#  $\tilde{B_{0}}$-invariant of groups

Zeinab Araghi Rostami  ,  Mohsen Parvizi  and  Peyman Niroomand  Department of
Pure Mathematics
Ferdowsi University of Mashhad, Mashhad, Iran<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of Pure Mathematics
Ferdowsi University of Mashhad, Mashhad, Iran<EMAIL_ADDRESS>Department of
Pure Mathematics
Damghan University, Damghan, Iran<EMAIL_ADDRESS>Department of
Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran

###### Abstract.

The Bogomolov multiplier of a group $G$ introduced by Bogomolov in $1988$.
After that in $2012$, Moravec introduced an equivalent definition of the
Bogomolov multiplier. In this paper we generalized the Bogomolov multiplier
with respect to a variety of groups. Then we give some new results on this
topic.

###### Key words and phrases:

Baer invariant, Bogomolov multiplier, $\tilde{B_{0}}$-invariant, Outer
commutator variety, Schur multiplier

##  1\. Introduction

For a group $G$, by a result of Hopf in [7], if $G\cong{F}/{R}$ be a free
presentation of $G$, then the groups $\frac{F^{\prime}}{[R,F]}$ and
$\frac{F^{\prime}\cap R}{[R,F]}$ are independent of the choice of $F$, i.e.,
are invariants of $G$. Moreover, he proved that $\frac{F^{\prime}\cap
R}{[R,F]}$ is isomorphic to the Schur multiplier
$H^{2}(G,{\mathbb{Q}}/{\mathbb{Z}})$ of $G$. Baer in [2] generalized these
concepts to an arbitrary variety of groups. Later, Frohlich in [4],
generalized the idea of Baer for any two varieties $\mathcal{V}$ and
$\mathcal{W}$ when $\mathcal{W}\subseteq\mathcal{V}$ and in the third article
[11], the Baer invariant of $G$ was reviewed with respect to two varieties
$\mathcal{V}$ and $\mathcal{W}$ in general.
Bogomolov in [3] proved that unramified cohomology group
${H_{nr}^{2}}({\mathbb{C}(V)}^{G},{\mathbb{Q}}/{\mathbb{Z}})$ is canonically
isomorphic to $B_{0}(G)=\bigcap_{A\leq G}\ \ker{{res}_{A}^{G}}$, where $A$ is
an abelian subgroup of $G$, and
${{res}_{A}^{G}}:H^{2}(G,{\mathbb{Q}}/{\mathbb{Z}})\longrightarrow
H^{2}(A,{\mathbb{Q}}/{\mathbb{Z}})$ is the usual cohomological restriction
map. $B_{0}(G)$ is defined as the subgroup of the so called the Schur
multiplier $\mathcal{M}(G)=H^{2}(G,{\mathbb{Q}}/{\mathbb{Z}})$ of $G$.
Kunyavskii in [9] named $B_{0}(G)$ the _Bogomolov multiplier_ of $G$. Moravec
in [12] showed that in the class of finite groups, the group
$\tilde{B_{0}}(G)=\mathcal{M}(G)/\mathcal{M}_{0}(G)$ is noncanonically
isomorphic to $B_{0}(G)$, where the Schur multiplier $\mathcal{M}(G)$ is
isomorphic to $\ker(G\wedge G\rightarrow[G,G])$ given by $x\wedge
y\rightarrow[x,y]$ and $\mathcal{M}_{0}(G)$ is a subgroup of $\mathcal{M}(G)$
defined as $\mathcal{M}_{0}(G)=<x\wedge y\ |\ [x,y]=0,\ x,y\in G>$. Moravec in
[12] showed that if ${F}/{R}$ be a free presentation of the group $G$, then
$\tilde{B_{0}}(G)\cong\frac{F^{\prime}\cap R}{<K(F)\cap R>}$, and it is an
invariant of $G$.

##  2\. Notations and preliminaries

In the following we give some preliminaries which are needed in the next.
Let $x$ and $y$ be two elements of a group $G$. Then $[x,y]$, the commutator
of $x$ and $y$, and $x^{y}$ denote the elements $x^{-1}y^{-1}xy$ and
$y^{-1}xy$, respectively. The left normed commutators of higher weights are
defined inductively as $[x_{1},x_{2},...,x_{n}]=[[x_{1},...,x_{n-1}],x_{n}]$.
If $M$ and $N$ are two subgroups of a group $G$ then $[M,N]$ denotes the
subgroup of $G$ generated by all the commutators $[m,n]$ with $m\in M$ and
$n\in N$. In particular, if $M=N=G$ then $[G,G]$, which is denoted by
$G^{\prime}$, is the derived subgroup of $G$. The lower and upper central
series are denoted by ${\gamma}_{c}(G)$, where $c\geqslant 1$. and $Z_{d}(G)$
where $d\geqslant 0$, respectively. Let $F_{\infty}$ be the free group freely
generated by an infinite countable set $\\{x_{1},x_{2},...\\}$. If
$u=u(x_{1},...,x_{s})$ and $v=v(x_{1},...,x_{t})$ are two words in
$F_{\infty}$, then the composite of $u$ and $v$, $u{\circ}v$ is defined as

$u{\circ}v=u(v(x_{1},...,x_{t}),...,v(x_{(s-1)t+1},...,x_{st})).$

In particular, the composite of some nilpotent words is called a polynilpotent
word, i.e.,
${\gamma}_{{c_{1}+1},...,{c_{t}+1}}={\gamma}_{c_{1}+1}{\circ}{\gamma}_{c_{2}+1}{\circ}...{\circ}{\gamma}_{c_{t}+1}$,
where ${\gamma}_{c_{i}+1}(1\leq i\leq t)$ is a nilpotent word in distinct
variables.
Outer commutator words are defined inductively, as follows. The word $x_{i}$
is an outer commutator word (or o.c word) of weight one. If
$u=u(x_{1},...,x_{s})$ and $v=v(x_{s+1},...,x_{s+t})$ are o.c words of weights
$s$ and $t$ respectively, then
$w(x_{1},...,x_{s+t})=[u(x_{1},...,x_{s}),v(x_{s+1},...,x_{s+t})]$ is an o.c
word of weight $s+t$.
Let $V$ be a subset of $F_{\infty}$ and $\mathcal{V}$ be the variety of groups
defined by the set of laws $V$, P. Hall in [5] introduced the notion of the
verbal subgroup $V(G)$ and the marginal subgroup $V^{*}(G)$, associated with
the variety $\mathcal{V}$ and the group $G$ (see [10, 13], for more
information).

######  Proposition 2.1.

If $N\unlhd G$ and $\mathcal{V},\mathcal{W}$ are varieties such that
$\mathcal{V}\subseteq[\mathcal{W},\mathcal{F}]$, where $\mathcal{F}$ be the
variety whose groups are all trivial, then

  1. (i)

$N\cap V(G)\supseteq[NV^{*}G]\supseteq[N,W(G)],$

  2. (ii)

$[V^{*}(G),W(G)]=1,$

where
$[NV^{*}G]=<v(g_{1},...,g_{i}{n},...,g_{s})v(g_{1},...,g_{s})^{-1}\ |\ v\in
V,g_{i}\in G,n\in N,1\leq i\leq s>$.

###### Proof.

See P. Hall [5]. ∎

Let $G$ be a group with a free presentation $G\cong{F}/{R}$, then the Baer
invariant of $G$, with respect to the variety $\mathcal{V}$, is defined to be

$\mathcal{V}\mathcal{M}(G)=\frac{R\cap V(F)}{[RV^{*}F]}.$

Leedham-Green and Mckay in [10] showed that the Baer invariant of the group
$G$ is always abelian and independent of the choice of a free presentation of
$G$. In particular, if $\mathcal{V}$ is the variety of abelian or nilpotent
groups of class at most $c$ $(c\geq 2)$, then the Baer invariant of $G$ is
isomorphic to the Schur multiplier of $G$ or
$\frac{R\cap{\gamma}_{c+1}(F)}{[R,cF]}$ where
$([R,cF]=[R,\underbrace{F,...,F}_{c-copies}])$, respectively.

In this paper, we are going to generalized the concept of Bogomolov multiplier
with respect to an arbitrary variety of groups.

######  Lemma 2.2.

(Hulse and Lennox 1976) Let $G$ be a group and $u$ and $w$ be two words and
$v=[u,w]$ and $K$ is a normal subgroup of a group $G$, then

$[Kv^{*}G]=[[Ku^{*}G],w(G)][u(G),[Kw^{*}G]].$

###### Proof.

See [6, Lemma2.6]. ∎

######  Proposition 2.3.

Let $N\unlhd G$, and $V,W$ be sets of o.c words, and $\mathcal{V},\mathcal{W}$
are the corresponding varieties such that
$\mathcal{V}\subseteq[\mathcal{W},\mathcal{F}]$, where $\mathcal{F}$ be the
variety of trivial group. Then

$[N,W(G)]\subseteq[NV^{*}G]\subseteq<T(G)\cap N>\subseteq N\cap V(G),$

where $T(G)=\\{v(g_{1},...,g_{s})\ |\ v\in V,g_{i}\in G,1\leq i\leq s\\}$.

###### Proof.

The result follows by using Proposition 2.1 and Lemma 2.2. ∎

##  3\. $\tilde{B_{0}}$-invariant

In this section, we want to introduce another invariant of $G$ which is
closely related to Baer invariant.

######  Definition 3.1.

Let $G$ be any group with a free presentation $G\cong{F}/{R}$ and $V$ be a set
of o. c. words. Then the $\tilde{B_{0}}$-invariant of $G$,
$\mathcal{V}\tilde{B_{0}}(G)$ with respect to the variety $\mathcal{V}$ is
defined by

$\mathcal{V}\tilde{B_{0}}(G)=\frac{R\cap V(F)}{<T(F)\cap R>},$

where $T(F)=\\{v(f_{1},...,f_{s})\ |\ v\in V,f_{i}\in F,1\leq i\leq s\\}$.

Since $[V(G),G]\subseteq V(G)$. Therefore
$\mathcal{V}\subseteq[\mathcal{V},\mathcal{F}]$. Thus by using Proposition
2.3, we have

$[R\cap V(F),R\cap V(F)]\subseteq[R,V(F)]\subseteq[RV^{*}F]\subseteq<T(F)\cap
R>.$

Hence the group $\mathcal{V}\tilde{B_{0}}(G)$ is abelian. Also we can write

$\mathcal{V}\tilde{B_{0}}(G)=\frac{R\cap V(F)}{<T(F)\cap R>}=\frac{(R\cap
V(F))/[RV^{*}F]}{<T(F)\cap R>/[RV^{*}F]}.$

We know $\dfrac{(R\cap V(F))}{[RV^{*}F]}$ is the Baer invarint. Now, if we
denote $\dfrac{<T(F)\cap R>}{[RV^{*}F]}$ with
$\mathcal{V}{\mathcal{M}}_{0}(G)$, then we have

$\mathcal{V}\tilde{B_{0}}(G)=\frac{\mathcal{V}\mathcal{M}(G)}{\mathcal{V}{\mathcal{M}}_{0}(G)}.$

The next lemma shows this notion is independent of the chosen free
presentation of $G$.

######  Lemma 3.2.

Let $\mathcal{V}$ be a variety of groups, $G,\bar{G}$ be two groups with free
presentations
$1\xrightarrow{}R\xrightarrow{}F\xrightarrow{\pi}G\xrightarrow{}1$,
$1\xrightarrow{}\bar{R}\xrightarrow{}\bar{F}\xrightarrow{\bar{\pi}}\bar{G}\xrightarrow{}1$,
respectively, and $\alpha:G\rightarrow\bar{G}$ be a homomorphism. Then

  1. (i)

there exists a homomorphism $\beta:F\rightarrow\bar{F}$ such that
${\alpha}{\pi}={\bar{\pi}}{\beta}$,

  2. (ii)

there exists a homomorphism
${\alpha}^{*}:\mathcal{V}\tilde{B_{0}}(G)\rightarrow\mathcal{V}\tilde{B_{0}}(\bar{G})$
which only dependst on $\alpha$,

  3. (iii)

if $1\rightarrow\hat{R}\rightarrow\hat{F}\rightarrow\hat{G}\rightarrow 1$ be a
free presentation for $\hat{G}$ and $\gamma:\bar{G}\rightarrow\hat{G}$ be a
homomorphism, then $({\gamma}{\alpha})^{*}={\gamma}^{*}{\alpha}^{*}$,

  4. (iv)

$\mathcal{V}\tilde{B_{0}}(G)$ is independent of the presentation of $G$.

###### Proof.

$(i)$ Since $\alpha:G\rightarrow\bar{G}$ is a homomorphism, there exists a not
necessarily unique homomorphism $\beta:F\rightarrow\bar{F}$ which is induced
by $\alpha$ and ${\alpha}{\pi}={\bar{\pi}}{\beta}$.
$(ii)$ $\beta$ induces a homomorphism ${\alpha}^{*}:\frac{R\cap
V(F)}{<T(F)\cap R>}\rightarrow\frac{\bar{R}\cap
V(\bar{F})}{<T(\bar{F})\cap\bar{R}>}$ given by
$x<T(F)\cap R>\mapsto\beta(x)<T(\bar{F})\cap\bar{R}>$, where $x\in R\cap
V(F)$. We claim that ${\alpha}^{*}$ is well-defined. Suppose that
${x_{1}}<T(F)\cap R>={x_{2}}<T(F)\cap R>$, where ${x_{1}},{x_{2}}\in R\cap
V(F)$. We have ${x_{1}}{x_{2}}^{-1}\in<T(F)\cap R>$ and so ${x_{1}}={x_{2}}t$,
for some $t\in<T(F)\cap R>$.

$\displaystyle{{\alpha}^{*}}({x_{1}}<T(F)\cap
R>)={\beta}(x_{1})<T(\bar{F})\cap\bar{R}>={\beta}({x_{2}}t)<T(\bar{F})\cap\bar{R}>$
$\displaystyle={\beta}(x_{2}){\beta}(t)<T(\bar{F})\cap\bar{R}>.$

It is enough to show that for $t\in<T(F)\cap R>$,
${\beta}(t)\in<T(\bar{F})\cap\bar{R}>$.
Since $t\in<R\cap T(F)>$, there exist ${x_{i}},{y_{i}}\in F$ such that
$t={\prod}_{i=1}^{n}[x_{i},y_{i}]^{{\alpha}_{i}}$ and $[x_{i},y_{i}]\in R\cap
T(F)$. So
${\beta}(t)=\prod_{i=1}^{n}[{\beta}(x_{i}),{\beta}(y_{i})]^{{\alpha}_{i}}$.
Also, we have

$\bar{\pi}({\beta}([x_{i},y_{i}]))={\alpha}{\pi}([x_{i},y_{i}])={\alpha}(1)=1\
\ \ ,\ \ \ {\beta}([x_{i},y_{i}])=[{\beta}(x_{i}),{\beta}(y_{i})]\in
T(\bar{F}).$

Therefore ${\beta}(t)\in<T(\bar{F})\cap\bar{R}>$ and ${\alpha}^{*}$ is well-
defined.
Let ${\beta}^{\prime}$ be another homomorphism induced by $\alpha$ such that,
${\bar{\pi}}{\beta}={\alpha}{\pi}={\bar{\pi}}{\beta}^{\prime}$. Then for any
$f\in F$, $\beta(f)\equiv{\beta}^{\prime}(f)$ $\pmod{\bar{R}}$, and so if
$v\in V(x_{i})$, then $v(\beta(f_{1}),...,\beta(f_{s}))\equiv
v({\beta}^{\prime}(f_{1}),...,{\beta}^{\prime}(f_{s}))$
$\pmod{<T(\bar{F})\cap\bar{R}>}$, for all $f_{1},...,f_{s}\in F$.
Thus, for all $x\in R\cap V(F)$, we have

$\displaystyle{\alpha}^{*}(x<T(F)\cap R>)={\beta}(x)<T(\bar{F})\cap\bar{R}>$
$\displaystyle={\beta}(v(f_{1},...,f_{s}))<T(\bar{F})\cap\bar{R}>=v({\beta}(f_{1}),...,{\beta}(f_{s}))<T(\bar{F})\cap\bar{R}>$
$\displaystyle=v({\beta}^{\prime}(f_{1}),...,{\beta}^{\prime}(f_{s}))<T(\bar{F})\cap\bar{R}>={\beta}^{\prime}(v(f_{1},...,f_{s}))<T(\bar{F})\cap\bar{R}>$
$\displaystyle={\beta}^{\prime}(x)<T(\bar{F})\cap\bar{R}>={({\alpha}^{\prime})}^{*}(x<T(\bar{F})\cap\bar{R}>).$

Hence ${\alpha}^{*}=({\alpha}^{\prime})^{*}$.
$(iii)$ By using $(i)$ and $(ii)$, there exist a homomorphisms
$\bar{\beta}:\bar{F}\to\hat{F}$ and
${\gamma}^{*}:\mathcal{V}\tilde{B_{0}}(\bar{G})\rightarrow\mathcal{V}\tilde{B_{0}}(\hat{G})$.
By using ${\gamma}{\alpha}:G\to\hat{G}$, $(i)$ and $(ii)$, there exist two
homomorphisms $\hat{\beta}=\bar{\beta}{\beta}:F\to\hat{F}$, respectively.
$({\gamma}{\alpha})^{*}:\mathcal{V}\tilde{B_{0}}({G})\rightarrow\mathcal{V}\tilde{B_{0}}(\hat{G})$
and $(x<T(F)\cap R>)\mapsto(\hat{\beta}(x)<T(\hat{F})\cap\hat{R}>)$, such that
$({\gamma}{\alpha})^{*}={\gamma}^{*}{\alpha}^{*}$.
$(iv)$ Let $G\cong{F}/{R}$ and $G\cong{\bar{F}}/{\bar{R}}$ be two
presentations for $G$. Then we have

$\textstyle{1\longrightarrow
R\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{\beta}$$\textstyle{{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\longrightarrow
1}$$\scriptstyle{1_{G}}$$\textstyle{1\longrightarrow\bar{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bar{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\pi}}$$\scriptstyle{\delta}$$\textstyle{{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\longrightarrow
1}$$\scriptstyle{1_{G}}$$\textstyle{1\longrightarrow{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\pi}}$$\textstyle{{G}\longrightarrow
1.}$

By using $(i)$, there exist homomorphisms $\beta$ and $\delta$, such that
$\pi={\bar{\pi}}{\beta}$ and $\bar{\pi}={\pi}{\delta}$. Now using $(ii)$,
there exist the homomorphisms ${\alpha}^{*}:\frac{R\cap V(F)}{<T(F)\cap
R>}\rightarrow\frac{\bar{R}\cap V(\bar{F})}{<T(\bar{F})\cap\bar{R}>}$ and
${\gamma}^{*}:\frac{\bar{R}\cap
V(\bar{F})}{<T(\bar{F})\cap\bar{R}>}\rightarrow\frac{R\cap V(F)}{<T(F)\cap
R>}$ which are independent of $\beta$ and $\delta$, respectively. Since the
above diagram is commutative and ${\delta}{\beta}:F\rightarrow F$,
$1_{G}:G\rightarrow G$ are two homomorphisms, by using $(ii)$, there exist the
homomorphism $({\gamma}{\alpha})^{*}:\frac{R\cap V(F)}{<T(F)\cap
R>}\rightarrow\frac{R\cap V(F)}{<T(F)\cap R>}$, given by $f<T(F)\cap
R>\mapsto{\delta}{\beta}(f)<T(F)\cap R>$, which is independent of
${\delta}{\beta}$. Also, we can take ${\delta}{\beta}=1_{F}:F\rightarrow F$,
so, $({\gamma}{\alpha})^{*}=1_{\mathcal{V}\tilde{B_{0}}(G)}$. By using
$(iii)$, $1=({\gamma}{\alpha})^{*}={\gamma}^{*}{\alpha}^{*}$. Similarly, we
can see that ${{\alpha}^{*}}{{\gamma}^{*}}=1$. Therefore ${\alpha}^{*}$ and
${\gamma}^{*}$ are isomorphism. ∎

As a special case, if $\mathcal{A}$ is the variety of abelian groups, then
$A(F)=F^{\prime}$ and $T(F)=\\{[x,y]\ |\ x,y\in F\\}$. So, in the finite case,
the $\tilde{B_{0}}$-invariant of $G$ is exactly the Bogomolov multiplier

$\tilde{B_{0}}(G)\cong\frac{R\cap F^{\prime}}{<T(F)\cap R>}$

If $\mathcal{N}_{c}$ is the variety of nilpotent groups of class at most
$c\geq 1$, then $N_{c}(F)={\gamma}_{c+1}(F)$ and $T(F)=\\{[x_{1},...,x_{c+1}]\
|\ x_{i}\in F,1\leq i\leq c+1\\}$. In this case, this multiplier is called the
$c$-nilpotent $\tilde{B_{0}}$-multiplier of $G$, and defined as follows

$\mathcal{N}_{c}\tilde{B_{0}}(G)={\tilde{B_{0}}}^{(c)}(G)=\frac{R\cap{\gamma}_{c+1}(F)}{<T(F)\cap
R>}.$

We need some elementrary results about $\tilde{B_{0}}$-invariant. In the rest,
$\mathcal{V}$ is an outer commutator variety and $T(F)=\\{v(f_{1},...,f_{s})\
|\ v\in V,f_{i}\in F,1\leq i\leq s\\}$.

######  Proposition 3.3.

Let $G$ be a group and $N$ be a normal subgroup of $G$. Then the following
sequence is exact.

$\mathcal{V}\tilde{B_{0}}(G)\longrightarrow\mathcal{V}{\tilde{B_{0}}}(\frac{G}{N})\longrightarrow\frac{N}{<T(G)\cap
N>}\longrightarrow\frac{G}{V(G)}\longrightarrow\frac{G/N}{V(G/N)}\longrightarrow
1$

###### Proof.

Let $1\xrightarrow{}R\xrightarrow{}F\xrightarrow{\pi}G\xrightarrow{}1$ be a
free presentation of $G$ and let
$K=\ker(F\to G/N)$. We have $N\cong K/R$. The inclusion maps
$R\cap{V(F)}\xrightarrow{f}K\cap{V(F)},\ \ $ $K\cap{V(F)}\xrightarrow{g}K,\ \
$ $K\xrightarrow{h}F$ and $F\xrightarrow{l}F$ induce the sequence of
homomorphism

$\textstyle{\frac{R\cap{V(F)}}{<T(F)\cap
R>}\xrightarrow{f^{*}}\frac{K\cap{V(F)}}{<T(F)\cap
K>}\xrightarrow{g^{*}}\frac{K}{<T(F)\cap
K>R}\xrightarrow{h^{*}}\frac{F}{R{V(F)}}\xrightarrow{l^{*}}{\frac{F}{K{V(F)}}}\rightarrow
1.}$

We can see that

$\displaystyle{\frac{K}{<T(F)\cap K>R}\cong\frac{N}{<T(G)\cap N>}}\ \ ,\ \
\displaystyle{\frac{F}{R{V(F)}}\cong\frac{G}{V(G)}}\ \ ,\ \
\displaystyle{\frac{F}{K{V(F)}}\cong\frac{{G}/{N}}{{V(G/N)}}}.$

Now we have

$\displaystyle{\mathcal{V}{\tilde{B_{0}}}(G)\cong\frac{R\cap{V(F)}}{<T(F)\cap
R>}}\ \ \ ,\ \ \
\displaystyle{\mathcal{V}{\tilde{B_{0}}}(\dfrac{G}{N})\cong\frac{K\cap{V(F)}}{<T(F)\cap
K>}}.$

Moreover,

$\displaystyle{\text{Im}{f^{*}}=\text{Ker}{g^{*}}=}\ \ ,\ \
\displaystyle{\frac{R\cap{V(F)}}{<T(F)\cap K>}},\ \ ,\ \
\displaystyle{\text{Im}{g^{*}}=\text{Ker}{h^{*}}=\frac{K\cap{V(F)}}{<T(F)\cap
K>R}},$ $\displaystyle{\text{Im}{h^{*}}=\text{Ker}{l^{*}}=\frac{K}{R{V(F)}}},$

and $l^{*}$ is epimorphism. Therefore the above sequence is exact. ∎

######  Proposition 3.4.

Let $G$ be a group with a free presentation $G\cong{F}/{R}$, and $N$ be a
normal subgroup of $G$, such that $K=\text{Ker}(F\to{G}/{N})$. Then the
sequence

$1\to\dfrac{R\ \cap<T(F)\cap K>}{<T(F)\cap
R>}\to\mathcal{V}{\tilde{B_{0}}}(G)\to\mathcal{V}{\tilde{B_{0}}}(\dfrac{G}{N})\to\dfrac{N\cap
V(G)}{<T(G)\cap N>}\to 1,$

is exact.

###### Proof.

Suppose $1\xrightarrow{}R\xrightarrow{}F\xrightarrow{\pi}G\xrightarrow{}1$ be
a free presentation of $G$ and let
$K=\text{Ker}(F\to G/N)$. We have $N\cong K/R$. The inclusion maps
$R\ \cap<T(F)\ \cap\ K>\xrightarrow{f}R\ \cap{V(F)},\ \ $ $R\
\cap{V(F)}\xrightarrow{g}K\cap{V(F)}$ and the map
$K\cap{V(F)}\xrightarrow{h}(K\cap{V(F)})R$ induce the sequence of homomorphism
$\displaystyle{1\to\frac{R\ \cap<T(F)\cap K>}{<T(F)\cap
R>}\xrightarrow{f^{*}}\frac{R\cap{V(F)}}{<T(F)\cap
R>}\xrightarrow{g^{*}}\frac{K\cap{V(F)}}{<T(F)\cap K>}\xrightarrow{h^{*}}}$
$\displaystyle{\frac{(K\cap{V(F)})R}{<T(F)\cap K>R}\to 1}$. It is
straightforward to verify that
$\displaystyle{<T(G)\cap N>=\frac{<T(F)\cap K>R}{R}}$ and
$\displaystyle{N\cap{V(G)}=\frac{K}{R}\cap\frac{{V(F)}R}{R}=\frac{(K\cap{V(F)})R}{R}}$.
Therefore we have

$\displaystyle{\frac{N\cap{V(G)}}{<T(G)\cap N>}=\frac{({(K\cap
V(F))R})/{R}}{({<T(F)\cap K>R})/{R}}\cong\frac{(K\cap{V(F)})R}{<T(F)\cap
K>R}}.$

Finally we have

$\displaystyle{\mathcal{V}{\tilde{B_{0}}}(G)\cong\frac{R\cap{V(F)}}{<T(F)\cap
R>}}\ \ \ ,\ \ \
\displaystyle{\mathcal{V}{\tilde{B_{0}}}(\dfrac{G}{N})\cong\frac{K\cap{V(F)}}{<T(F)\cap
K>}},$

and

$\displaystyle{\text{Im}{f^{*}}=\text{Ker}{g^{*}}=\frac{R\ \cap<T(F)\cap
K>}{<T(F)\cap R>}},\ \ \ ,\ \ \
\displaystyle{\text{Im}{g^{*}}=\text{Ker}{h^{*}}=\frac{R\cap{V(F)}}{<T(F)\cap
K>}}.$

Moreover $h^{*}$ is epimorphism. Therefore the above sequence is exact. ∎

######  Lemma 3.5.

Let $G$ be a group with a normal subgroup $N$. Then there exists a group $H$
with a normal subgroup $B$ such that

  1. (i)

$V(G)\cap N\cong{H}/{B}$,

  2. (ii)

$B\cong\mathcal{V}\tilde{B_{0}}(G)$,

  3. (iii)

$\mathcal{V}\tilde{B_{0}}(\dfrac{G}{N})$ is an epimorphic image of $H$.

###### Proof.

Let $G\cong F/R$ be a free presentation of $G$ and suppose $N\cong K/R$. So
$G/N\cong F/K$. Let

$H=\dfrac{V(F)\cap K}{<T(F)\cap R>}\ \ \ \ ,\ \ \ \ B=\dfrac{V(F)\cap
R}{<T(F)\cap R>}.$

We have

$\displaystyle V(G)\cap N$
$\displaystyle\cong\dfrac{V(F)R}{R}\cap\dfrac{K}{R}=\dfrac{V(F)R\cap
K}{R}=\dfrac{(V(F)\cap K)R}{R}$ $\displaystyle\cong\dfrac{V(F)\cap K}{V(F)\cap
R}\cong\dfrac{(V(F)\cap K)/<T(F)\cap R>}{(V(F)\cap R)/<T(F)\cap R>}$
$\displaystyle=\dfrac{H}{B}.$

Also,

$\mathcal{V}{\tilde{B_{0}}}(\frac{G}{N})\cong\dfrac{V(F)\cap K}{<T(F)\cap
K>}\cong\dfrac{(V(F)\cap K)/<T(F)\cap R>}{<T(F)\cap K>/<T(F)\cap R>}.$

Hence, $(i)$, $(ii)$ and $(iii)$ are obtained. ∎

Now we obtain an explicit formula for the $\tilde{B_{0}}$-invariant of a
direct product of two groups.

It is easy to see that if $G_{1}$ and $G_{2}$ are groups with free
presentations ${F_{1}}/{R_{1}}$ and ${F_{2}}/{R_{2}}$, respectively. Let
$F=F_{1}\ast F_{2}$ be the free product of $F_{1}$ and $F_{2}$. Then $1\to
R\to F\to G_{1}\times G_{2}\to 1$ is a free presentation for $G_{1}\times
G_{2}$, where $R={R_{1}}{R_{2}}[F_{2},F_{1}]$.

######  Proposition 3.6.

Let $\mathcal{V}$ be a variety of groups defined by the set of laws $V$ and
let $G_{1}$ and $G_{2}$ be groups. Then

$\mathcal{V}{\tilde{B_{0}}}(G_{1}\times
G_{2})\cong\mathcal{V}{\tilde{B_{0}}}(G_{1})\times\mathcal{V}{\tilde{B_{0}}}(G_{2}).$

###### Proof.

By using Definition 3.1, we have

$\mathcal{V}{\tilde{B_{0}}}(G_{1}\times
G_{2})=\dfrac{({R_{1}}{R_{2}}[F_{2},F_{1}])\cap V(F_{1}\ast
F_{2})}{<T(F_{1}\ast F_{2})\cap({R_{1}}{R_{2}}[F_{2},F_{1}])>}.$

where $V$ is the set of o.c words of variety $\mathcal{V}$ and
$T(F)=\\{v(f_{1},...,f_{s})\ |\ v\in V,f_{i}\in F,1\leq i\leq s\\}$.
Let $F=F_{1}\ast F_{2}$, then the epimorphism $F\to F_{1}\times F_{2}$ induces
the following epimorphism

$\alpha:\dfrac{R\cap V(F)}{<T(F)\cap R>}\to\dfrac{R_{1}\cap
V(F_{1})}{<T(F_{1})\cap R_{1}>}\times\dfrac{R_{2}\cap
V({F_{2}})}{<T(F_{2})\cap R_{2}>}$ $x<T(F)\cap
R>\longmapsto({x_{1}}<T(F_{1})\cap R_{1}>\ ,\ {x_{2}}<T(F_{2})\cap R_{2}>)$

where $x={x_{1}}{x_{2}}$, $x_{1}\in R_{1}\cap V({F_{1}})$ and $x_{2}\in
R_{2}\cap V({F_{2}})$.
On the other hand

$\beta:\dfrac{R_{1}\cap V({F_{1}})}{<T(F_{1})\cap
R_{1}>}\times\dfrac{R_{2}\cap V({F_{2}})}{<T(F_{2})\cap R_{2}>}\to\dfrac{R\cap
V(F)}{<T(F)\cap R>}$

given by

$({x_{1}}<T(F_{1})\cap R_{1}>\ ,\ {x_{2}}<T(F_{2})\cap R_{2}>)\longmapsto
x<T(F)\cap R>$

is a well-defined homomorphism. It is easy to check that $\beta$ is a right
inverse to $\alpha$, so $\alpha$ is an epimorphism. Now, we show that $\alpha$
is a monomorphism. Let $x<T(F)\cap R>\ \in\ker\alpha$, such that,
$x={t_{1}}{t_{2}}$. So we have $t_{1}\in<T(F_{1})\cap R_{1}>$ and
$t_{2}\in<T(F_{2})\cap R_{2}>$. Since $t_{1},t_{2}\in<R\cap T(F)>$, then
$x\in<T(F)\cap R>$. Thus $\alpha$ is a monomorphism. ∎

##  4\. Verbal preserving extension

In this section we introduce a large class of commutativity preserving group
extensions.
As $\tilde{B_{0}}(G)$ plays an important role in describing commutativity
preserving extension (CP extension), $\mathcal{V}\tilde{B_{0}}(G)$ also plays
in a same role. It is called verbal preserving extensions (VP extension) of
groups.

######  Definition 4.1.

Let $Q$ be a group and $N$ be a $Q$-module. The extension
$1\xrightarrow{}N\xrightarrow{\chi}G\xrightarrow{\pi}Q\xrightarrow{}1$ of $N$
by $Q$ is a VP extension if trivial words of elements of $Q$ have trivial
lifts in $G$.

######  Definition 4.2.

[8] The extensions
$1\xrightarrow{}K\xrightarrow{i}G\xrightarrow{\pi}H\xrightarrow{}1$ and
$1\xrightarrow{}K\xrightarrow{i^{\prime}}G\xrightarrow{{\pi}^{\prime}}H\xrightarrow{}1$
are equivalent if there exists a group isomorphism $T:G\rightarrow G^{\prime}$
such that the following diagram is commutative.

$\textstyle{1\longrightarrow
N\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}$$\scriptstyle{\mu}$$\scriptstyle{1_{N}}$$\textstyle{{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{T}$$\textstyle{{Q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\longrightarrow
1}$$\scriptstyle{1_{Q}}$$\textstyle{1\longrightarrow{N}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mu}^{\prime}}$$\textstyle{{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\pi}^{\prime}}$$\textstyle{{Q}\longrightarrow
1.}$

######  Lemma 4.3.

The class of VP extension is closed under equivalence of extensions.

###### Proof.

Let

$\textstyle{1\longrightarrow
N\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\scriptstyle{1_{N}}$$\textstyle{{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{T}$$\textstyle{{Q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\longrightarrow
1}$$\scriptstyle{1_{Q}}$$\textstyle{1\longrightarrow
N\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mu}^{\prime}}$$\textstyle{{G^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\pi}^{\prime}}$$\textstyle{{Q}\longrightarrow
1}$

be equivalent extensions with a marginal kernel. Suppose that $G$ is a VP
extension of $N$ by $Q$. Choose $q_{1},...,q_{s}\in Q$ and $v\in V$, where
$v(q_{1},...,q_{s})=1$. Then there exists $g_{1},...,g_{s}\in G$ such that
$v(g_{1},...,g_{s})=1$ and ${{\pi}_{1}}(g_{i})=q_{i}$, where $i=1,...,s$. Take
${g^{\prime}_{i}}=T(g_{i})$. Then $v(g^{\prime}_{1},...,g^{\prime}_{s})=1$ and
${{\pi}_{2}}(g^{\prime}_{i})=q_{i}$. Thus $G^{\prime}$ is a VP extension of
$N$ by $Q$. ∎

Now we want to introduce a special version of VP extension with a marginal
kernel.

######  Definition 4.4.

The VP extension
$1\xrightarrow{}N\xrightarrow{\chi}G\xrightarrow{\pi}Q\xrightarrow{}1$ of $N$
by $Q$ is marginal, if $\chi(N)\subseteq V^{*}(G)$.

######  Proposition 4.5.

Let $1\xrightarrow{}N\xrightarrow{\chi}G\xrightarrow{\pi}Q\xrightarrow{}1$ be
a marginal extension. This sequence is a VP extension if and only if
$\chi(N)\cap T(G)=1$, where $T(G)=\\{v(g_{1},...,g_{s})\ |\ v\in V,g_{i}\in
G,i=1,...,s\\}$.

###### Proof.

Let $\chi(N)\cap T(G)=1$. Choose $q_{1},...,q_{s}\in Q$ and $v\in V$ with
$v(q_{1},...,q_{s})=1$. For some $g_{i}\in G$, we have $q_{i}=\pi(g_{i})$.
Then $\pi(v(g_{1},...,g_{s}))=1$, hence $v(g_{1},...,g_{s})\in\chi(N)\cap
T(G)$. Thus $v(g_{1},...,g_{s})=1$. Conversely, suppose that the above
sequence is a VP marginal extension. Choose $v(g_{1},...,g_{s})\in\chi(N)\cap
T(G)$. We know $\chi(N)=\ker{\pi}$, so $\pi(v(g_{1},...g_{s}))=1$ and
$v(\pi(g_{1}),...,\pi(g_{s}))=1$. By assumption, there exist
$g^{\prime}_{1},...,g^{\prime}_{s}\in G$ such that
$\pi(g_{i})=\pi(g^{\prime}_{i})$ and $v(g^{\prime}_{1},...,g^{\prime}_{s})=1$.
Since $\pi(g_{i})=\pi(g^{\prime}_{i})$,
${g_{i}}{g^{\prime}_{i}}^{-1}\in\ker{\pi}=\chi(N)$. So
$g_{i}=g^{\prime}_{i}{a_{i}}$, where $a_{i}\in\chi(N)$.
$1\xrightarrow{}N\xrightarrow{\chi}G\xrightarrow{\pi}Q\xrightarrow{}1$ be a
marginal extension. Thus we have
$v(g_{1},...,g_{s})=v(g^{\prime}_{1}a_{1},...,g^{\prime}_{s}a_{s})=v(g^{\prime}_{1},...,g^{\prime}_{s})=1$,
as required. ∎

######  Definition 4.6.

The normal subgroup $N$ of a group $G$ is a VP subgroup of $G$, if the
extension $1\xrightarrow{}N\xrightarrow{}G\xrightarrow{}G/N\xrightarrow{}1$ is
a VP extension.

Note that, when $N$ is a marginal subgroup of $G$, Proposition 4.5 shows that
$N$ is a VP subgroup if and only if $N\cap T(G)=1$.

######  Lemma 4.7.

Let $N$ be a marginal VP subgroup of $G$. Then the sequence

$1\xrightarrow{}\mathcal{V}\tilde{B_{0}}(G)\xrightarrow{}\mathcal{V}\tilde{B_{0}}(G/N)\xrightarrow{}N\cap
V(G)\xrightarrow{}1$

is exact.

###### Proof.

Let $G\cong F/R$ be a free presentation of $G$ and suppose
$K=\ker(F\rightarrow G/N)$. We have $N\cong K/R$, so $G/N\cong F/K$. Since $N$
is a marginal VP subgroup of $G$, $<T(F)\cap K>\leq R$. Thus, $<T(F)\cap
K>=<T(F)\cap R>$. Now the result is obtained by using Lemma 3.5. ∎

## References

  * [1] M. Artin and D. Mumford, _Some elementary examples of unirational varieties which are not rational_ , Proc. London. Math. Soc. (3) 25 (1972), 75–95.
  * [2] R. Baer, _Representations of groups as quotient groups_ , I-III, Trans. Amer. Math. Soc. 58 (1945), 295–419.
  * [3] F. A. Bogomolov, _The Brauer group of quotient spaces by linear group actions_ , Math. USSRIzv. 30 (1988), 455–485.
  * [4] A. Frohlich, _Baer-invariants of algebras_ , Trans. Amer. Math. See, 109 (1963), 221-244.
  * [5] P. Hall, ‘ _Nilpotent Groups_ , Canad. Math. Congress, Univ. of Alberta, 1957, Queen Mary College Math. Notes, Queen Mary College, London, 1970.
  * [6] M. Hall, J. K. Senior, _The Groups of Order $2^{n}(n\leq 6)$_, Macmillan, New York, 1964.
  * [7] H. Hopf, _Fundamental gruppe und zweite Bettische Gruppe_ , Comment. Math. Helv. 14 (1942), 257-309.
  * [8] U. Jezernik, P. Moravec, _Commutativity preserving extensions of groups_ , Proc. Roy. Soc. Edinburgh Sect. A 148 (2018), no. 3, 575-592.
  * [9] B. Kunyavskii, _The Bogomolov multiplier of finite simple groups_ , Cohomological and geometric approaches to rationality problems, 209-217, Progr. Math, 282, Birkhauser Boston, Inc, Boston, MA, 2010.
  * [10] C. R. Leedham-Green, S. Mckay, _Baer-invariants, Isologism, Varietal laws and Homology_ , Acta Math, 137, (1976), 99-150.
  * [11] M. R. Moghaddam, A. Salemkar, P. Niroomand, _Covering groups with respect to two varieties of groups_ , Southeast Asian Bull. Math-31, 2, (2007), 313-320.
  * [12] P. Moravec, _Unramified brauer groups of finite and infinite groups_ , Amer. J. Math. 134 (2012), 1679–1704.
  * [13] H. Neumann, _Varieties of Groups_ , Berlin-Heidelberg-New York: Springer. 1967.
  * [14] E. Noether, _Gleichungen mit vorgeschriebener Gruppe_ , Math. Ann. 78 (1916), 221–229.
  * [15] J. Reine Angew, _Verbal and marginal subgroups_ , 182 (1940), 156-157.
  * [16] J. Saltman, _Noether’s problem over an algebraically closed field_ , Invent. Math. 77 (1984), 71–84.
  * [17] R. G. Swan, _Noether’s problem in Galois theory_ , in ‘Emmy Noether in Bryn Mawr’, SpringerVerlag, Berlin, 1983.