}=\\{\Phi({\mathcal{G}}),\sigma\Phi({\mathcal{G}})\\}$.

Claim: To have that $\bar{T}^{\perp}\subseteq G$ is a saturated subgroup it is
enough to prove the following: $\forall x,y\in G$ if $<x,y>\equiv<1,xy>$, then
$x\in\bar{T}^{\perp}$ or $y\in\bar{T}^{\perp}$ .

Proof of Claim: Firstly we prove that $-1\notin\bar{T}^{\perp}$: take any
$x\notin\bar{T}^{\perp}$ (there is some $x$, as $\bar{T}^{\perp}\subseteq G$
has index 2) then as $<x,-x>\equiv<1,-1>$ it follows from assumption in the
claim that $-x\in\bar{T}^{\perp}$ so if $-1\in\bar{T}^{\perp}$ then
$x=-1.(-x)\in\bar{T}^{\perp}$, a contradiction. Now let us prove that
$\bar{T}^{\perp}$ is saturated: take any $a,b\in G$ such that $b\in
D_{G}(<1,a>)$, assume $a\in\bar{T}^{\perp}$ then we have to prove that
$b\in\bar{T}^{\perp}$: as $(-a).a=-1\notin\bar{T}^{\perp}$ then
$-a\notin\bar{T}^{\perp}$ and as $<b,ba>\equiv<1,a>$ we have
$<b,-a>\equiv<1,-ba>$ so, by the assumption in the claim, we get
$b\in{\bar{T}}^{\perp}$.
Now we will prove that $\sigma^{2}=id$ entails $\forall x,y\in G$ if
$<x,y>\equiv<1,xy>$ then $x\in\bar{T}^{\perp}$ or $y\in\bar{T}^{\perp}$:
We have three cases:
$\ast$ $x$ (or $y$) is $1$;
$\ast$ $x=y\neq 1$;
$\ast$ $x,y\neq 1$ and $x\neq y$.
There is nothing to proof in the first case. Now consider $x\in
G\setminus\\{1\\}$ such that $<x,x>\equiv<1,1>$: we must prove that
$x\in{\bar{T}}^{\perp}$. Since $G$ is $k$-stable, we have
$l(x)l(x)=l(1)l(1)=0$ then, by Theorem 4.8(i), there is a
$S\subseteq{\mathcal{G}}$ a clopen normal subgroup such that
${\mathcal{G}}/S\cong{\mathbb{Z}_{4}}$ and $S\subseteq M_{x}$. Consider the
quotient map $p_{S}:{\mathcal{G}}\twoheadrightarrow{\mathcal{G}}/S$ and write
${\mathcal{G}}/S=\\{1/S,r/S,r^{2}/S,r^{3}/S\\}$ then as $\sigma^{2}=id$ we
must have $\sigma/S\in\\{1/S,r^{2}/S\\}$. If $\sigma/S=1/S$ then $\sigma\in
S\subseteq M_{x}$ i.e. $<\sigma/\Phi({\mathcal{G}}),x>=0$ so
$x\in\\{\Phi({\mathcal{G}}),\sigma\Phi({\mathcal{G}})\\}^{\perp}$. If
$\sigma/S=r^{2}/S$ then $\sigma.r^{2}=\sigma.r^{-2}\in S\subseteq M_{x}$ i.e.
$<(\sigma.r^{2})/\Phi({\mathcal{G}}),x>=0$ but

$<(\sigma.r^{2})/\Phi({\mathcal{G}}),x>=<\sigma/\Phi({\mathcal{G}}),x>+<r/\Phi({\mathcal{G}}),x>+<r/\Phi({\mathcal{G}}),x>=<\sigma/\Phi({\mathcal{G}}),x>$

then $<\sigma/\Phi({\mathcal{G}}),x>=0$ so
$x\in\\{\Phi({\mathcal{G}}),\sigma\Phi({\mathcal{G}})\\}^{\perp}$. Now take
$x,y\in G\setminus\\{1\\}$ with $x\neq y$ and $<x,y>\equiv<1,xy>$ and supose
$x\notin\bar{T}^{\perp}$ then we must prove that $y\in\bar{T}^{\perp}$. As
$\\{x,y\\}$ is a two element l.i. set and $l(x)l(y)=l(1)l(xy)=0$ we have, by
Theorem 4.8(ii), some $D\subseteq{\mathcal{G}}$ a clopen normal subgroup such
that ${\mathcal{G}}/D\cong{\mathbb{D}}_{4}$, $D\subseteq M_{x}\cap M_{y}$ and
$M_{x.y}/D\cong{\mathbb{Z}_{4}}$. Consider the quotient morphism
$p_{D}:{\mathcal{G}}\twoheadrightarrow{\mathcal{G}}/D$ and write

${\mathcal{G}}/D=\\{1/D,r/D,r^{2}/D,r^{3}/D,s/D,sr/D,sr^{2}/D,sr^{3}/D\\}$

then, as $\sigma^{2}=id$, we have $\sigma/D\notin\\{r/D,r^{3}/D\\}$. Let us
prove that $\sigma/D\notin\\{1/D,r^{2}/D\\}$: as
$<1/\Phi({\mathcal{G}}),x>=<r^{2}/\Phi({\mathcal{G}}),x>=0$ we have
$\\{id,r^{2}\\}\subseteq M_{x}$ and as we selected
$x\notin\\{\Phi,\sigma\Phi\\}^{\perp}$ we have $\sigma\notin M_{x}$ then if
$\sigma/D=r^{2}/D$ then $\sigma.r^{-2}\in D\subseteq M_{x}$ so

$\sigma=(\sigma.r^{-2}).r^{2}\in D.M_{x}\subseteq M_{x}.M_{x}\subseteq M_{x},$

a contradiction; similarly $\sigma/D\neq 1/D$. So we have
$\sigma/D\in\\{s/D,sr/D,sr^{2}/D,sr^{3}/D\\}$. Now, as
$M_{x.y}/D\cong{\mathbb{Z}_{4}}$ we have $M_{x.y}=1D\cup r^{2}D\cup rD\cup
r^{3}D$ (see the proof of Theorem 4.8(ii)) and

$\\{M_{x},M_{y}\\}=\\{1D\cup r^{2}D\cup sD\cup sr^{2}D,1D\cup r^{2}D\cup
srD\cup sr^{3}D\\}.$

If $M_{x}=1D\cup r^{2}D\cup sD\cup sr^{2}D$ then as $\sigma\notin M_{x}$ we
have $\sigma/D\notin\\{s/D,sr^{2}/D\\}$ so

$\sigma/D\in\\{sr/D,sr^{3}/D\\}\subseteq M_{y}/D$

then $\sigma\in M_{y}$ that is
$y\in\\{\Phi({\mathcal{G}}),\sigma\Phi({\mathcal{G}})\\}^{\perp}$; similarly
if $M_{x}=1D\cup r^{2}D\cup srD\cup sr^{3}D$ then
$y\in\\{\Phi({\mathcal{G}}),\sigma\Phi({\mathcal{G}})\\}^{\perp}$.

  2. ii-

Let $\bar{T}=\\{\Phi({\mathcal{G}}),\sigma\Phi({\mathcal{G}})\\}$.

Claim: To have that $\bar{T}^{\perp}\subseteq G$ is not a saturated subgroup
it is enough to prove the following: $\exists x,c\in
G\setminus\bar{T}^{\perp}$ such that $<x,c>\equiv<1,xc>$.

Proof of Claim: If $-1\in\bar{T}^{\perp}$ then $\bar{T}^{\perp}\subsetneq G$
so $G=D_{G}<1,-1>\nsubseteq\bar{T}^{\perp}$ so $\bar{T}^{\perp}$ is not a
saturated subgroup. If $-1\notin\bar{T}^{\perp}$ then take $x,c\in
G\setminus\bar{T}^{\perp}$ such that $<x,c>\equiv<1,xc>$ so we have
$<c,-xc>\equiv<1,-x>$, that is $c\in D_{G}<1,-x>$ and $-x\in\bar{T}^{\perp}$:
if $-x\notin\bar{T}^{\perp}$ then as $\bar{T}^{\perp}\subseteq G$ has index 2
$-1.\bar{T}^{\perp}=-x.\bar{T}^{\perp}$ so $x=-1.-x\in\bar{T}^{\perp}$; that
is we established that there are $a(=-x)\in\bar{T}^{\perp}$ and $c\in
D_{G}<1,a>$ with $c\notin\bar{T}^{\perp}$: this means that $\bar{T}^{\perp}$
is not saturated.
Now we will prove that $\sigma^{2}\neq id$ entails $\exists x,c\in
G\setminus\bar{T}^{\perp}$ such that $<x,c>\equiv<1,xc>$:
Take $B$ any well ordered base of $G$ and consider the composition

$\mathcal{W}(B)\twoheadrightarrow\mathcal{W}(B)/\mathcal{V}(B){\overset{\cong}{\rightarrow}}{\mathcal{G}}$

and, by Lemma 2.10, choose any lifting $\widetilde{\sigma}\in\mathcal{W}(B)$
of $\sigma\in{\mathcal{G}}$. Since $\sigma^{2}\neq id\in{\mathcal{G}}$ we get
$\widetilde{\sigma}^{2}\in\Phi(B)\setminus{\mathcal{V}}(B)$. Since

$\displaystyle{\mathcal{V}}(B)$ $\displaystyle=Q_{B}^{\perp}=[\\{q_{ab}^{B}\in
P_{2}(B):l(a)l(b)=0\in k_{2}(G)\\}]^{\perp}$
$\displaystyle=\bigcap\\{(q_{ab}^{B})^{\perp}\subseteq\Phi(B):a,b\neq
1,l(a)l(b)=0\\}$

we get $a,b\neq 1$ with $l(a)l(b)=0\in k_{2}(G)$ and
$\widetilde{\sigma}^{2}\in\Phi(B)\setminus(q^{B}_{ab})^{\perp}$. There are two
cases to consider: $a=b$ and $a\neq b$. In the first case $\\{a\\}$ is a
singleton l.i. set and in the second $\\{a,b\\}$ is a two element l.i. set:
consider any well ordered basis $B^{\prime}=\\{a^{\prime}_{i}:i\in I\\}$ such
that $a=a_{i}$ for some $i\in I$ in the first case and, $a=a_{i},b=a_{j}$ for
some $i<j\in I$ in the second case. Now consider the isomorphism of change of
basis
$\mu_{B^{\prime}B}:\mathcal{W}(B)\overset{\cong}{\rightarrow}\mathcal{W}(B^{\prime})$
(see Lemma 3.3)) and take
$\sigma^{\prime}=\mu_{B^{\prime}B}(\widetilde{\sigma})\in\mathcal{W}(B^{\prime})$.
Then $\sigma^{\prime}\in\mathcal{W}(B^{\prime})$ is a lifting of
$\sigma\in{\mathcal{G}}$ with respect to the epimorphism
$\mathcal{W}(B^{\prime})\twoheadrightarrow\mathcal{W}(B^{\prime})/{\mathcal{V}}(B^{\prime})$
${\overset{\cong}{\rightarrow}}$ ${\mathcal{G}}$ (Lemma 2.10, again) and
$\sigma^{\prime
2}\in\Phi(B^{\prime})\setminus\mu_{B^{\prime}B}[(q^{B}_{a,b})^{\perp}]=\Phi(B^{\prime})\setminus(q^{B^{\prime}}_{a,b})^{\perp}$
(see Remark 3.4). As $l(a)l(b)=0$, by the proof of the Theorem 4.8, we have
$(q^{B}_{a_{i},a_{i}})^{\perp}=S^{\prime}_{i}\cap\Phi(B^{\prime})$ in first
case and $(q^{B}_{a_{i},a_{j}})^{\perp}=D^{\prime}_{ij}\cap\Phi(B^{\prime})$
in the second case, then $\sigma^{\prime 2}\in\Phi(B^{\prime})\setminus
S^{\prime}_{i}$ (resp. $\sigma^{\prime 2}\in\Phi(B^{\prime})\setminus
D^{\prime}_{ij}$). A straightforward calculation with the group operation in
$\mathcal{W}(B^{\prime})$ gives $\sigma^{\prime}\notin
M^{\prime}_{i}\subseteq\mathcal{W}(B^{\prime})$ in the first case and
$\sigma^{\prime}\notin M^{\prime}_{i}\cup
M^{\prime}_{j}\subseteq\mathcal{W}(B^{\prime})$ in the second case, then
applying $\mathcal{W}(B^{\prime})\twoheadrightarrow{\mathcal{G}}$ we have
$\sigma\notin M_{a_{i}}\subseteq{\mathcal{G}}$ (resp. $\sigma\notin
M_{a_{i}}\cup M_{a_{j}}\subseteq{\mathcal{G}}$). Now recall that for each
$y\in G\setminus\\{1\\}$ and each
$\theta\in{\mathcal{G}}\setminus\Phi({\mathcal{G}})$, $\theta\notin M_{y}$ iff
$<\\{\Phi({\mathcal{G}}),\theta.\Phi({\mathcal{G}})\\},y>=1$ iff
$y\notin\\{\Phi({\mathcal{G}}),\theta.\Phi({\mathcal{G}})\\}^{\perp}$. Then,
since $G$ is a standard pre-special group we have, in both cases, $1\neq a,b$
, $l(a)l(b)=0\in k_{2}(G)$,
$a,b\notin\\{\Phi({\mathcal{G}}),\sigma.\Phi({\mathcal{G}})\\}^{\perp}$ and,
in particular, since $G$ is a $k$-stable pre-special group, $1\in
D_{G}(<a,b>)$ or, equivalently, $<a,b>\equiv<1,ab>$.

  3. iii-

Recall that for special groups the maximal saturated subgroups are precisely
the index 2 saturated subgroups so the result follows from items (i) and (ii).

  4. iv-

Let $\Delta\subseteq G$ be a saturated subgroup: as $G$ is a special group
$\Delta=\bigcap\\{\Sigma\subseteq G:\Sigma\in X_{\Delta}\\}$ where

$X_{\Delta}=\\{\Sigma\subseteq G:\Sigma\mbox{ is a maximal saturated subgroup
and }\Delta\subseteq\Sigma\\}$

then, by Proposition 4.4,

$\Delta^{\perp}=\bigvee\\{\Sigma^{\perp}\subseteq{\mathcal{G}}/{\Phi(\mathcal{G})}:\Sigma\in
X_{\Delta}\\};$

by item (iii)
$\Sigma^{\perp}=\\{\Phi({\mathcal{G}}),\sigma\Phi({\mathcal{G}})\\}$ for some
$\sigma\in{\mathcal{G}}\setminus\Phi({\mathcal{G}})$, $\sigma^{2}=id$; take
$T_{\Sigma}=\Phi({\mathcal{G}})\cup\sigma\Phi({\mathcal{G}})$ then
$T_{\Sigma}\subseteq{\mathcal{G}}$ is a closed (normal) subgroup such that
$\Phi({\mathcal{G}})\subseteq T_{\Sigma}$ and all elements of
$T_{\Sigma}\setminus\\{1\\}$ are involutions so

$\bigvee\\{T_{\Sigma}:\Sigma\in X_{\Delta}\\}=closure([\\{T_{\Sigma}:\Sigma\in
X_{\Delta}\\}])$

is a closed subgroup of ${\mathcal{G}}$ that contains $\Phi({\mathcal{G}})$
and is (topologically) generated by involutions. Now note that
$T_{\Sigma}/\Phi({\mathcal{G}})=\\{\Phi({\mathcal{G}}),\sigma\Phi({\mathcal{G}})\\}=\Sigma^{\perp}$
and then

$\displaystyle\Delta^{\perp}$
$\displaystyle=\bigvee\\{\Sigma^{\perp}\subseteq{\mathcal{G}}/{\Phi(\mathcal{G})}:\Sigma\in
X_{\Delta}\\}=\bigvee\\{T_{\Sigma}/\Phi({\mathcal{G}})\subseteq{\mathcal{G}}/{\Phi(\mathcal{G})}:\Sigma\in
X_{\Delta}\\}$
$\displaystyle=(\bigvee\\{T_{\Sigma}\subseteq{\mathcal{G}}:\Sigma\in
X_{\Delta}\\})/\Phi({\mathcal{G}})$

as $q:{\mathcal{G}}\twoheadrightarrow{\mathcal{G}}/{\Phi(\mathcal{G})}$ gives
an isomorphism of complete lattices between the set of closed subgroups of
${\mathcal{G}}$ which contains $\Phi({\mathcal{G}})$ and the set of closed
subgroups of ${\mathcal{G}}/{\Phi(\mathcal{G})}$.

Now take $T\subseteq{\mathcal{G}}$ a closed subgroup of ${\mathcal{G}}$ such
that $\Phi({\mathcal{G}})\subseteq T$ and $T$ is topologically generated by
involutions. Write $I_{T}=\\{\sigma\in
T:\sigma\in{\mathcal{G}}\setminus\Phi({\mathcal{G}}),\sigma^{2}=id\\}$ then,
for each $\sigma\in I_{T}$, $\sigma\Phi({\mathcal{G}})\subseteq T$ ,
$T_{\sigma}=\Phi({\mathcal{G}})\cup\sigma\Phi({\mathcal{G}})$ is a closed
(normal) subgroup of ${\mathcal{G}}$ and

$T=closure([\bigcup\\{T_{\sigma}:\sigma\in
I_{T}\\}])=\bigvee\\{T_{\sigma}:\sigma\in I_{T}\\},$

also
$T_{\sigma}/\Phi({\mathcal{G}})=\\{\Phi({\mathcal{G}}),\sigma\Phi({\mathcal{G}})\\}$
and, by item (iv), $(T_{\sigma}/\Phi({\mathcal{G}}))^{\perp}\subseteq G$ is a
maximal saturated subgroup of $G$. Then we have

$\displaystyle(T/\Phi({\mathcal{G}}))^{\perp}$
$\displaystyle=((\bigvee\\{T_{\sigma}:\sigma\in
I_{T})/\Phi({\mathcal{G}}))^{\perp}=(\bigvee\\{T_{\sigma}/\Phi({\mathcal{G}}):\sigma\in
I_{T}\\})^{\perp}$
$\displaystyle=\bigcap\\{(T_{\sigma}/\Phi({\mathcal{G}}))^{\perp}:\sigma\in
I_{T}\\}$

which is a saturated subgroup of $G$.

∎

######  Theorem 4.13.

Let $G$ be a standard special group. Are equivalent

  1. i -

$G$ is "Pythagorean" or "almost reduced"333I.e. for all $a\in G$,
$<a,a>\equiv<1,1>$ iff $a=1$, but eventually $-1=1$..

  2. ii -

$\mbox{Gal}(G)$ is generated by involutions.

###### Proof.

Note that the unique non-formally real Pythagorean special group
(equivalently, $-1\neq 1$) is $G=\\{1\\}$ and thus $Gal(G)=\\{1\\}$.

$(i)\Rightarrow(ii)$: The hypothesis means that $\\{1\\}\subseteq G$ is a
saturated subset of $G$, then by item (iv) of the previous Proposition,
$Gal(G)$ is generated by involutions.

$(ii)\Rightarrow(i)$: It follows the hypothesis that there is no continuous
epimorphism
${\mathcal{G}}/\Phi({\mathcal{G}})\twoheadrightarrow{\mathbb{Z}_{4}}$. Since
$G$ is standard SG, for all $a\in G\setminus\\{1\\}$, $l(a)l(a)\neq
0=l(1)l(1)$ and, since $G$ is in particular $k$-stable, then for all $a\in
G\setminus\\{1\\}$, is not the case $<a,a>\equiv<1,1>$, that is: $G$ is
Pythagorean. ∎

######  Remark 4.14.

Another Galois theoretic characterization of the Pythagoreaness of $G$ is

$\Phi(\mbox{Gal}(G))=[\mbox{Gal}(G),\mbox{Gal}(G)].$

######  Theorem 4.15.

Let $G$ be a standard special group. Consider the following

  1. i-

$G$ is not formally real.

  2. ii-

Every involution is is $\Phi(Gal(G))$.

  3. iii-

Every involution in ${Gal(G)}$ is central.

Then $(i)\Rightarrow(ii)\Rightarrow(iii)$ and if $card({Gal(G)})>2$, then all
are equivalent.

###### Proof.

By Proposition 4.12(iii) $G$ is formally real iff there is an involution
$\sigma\in{Gal(G)}\setminus\Phi({Gal(G)})$, so we get $(i)\Rightarrow(ii)$. As
${Gal(G)}$ is a ${\mathcal{C}}$-group we have $[\sigma^{2},\tau]=1$ and, since
$\Phi({Gal(G)})={Gal(G)}^{2}$ (pro-2-group), then $\Phi({Gal(G)})\subseteq
center({Gal(G)})$ so $(ii)\Rightarrow(iii)$. Now suppose $card({Gal(G)})>2$:
to prove $(iii)\Rightarrow(i)$ let us assume $G$ formally real and note that
any involution $\notin\Phi({Gal(G)})$ is not in the center of ${Gal(G)}$. ∎

##  5 The functorial behavior of $Gal$ and SG-cohomology

We have developed the theme "basis change induced isomorphisms" in the general
context of pre-Special Groups, as the fundamental step to get a single Galois
group of a pre-special group. In this final section, we analyze some
functorial behavior of the $Gal$ construction of $SG$-theory and provide the
first steps to a Galoisian cohomology for the $SG$-theory, in an attempt to
complete the "Milnor scenario" of Igr’s ([13]) in abstract theories of
quadratic forms.

###  5.1 From PSG to Galois groups

######  Remark 5.1.

Construction:

Let $f:G\rightarrow G^{\prime}$ a $pSG$-homomorphism of pre-special groups.

Let $B_{1}^{\prime}=\\{a^{\prime}_{k}:k\in I_{1}^{\prime}\\}$ be an well
ordered basis of $f[G]$ and extends it to $B^{\prime}=\\{a^{\prime}_{k}:k\in
I^{\prime}\\}$, an well ordered basis of $G^{\prime}$. Now select $a_{k}\in
f^{-1}[\\{a^{\prime}_{k}\\}],k\in I^{\prime}_{1}$. Then the set
$B_{1}=\\{a_{i}:i\in I^{\prime}_{1}\\}\subseteq G$ is linearly independent,
now complete this to basis of $G$, $B=\\{a_{i}:i\in I\\}$: we just need to
glue a well ordered basis of $ker(f)$.

We have some induced functions:

(0) ${f}^{0}_{B,B^{\prime}}:B^{\prime}\rightarrow{\mathcal{W}}(B)$ is such
that ${f}^{0}_{B,B^{\prime}}(a^{\prime}_{k})=a_{k}$, if $a^{\prime}_{k}\in
B^{\prime}_{1}$ and ${f}^{0}_{B,B^{\prime}}(a^{\prime}_{k})=1$, if
$a^{\prime}_{k}\in B^{\prime}\setminus B^{\prime}_{1}$.

(1) $f^{(1)}_{B,B^{\prime}}:B\rightarrow P_{2}(B^{\prime})$ is such that
${f}^{1}_{B,B^{\prime}}(a_{k})=a^{\prime}_{k}$, if $a_{k}\in B_{1}$ and
${f}^{1}_{B,B^{\prime}}(a_{k})=0$, if $a_{k}\in B\setminus B_{1}$.

(2) $f^{(2)}_{B,B^{\prime}}:\\{(a_{i},a_{j})\in B\times B:i,j\in I,i\leq
j\\}\rightarrow P_{2}(B^{\prime})$ is such that
${f}^{2}_{B,B^{\prime}}(a_{i},a_{j})=a^{\prime}_{i}.a^{\prime}_{j}$, if
$(a_{i},a_{j})\in B_{1}\times B_{1}$ and
${f}^{2}_{B,B^{\prime}}(a_{i},a_{j})=0$, if $(a_{i},a_{j})\in B\times
B\setminus B_{1}\times B_{1}$.

Keeping the notation above, we have

######  Proposition 5.2.

The function ${f}^{0}_{B,B^{\prime}}:B^{\prime}\rightarrow{\mathcal{W}}(B)$
induces a continuous homomorphism
$\bar{f}_{B,B^{\prime}}:{\mathcal{W}}(B^{\prime})/{\mathcal{V}}(B^{\prime})\rightarrow{\mathcal{W}}(B)/{\mathcal{V}}(B)$.

###### Proof.

It follows from Proposition 2.6 and the definition of
${f}^{0}_{B,B^{\prime}}:B^{\prime}\rightarrow{\mathcal{W}}(B)$, that its image
converges to $1\in{\mathcal{W}}(B)$. Thus, by the universal property
${\mathcal{W}}(B^{\prime})$ (Theorem 2.11), ${f}^{0}_{B,B^{\prime}}$ extends
uniquely to a continuous homomorphism of pro-2-groups
$\hat{f}^{0}_{B,B^{\prime}}:{\mathcal{W}}(B^{\prime})\rightarrow{\mathcal{W}}(B)$.

Now, $f:G\to G^{\prime}$ also induces a $\mathbb{Z}_{2}$-module homomorphism
$\hat{f}^{(2)}_{B,B^{\prime}}:P_{2}(B)\rightarrow P_{2}(B^{\prime})$: this is
just the unique $\mathbb{Z}_{2}$-linear extension of the induced map
$f^{(2)}_{B,B^{\prime}}:\\{(a_{i},a_{j})\in B\times B:i,j\in I,i\leq
j\\}\rightarrow P_{2}(B^{\prime})$. Moreover, since $k_{*}(f):k_{*}(G)\to
k_{*}(G^{\prime})$ is an Igr-morphism, then for each $a,b\in G$ such that
$l(a)l(b)=0\in k_{2}(G)$, we have $l(fa).l(fb)=0\in k_{2}(G^{\prime})$. From
this we obtain that
$\hat{f}^{(2)}_{B,B^{\prime}}(q^{B}_{a,b})=q^{B^{\prime}}_{fa,fb}$ (see
Proposition 3.2) and, therefore, $f^{(2)}_{B,B^{\prime}}[Q(B)]\subseteq
Q(B^{\prime})$.

For each $a,b\in G$ and $\sigma^{\prime}\in{\mathcal{W}}(B^{\prime})$, we have

$<\hat{f}^{(0)}_{B,B^{\prime}}(\sigma^{\prime}),q^{B}_{a,b}>=<\sigma^{\prime},\hat{f}^{(2)}_{B,B^{\prime}}(q^{B}_{a,b}>=<\sigma^{\prime},q^{B^{\prime}}_{fa,fb}>.$

Thus
$\hat{f}^{(0)}_{B,B^{\prime}}[{\mathcal{V}}(B^{\prime})]\subseteq{\mathcal{V}}(B)$
and then,
$\hat{f}^{0}_{B,B^{\prime}}:{\mathcal{W}}(B^{\prime})\rightarrow{\mathcal{W}}(B)$
induces a unique continuous homomorphism of pro-2-groups
$\bar{f}_{B,B^{\prime}}:{\mathcal{W}}(B^{\prime})/{\mathcal{V}}(B^{\prime})\rightarrow{\mathcal{W}}(B)/{\mathcal{V}}(B)$.

∎

######  Proposition 5.3.

Let $f:G\rightarrow G^{\prime}$ be an injective $pSG$-morphism.

  1. i -

Then
$\hat{f}^{0}_{B,B^{\prime}}:{\mathcal{W}}(B^{\prime})\rightarrow{\mathcal{W}}(B)$
and
$\bar{f}_{B,B^{\prime}}:{\mathcal{W}}(B^{\prime})/{\mathcal{V}}(B^{\prime})\rightarrow{\mathcal{W}}(B)/{\mathcal{V}}(B)$
are surjective morphisms of pro-2-groups (thus they can be identified with
projections).

  2. ii -

Let $f^{\prime}:G^{\prime\prime}\rightarrow G$ is an injective $pSG$-morphism.
If $B^{\prime}$ is an well ordered basis of $G^{\prime}$ obtained by
successive extensions of an well ordered basis $B^{\prime}_{2}$ of $f\circ
f^{\prime}[G^{\prime\prime}]$ to $B^{\prime}_{1}$, an well ordered basis of
$f[G]\supseteq f\circ f^{\prime}[G^{\prime\prime}]$, then applying the
construction above described, we obtain
$\hat{f}^{0}_{B^{\prime\prime},B^{\prime}}=\hat{f}^{0}_{B^{\prime\prime},B}\circ\hat{f}^{0}_{B,B^{\prime}}:{\mathcal{W}}(B^{\prime})\rightarrow{\mathcal{W}}(B^{\prime\prime})$
and
$\bar{f}^{0}_{B^{\prime\prime},B^{\prime}}=\bar{f}^{0}_{B^{\prime\prime},B}\circ\bar{f}^{0}_{B,B^{\prime}}:{\mathcal{W}}(B^{\prime})/{\mathcal{V}}(B^{\prime})\rightarrow{\mathcal{W}}(B^{\prime\prime})/{\mathcal{V}}(B^{\prime\prime})$.

###### Proof.

  1. i -

By the injectivity hypothesis, we have
${f}^{0}_{B,B^{\prime}}[B^{\prime}]=B\cup\\{1\\}\subseteq{\mathcal{W}}(B)$,
thus
$\hat{f}^{0}_{B,B^{\prime}}:{\mathcal{W}}(B^{\prime})\rightarrow{\mathcal{W}}(B)$
is a continuous function with dense image from a compact space into a
Hausdorff space. Therefore $\hat{f}^{0}_{B,B^{\prime}}$ and
$\bar{f}^{0}_{B,B^{\prime}}$ are surjective continuous homomorphisms.

  2. ii -

It follows from a straightforward calculation that
${f}^{0}_{B^{\prime\prime},B^{\prime}}=\hat{f}^{0}_{B^{\prime\prime},B}\circ{f}^{0}_{B,B^{\prime}}$.
Therefore, the uniqueness of extensions and the homomorphism theorem
guarantees that
$\hat{f}^{0}_{B^{\prime\prime},B^{\prime}}=\hat{f}^{0}_{B^{\prime\prime},B}\circ\hat{f}^{0}_{B,B^{\prime}}$
and
$\bar{f}^{0}_{B^{\prime\prime},B^{\prime}}=\bar{f}^{0}_{B^{\prime\prime},B}\circ\bar{f}^{0}_{B,B^{\prime}}$.

∎

###  5.2 From Galois Groups to PSG

Let $G$ be a pre-special group an denote $\mathcal{G}=Gal(G)$. We have seen in
Proposition 4.2 that there is a canonical isomorphism
$\phi_{G}:{\mathcal{G}}/\Phi({\mathcal{G}})\overset{\cong}{\rightarrow}Hom(G,{\mathbb{Z}_{2}})$
so we get a “perfect pairing”
$\hat{\phi}_{G}:{\mathcal{G}}/\Phi({\mathcal{G}})\times
G\rightarrow{\mathbb{Z}_{2}}$ and there is also a canonical bijection
$G\cong\\{T\subseteq{\mathcal{G}}:T$ is a closed normal subgroup of index
$\leq 2\\}$.

We will explain now the term “canonical” employed, starting with the following

######  Lemma 5.4.

  1. i -

Let $G,G^{\prime}$ be pre-special groups Then each continuous homomorphism
$\theta:Gal(G^{\prime})\ \rightarrow\ Gal(G)$ induces a
$\mathbb{Z}_{2}$-module homomorphism $\check{\theta}:G\to G^{\prime}$.

  2. ii -

The association above, $\theta\mapsto\check{\theta}$, determines a
contravariant functor from the category from all pairs $(G,Gal(G))$, $G$ a
pre-special group, and continuous homomorphisms, into the category of
${\mathbb{Z}_{2}}$-modules.

###### Proof.

  1. i -

We have a ${\mathbb{Z}_{2}}$-homomorphism
$\theta^{*}:Homcont(Gal(G),{\mathbb{Z}_{2}})\to
Homcont(Gal(G^{\prime}),{\mathbb{Z}_{2}})$, $\mu\mapsto\mu\circ\theta$. By
Proposition 4.2(iii), we have ${\mathbb{Z}_{2}}$-isomorphisms
$\psi_{G},\psi_{G^{\prime}}$. Combining the information we define the
${\mathbb{Z}_{2}}$-homomorphism
$\check{\theta}:=\psi_{G^{\prime}}^{-1}\circ\theta^{*}\circ\psi_{G}:G\
\rightarrow\ G^{\prime}$.

  2. ii -

Note that $id_{\mathcal{G}}^{*}=id$, thus $\check{id}_{\mathcal{G}}=id_{G}$.
Let $\theta^{\prime}:Gal(G^{\prime\prime})\to Gal(G^{\prime})$ be a continuous
homomorphism. Then we have
$(\theta\circ\theta^{\prime})^{*}={\theta^{\prime}}^{*}\circ\theta^{*}$, thus
$(\theta\circ\theta^{\prime})\check{}=\check{\theta^{\prime}}\circ\check{\theta}$.

∎

######  Remark 5.5.

Note that for any surjective continuous homomorphism
$\theta:{\mathcal{G}}^{\prime}\rightarrow{\mathcal{G}}$, we have that
$\check{\theta}:G\to G^{\prime}$ is an injective
${\mathbb{Z}_{2}}$-homomorphism.

This suggest that the (sub)category of Galois groups and continuous
epimorphisms is the “right” domain category of Galois groups. We have the
following:

######  Proposition 5.6.

The functor described in the Lemma above restricts to a functor from the
subcategory of Galois groups of standard pre-special groups (Definition above
4.9) and continuous epimorphisms to the category of standard pre-special
groups and injective $qSG$-morphisms (i.e., the group homomorphisms that
preserves $\equiv$, but that eventually does not preserves -1).

###### Proof.

Assume that $G$ is a $k$-stable pre-special group and that $G^{\prime}$ is a
standard pre-special group, we will prove that $\check{\theta}$ is a injective
$qSG$-homomorphism from $G$ to $G^{\prime}$.

Since $\check{\theta}:G\to G^{\prime}$ is a group homomorphism, it is enough
to show that, for each $a,b\in G\setminus\\{1\\}$, $1\in D_{G}<a,b>\Rightarrow
1^{\prime}\in D_{G}<\check{\theta}(a),\check{\theta}(b)>$ and, since
$G,G^{\prime}$ are $k$-stable pre-special group, this is equivalent to show
$l(a)l(b)=0\in k_{2}(G)\Rightarrow
l(\check{\theta}(a))l(\check{\theta}(b))=0\in k_{2}(G^{\prime})$. Now we have
2 cases to consider:

$a=b$ : Since $G$ is a $k$-stable pre-special group then, by Theorem 4.8(i),
there exists $S\subseteq{\mathcal{G}}$ a closed normal subgroup with
${\mathcal{G}}/S\cong{\mathbb{Z}_{4}}$ and $S\subseteq M_{a}$. As
$\theta:{\mathcal{G}}^{\prime}\twoheadrightarrow{\mathcal{G}}$ is an
surjective continuous homomorphism we have that the quotient map
$\theta_{S}:{\mathcal{G}}^{\prime}/\theta^{-1}[S]\rightarrow{\mathcal{G}}/S$
is an isomorphism so $\theta^{-1}[S]\subseteq{\mathcal{G}}^{\prime}$ is a
closed normal subgroup such that
${\mathcal{G}}^{\prime}/\theta^{-1}[S]\cong{\mathcal{G}}/S\cong{\mathbb{Z}_{4}}$
and
$\theta^{-1}[S]\subseteq\theta^{-1}[M_{a}]=M^{\prime}_{\theta^{\star}(a)}$,
where this last equality holds by the bijections in items (ii) and (iii) in
Proposition 4.2 and the definition of $\check{\theta}$. Now, since
$G^{\prime}$ is a standard pre-special group, we have
$l(\check{\theta}(a))l(\check{\theta}(a))=0\in k_{2}(G^{\prime})$

$a\neq b$ : As $\check{\theta}$ is an injective
${\mathbb{Z}_{2}}$-homomorphism, $\check{\theta}(a)\neq\check{\theta}(b)$.
Since $G$ is a $k$-stable pre-special group then, by Theorem 4.8(ii), there
exists $D\subseteq{\mathcal{G}}$ a closed normal subgroup with
${\mathcal{G}}/D\cong\mathbb{D}_{4}$, $D\subseteq M_{a}\cap M_{b}$ and
$M_{ab}/D\cong{\mathbb{Z}_{4}}$. As
${\theta}:{\mathcal{G}}^{\prime}\twoheadrightarrow{\mathcal{G}}$ is an
surjective continuous homomorphism we have that the quotient map
$\theta_{D}:{\mathcal{G}}^{\prime}/\theta^{-1}[D]\rightarrow{\mathcal{G}}/D$
is an isomorphism so $\theta^{-1}[D]\subseteq{\mathcal{G}}^{\prime}$ is a
closed normal subgroup such that
${\mathcal{G}}^{\prime}/\theta^{-1}[D]\cong{\mathcal{G}}/D\cong{\mathbb{D}_{4}}$
and $\theta^{-1}[D]\subseteq\theta^{-1}[M_{a}\cap
M_{b}]=M^{\prime}_{\check{\theta}(a)}\cap M^{\prime}_{\check{\theta}(b)}$. Now
we check that
$M^{\prime}_{\check{\theta}(a)\check{\theta}(b)}/\theta^{-1}[D]\cong{\mathbb{Z}_{4}}$:
as $\theta$ is an epimorphism $\theta[theta^{-1}[M_{ab}]]=M_{ab}$ so, as
$M^{\prime}_{\check{\theta}(a)\check{\theta}(b)}=M^{\prime}_{\check{\theta}(ab)}=\theta^{-1}[M_{ab}]$
(because $\check{\theta}(a)\neq\check{\theta}(b)$ and there are exactly three
maximal above $M^{\prime}_{\check{\theta}(a)}\cap
M^{\prime}_{\check{\theta}(b)}$), we have an epimorphism
$\theta_{|}:M^{\prime}_{\check{\theta}(ab)}\twoheadrightarrow M_{ab}$ thus
$ker(\theta_{|})=\theta^{-1}[D]\subseteq M^{\prime}_{\check{\theta}(ab)}$ and
the quotient map
$\theta_{|{D}}:M^{\prime}_{\check{\theta}(ab)}/\theta^{-1}[D]\rightarrow
M_{ab}/D$ is an isomorphism so
$M^{\prime}_{\check{\theta}(ab)}/\theta^{-1}[D]\cong
M_{ab}/D\cong{\mathbb{Z}_{4}}$. Now, since $G^{\prime}$ is a standard pre-
special group, we have $l(\check{\theta}(a))l(\check{\theta}(b))=0\in
k_{2}(G^{\prime})$. ∎

###  5.3 Towards a galoisian cohomology for $SG$-theory

Let $G$ be a standard pre-special group. Since $\mathcal{G}=\mbox{Gal}(G)$ is
a profinite group, the Galois Cohomology is available for this subclass of
pre-special groups. In particular, there is the graded cohomology ring
$H^{*}(G):=H^{*}(\mathcal{G},\\{\pm 1\\})$ where $\mathcal{G}$ act trivially
on $\mathbb{F}_{2}=\mathbb{Z}/2\mathbb{Z}$. Therefore, at least some parts of
Milnor’s scenario for containing 3 graded rings related to quadratic forms
theory of fields (with $char\neq 2$) is available for (standard) special
groups: $W_{*}(G),k_{*}(G),$ and $H^{*}(G)$.

The result above just provides the initial step to establish cohomological
methods in SG-theory.

######  Theorem 5.7.

As in the field case, consider $\mathbb{Z}_{2}\cong\\{\pm 1\\}$ as a discrete
$Gal(G)$-module endowed with the trivial action, i.e., $\sigma.a=a$, for all
$\sigma\in Gal(G)$ and $a\in\mathbb{Z}_{2}$. Then
$H_{*}(G):=H_{*}(Gal(G),\mathbb{Z}_{2})$, is a graded ring endowed with the
cup product. Moreover, there are canonical isomorphisms of pointed $2$-groups:
(i) $(\mathbb{Z}_{2},-1)\cong(H^{0}(G),(-1))$;
(ii) $(G,-1)\cong(H^{1}(G),(-1))$.

###### Proof.

We write $\mathcal{G}:=Gal(G)$. Just recall that:

$H^{0}(\mathcal{G},\mathbb{Z}_{2})=(\mathbb{Z}_{2})^{\mathcal{G}}=Fix(\mathbb{Z}_{2})={\mathbb{Z}_{2}}$,
since $\mathcal{G}$ is acting trivially on $\mathbb{Z}_{2}$.

For
$H^{1}(\mathcal{G},\mathbb{Z}_{2}):=CrossedHom(\mathcal{G},\mathbb{Z}_{2})/principalCrossedHom(\mathcal{G},\mathbb{Z}_{2})$,
since $\mathcal{G}$ is acting trivially on $\mathbb{Z}_{2}$, we get

$\displaystyle principalCrossedHom(\mathcal{G},\mathbb{Z}_{2})$
$\displaystyle:=\ \mbox{Im}(\overline{\partial}_{1})$
$\displaystyle:=\\{x:\mathcal{G}\rightarrow\mathbb{Z}_{2}:x=\overline{\partial}_{1}a\mbox{
for some }a\in\mathbb{F}_{2}\\}$
$\displaystyle=\\{x:\mathcal{G}\rightarrow\mathbb{Z}_{2}:\mbox{ there exist
}\mathbb{F}_{2}\in\mathbb{F}_{2}\mbox{ such that }x(\sigma)=\sigma a-a\mbox{
for all }\sigma\in\mathcal{G}\\}$
$\displaystyle=\\{x:\mathcal{G}\rightarrow\mathbb{Z}_{2}:\mbox{ there exist
}a\in\mathbb{F}_{2}\mbox{ such that }x(\sigma)=0\mbox{ for all
}\sigma\in\mathcal{G}\\}$ $\displaystyle=\\{0\\};$

and

$\displaystyle CrossedHom(\mathcal{G},\mathbb{Z}_{2})$ $\displaystyle:=\
\mbox{Ker}(\overline{\partial}_{2})$
$\displaystyle:=\\{x:\mathcal{G}\rightarrow\mathbb{Z}_{2}:x\mbox{ is
continuous and }x(\sigma\tau)=\sigma x(\tau)+x(\sigma)\mbox{ for all
}\sigma,\tau\in\mathcal{G}\\}$
$\displaystyle=\\{x:\mathcal{G}\rightarrow\mathbb{Z}_{2}:x\mbox{ is continuous
and }x(\sigma\tau)=x(\tau)+x(\sigma)\mbox{ for all
}\sigma,\tau\in\mathcal{G}\\}$
$\displaystyle={Homcont}(\mathcal{G},\mathbb{Z}_{2}).$

Therefore
$H^{1}(\mathcal{G},\mathbb{Z}_{2})=Homcont(\mathcal{G},\mathbb{Z}_{2})/\\{0\\}\cong