# Simple Section Biset Functors

Olcay Coşkun
Department of Mathematics
Boğaziçi University
Bebek, 34342 Istanbul
Turkey
<EMAIL_ADDRESS>Both authors are supported by Tübitak-1001-119F422.
Ruslan Muslumov
Department of Mathematics
Boğaziçi University
Bebek, 34342 Istanbul
Turkey
<EMAIL_ADDRESS>
###### Abstract

We classify simple modules over the Green biset functor of section Burnside
rings.

##  1 Introduction

Green biset functors, as introduced by Serge Bouc [4], are rings in the
category of biset functors. The initial example is the Burnside functor. Its
module category is equivalent to the category of all biset functors. Other
examples of Green biset functors include the functor of fibered Burnside rings
[2], the functor of trivial source modules [7], the functor of characters [8]
and the functors of slice and section Burnside rings [5]. It is also possible
to produce new Green biset functors by applying the Yoneda-Dress construction,
as done in [4]. One of the main problems regarding Green biset functors is the
classification of their simple modules. The case of biset functors is solved
by Bouc in his fundamental paper [3]. Later simple functors over the functor
of rational characters is classified by Barker in [1], simple fibered biset
functors are described in [2], see also [8], and simple functors over the
trivial source ring in [7]. There are some partial results about the functor
of slice Burnside rings in [9]. In contrast to all these examples, there is no
general classification theorem or conjecture for simple modules over Green
biset functors.

The aim of this paper is to solve the classification problem for the functor
of section Burnside rings. Section Burnside rings are introduced by Bouc in
[5] as the Grothendieck ring of the subcategory of ‘Galois’ morphisms in the
category of morphisms of $G$-sets. It is shown in [5] that basic morphisms of
$G$-sets are given by the natural maps $G/S\to G/T$ associated to subgroups
$S\leqslant T\leqslant G$ and basic Galois morphisms are those associated to
subgroups $S\unlhd T\leqslant G$ (a pair of this from is called a section of
$G$). With this result, the section Burnside ring $\Gamma(G)$ is free as an
abelian group on the $G$-conjugacy classes of sections of $G$. It is also
shown in [5] that the functor $\Gamma$ associating $\Gamma(G)$ to the finite
group $G$ induces a Green biset functor. We summarize these results in Section
3.

In this paper we use techniques from [2] to analyze the structure of the Green
biset functor $\Gamma$ and classify its simple modules. For this aim, we
introduce a composition product for sections of direct products of groups.
More precisely, given finite groups $G,H$ and $K$ and sections $S\unlhd
T\leqslant G\times H$ and $U\unlhd V\leqslant H\times K$, we define

$\Big{(}\frac{G\times H}{S\unlhd T}\Big{)}\times_{H}\Big{(}\frac{H\times
K}{U\unlhd V}\Big{)}=\bigsqcup_{t\in p_{2}(S)\backslash
H/p_{1}(U)}\Big{(}\frac{G\times K}{S\ast{}^{(t,1)}U\unlhd
T\ast{}^{(t,1)}V}\Big{)}$

where we denote the projection morphism $(G\times H)/S\to(G\times H)/T$ by
$\big{(}\frac{G\times H}{S\unlhd T}\big{)}$ and the $\ast$-product is the
usual composition of relations. In Section 4 we show that the category of
modules over the Green biset functor $\Gamma$ is equivalent to the category of
functors from the category $\mathcal{P}_{k\Gamma}$ of finite groups in which
composition is given by the linear extension of the above composition product.
Here $k$ is a commutative ring of coefficients for $\Gamma$. We call a module
over $k\Gamma$ a section biset functor (over $k$).

By the general theory described in [4], evaluation of a (simple) section biset
functor at a finite group $G$ is (zero or) a (simple) module over the
endomorphism ring $E_{G}:=\Gamma(G,G)$ of $G$ in $\mathcal{P}_{\Gamma}$.
Moreover for a simple section biset functor $S$, if $G$ is a group of smallest
order with $S(G)\not=0$, then $S(G)$ is a simple module over the algebra
$\bar{E}_{G}$. Here $\bar{E}_{G}$ is the quotient of $E_{G}$ by the ideal
generated by all the morphisms that factor through a group of smaller order.

As also discussed in [8], the above observation is valid for a simple module
over an arbitrary Green biset functor $A$. Indeed if we define $\bar{A}_{G}$
as the quotient of $A(G,G)$ by the ideal generated by all morphisms in
$\mathcal{P}_{A}$ that factor through a group of smaller order, then there is
a simple $A$-module corresponding to any simple $\bar{A}_{G}$-module. To
complete the classification of simple $A$-modules, one needs a classification
of simple $\bar{A}_{G}$-modules for all $G$ and an equivalence relation on all
these simple modules to determine isomorphism types of simple $A$-modules
associated to them.

Following [2] we use certain idempotent elements in $E_{G}$ to prove that the
algebra $\bar{E}_{G}$ is isomorphic to a product of matrix algebras over
certain group algebras associated to these idempotents. In our case, the
idempotents of interest in $E_{G}$ are parameterized by pairs $(K,P)$ of
normal subgroups of $G$ that centralizes each other which are also reduced in
the sense of Section 6. Given such a pair $(K,P)$, the section
$(\Delta_{K}(G),\Delta(P))$ of $G\times G$ gives the idempotent element
$e_{(K,P)}$ in $E_{G}$. Here $\Delta_{K}(G)=\\{(g,h)\in G\times G|h^{-1}g\in
K\\}$ and $\Delta(P)$ is the diagonal inclusion of $P$ in $G\times G$. In
particular we associate simple $E_{G}$-modules to each reduced pair $(K,P)$.
As mentioned in Section 3, it is also possible to associate crossed modules to
pairs $(K,P)$ in a natural way and it turns out that two reduced pairs induce
isomorphic $E_{G}$-modules if and only if the corresponding crossed modules
are isomorphic. With this result, we classify simple modules over the algebra
$E_{G}$, see Section 6 for details. Finally, in Section 7, we show that the
relation defined on reduced pairs can be extended to an equivalence relation
on the set of quadruples $(G,K,P,[V])$ where $G$ is a finite group, $(K,P)$ is
a reduced pair for $G$ and $V$ is an irreducible $E_{G}$-module associated to
the pair $(K,P)$. Moreover there is a bijective correspondence between the
equivalence classes of these quadruples and the isomorphism classes of simple
section biset functors.

In order to complete the parametrization one needs to know the complete set of
reduced pairs for each finite group $G$. Although we do not know the complete
answer, we have some necessary and also some sufficient conditions to ensure
that a pair $(K,P)$ is reduced. For example $(K,P)$ is reduced if $K\leqslant
P$. Hence the pairs $(1,P)$ for any $P\unlhd G$ and the pairs $(K,G)$ for any
$K\leqslant Z(G)$ are reduced. In particular, the algebra $\bar{E}_{G}$ is
non-zero for any finite group $G$.

Our reason for attempting to classify simple modules over the Green biset
functor $k\Gamma$ is that it is different from the previous known examples in
the following sense. First the previous examples are known to have strong
connections with the theory of representations of finite groups and in the
classification, these connections are used in crucial ways. On the other hand
section Burnside rings are relatively new and these kind of deep connections
are not known yet. Therefore it is close to be an abstract example. Having a
successful application of the techniques from [2] signals a path towards a
more general theory of simple modules over Green biset functors.

A result that might be of general interest is the version of Goursat’s Theorem
for sections. The well-known Goursat’s Theorem for subgroups states that there
is a bijective correspondence between subgroups of a direct product $G\times
H$ and the quintuples $(P,K,\eta,L,Q)$ where $K\unlhd P\leqslant G$ and
$L\unlhd Q\leqslant H$ and $\eta:Q/L\to P/K$ a group isomorphism. Our version,
given as Theorem 3.6, determines exact conditions on the pairs of quintuples
coming from Goursat’s Theorem so that the induced map between sections of
$G\times H$ and the pairs satisfying these conditions is bijective.

##  2 Green biset functors

Let $k$ be a commutative ring with unity and let $\mathcal{C}_{k}$ denote the
biset category over $k$, that is, the category with objects consisting of
finite groups and morphisms from $G$ to $H$ are given by the double Burnside
group $kB(H,G)$. A $k$-linear functor $F:\mathcal{C}_{k}\to k$-mod is called a
_biset functor over $k$_. We denote the functor category of all biset functors
over $k$ by $\mathcal{F}_{k}$.

###### 2.1 Definition

Let $A$ be a biset functor over $k$. We call $A$ a _Green biset functor_ over
$k$ if there are bilinear maps $A(G)\times A(H)\to A(G\times H),(a,b)\mapsto
a\times b$ for each pair of finite groups $G$ and $H$ and an element
$\epsilon_{A}\in A(1)$ with the following properties.

  1. (i)

For any finite groups $G,H$ and $K$ and any $(a,b,c)\in A(G)\times A(H)\times
A(K)$, we have

$(a\times b)\times c=A(\mathrm{Iso}^{\alpha})(a\times(b\times c))$

where $\alpha$ is the canonical isomorphism from $G\times(H\times K)$ to
$(G\times H)\times K$.

  2. (ii)

For any finite group $G$ and any $a\in A(G)$, we have

$A(\mathrm{Iso}^{\lambda})(\epsilon_{A}\times
a)=a=A(\mathrm{Iso}^{\lambda^{\prime}})(a\times\epsilon_{A})$

where $\lambda:1\times G\to G$ and $\lambda^{\prime}:G\times 1\to G$ denote
the canonical isomorphisms.

  3. (iii)

Let $X\in kB(G^{\prime},G)$ and $Y\in kB(H^{\prime},H)$. For any $(a,b)\in
A(G)\times A(H)$, we have

$A(X\times Y)(a\times b)=A(X)(a)\times A(Y)(b).$

In other words, a Green biset functor is a monoid object in the category
$\mathcal{F}_{k}$ of all biset functors. The bilinear maps defining the Green
biset functor structure on $A$ gives a ring structure on $A(G)$ for any finite
group $G$ and these structures are compatible with the biset functor structure
on $A$ via the above properties. Conversely, if $A$ is any biset functor such
that $A(G)$ has a ring structure for each $G$ compatible with the biset
functor structure, one can make $A$ a Green biset functor. See [8, 3.2.7 and
4.2.3].

###### 2.2 Example

The Burnside biset functor is a Green biset functor and it is an initial
object in the category of all Green biset functors [4, 8.5.3] . Given a Green
biset functor $A$, the unique morphism $kB\to A$ of Green biset functors is
given by mapping a $G$-set $X$ to $A(X)(\epsilon_{A})$ in $A(G)$. Here we
regard $X$ as a $(G,1)$-biset. Other known examples of Green biset functors
include the functor of representation rings, the functor of fibered Burnside
rings and the functor of trivial source rings.

###### 2.3

The category $\mathcal{P}_{A}$. Let $A$ be a Green biset functor. We associate
a category $\mathcal{P}_{A}$ to $A$ as follows ([4, Section 8.6]). The objects
of $\mathcal{P}_{A}$ are all finite groups.

  1. (i)

Given finite groups $G$ and $H$, we put
$\mathrm{Hom}_{\mathcal{P}_{A}}(H,G)=A(G\times H)$.

  2. (ii)

Given finite groups $G,H,K$, and morphisms $\alpha\in A(H\times K)$ and
$\beta\in A(G\times H)$, the composition is defined by

$\beta\circ\alpha=A(\mathrm{Def}^{G\times\Delta(H)\times K}_{G\times
K}\mathrm{Res}^{G\times H\times H\times K}_{G\times\Delta(H)\times
K})(\beta\times\alpha).$

Here we identify $G\times K$ with $(G\times\Delta(H)\times
K)/(1\times\Delta(H)\times 1)$ via the obvious canonical isomorphism.

  3. (iii)

Given a finite group $G$, the identity morphism of $G$ in $\mathcal{P}_{A}$ is
$A(\mathrm{Ind}_{\Delta(G)}^{G\times
G}\mathrm{Inf}_{1}^{\Delta(G)})(\epsilon_{A})$.

Note that the unique morphism $kB\to A$ of Green biset functors induces a
functor $\mathcal{C}_{k}\to\mathcal{P}_{A}$. In particular we may consider the
basic operations associated to bisets as operations in $\mathcal{P}_{A}$.

###### 2.4

$A$-modules. With this definition, an _$A$ -module_ is a $k$-linear functor
$\mathcal{P}_{A}\to k$-mod. We denote the (abelian) functor category of all
$A$-modules by $A$-Mod. When $A=kB$, the category $kB$-Mod is the same as the
category $\mathcal{F}_{k}$ of all biset functors. Moreover for any Green biset
functor $A$, via the unique morphism $kB\to A$ of Green biset functors, any
$A$-module is a biset functor.

Let $A$ be a Green biset functor and $F$ be an $A$-module. For any finite
group $G$, the evaluation $F(G)$ is a module over the endomorphism ring
$E_{G}^{A}=\mathrm{End}_{\mathcal{P}_{A}}(G)$ of $G$ in $\mathcal{P}_{A}$.
Moreover if $G$ is a group of minimal order such that $F(G)\not=0$, then the
$E_{G}^{A}$-module $F(G)$ is annihilated by all the endomorphisms that factor
through a group of smaller order. We denote the ideal of $E_{G}^{A}$ generated
by such morphism by $I_{G}^{A}$ and the quotient $E_{G}^{A}/I_{G}^{A}$ by
$\hat{A}(G)$ and call it the _essential algebra of $A$ at $G$_. In particular
when $G$ is minimal for $F$, then $F(G)$ becomes an $\hat{A}(G)$-module.

Let $S$ be a simple $A$-module. Then for any group $G$, the $E_{G}^{A}$-module
$S(G)$ is either zero or a simple $E_{G}^{A}$-module. If $G$ is minimal for
$S$ then $S(G)$ becomes a simple $\hat{A}(G)$-module. By the general results
in [4], the simple $A$-module $S$ is determined by the pair $(G,S(G))$ for any
finite group $G$. When $G$ is also minimal for $S$, we call the pair
$(G,S(G))$ a _seed_ for $S$. If $(G,V)$ is a seed for $S$, we also denote $S$
by $S_{G,V}^{A}$. Conversely, if a pair $(G,V)$ is given where $V$ is a simple
$\hat{A}(G)$-module, there is a unique simple $A$-module with seed $(G,V)$. We
denote it by $S_{G,V}^{A}$. In general two seeds $(G,V)$ and
$(G^{\prime},V^{\prime})$ may induce isomorphic simple $A$-modules. In this
case, we say that $(G,V)$ and $(G^{\prime},V^{\prime})$ are equivalent. Then
we obtain a correspondence between isomorphism classes of simple $A$-modules
and equivalence classes of seeds. Note that two seeds $(G,V)$ and
$(G^{\prime},V^{\prime})$ might be equivalent without $G$ and $G^{\prime}$
being isomorphic, see [2].

Finally we recall from [2] and [4] the following construction of the simple
functors from seeds. Let $V$ be an $E_{G}$-module. We denote by $L_{G,V}$ the
$A$-module given by

$L_{G,V}(H)=A(H\times G)\otimes_{E_{G}}V$

for any finite group $H$. Here the right action of $E_{G}$ is given by
composition on the right and the action of $A$ is given by composition on the
left.

Now suppose $V$ is a simple $E_{G}$-module. Then by [3, Lemme 1], the
$A$-module $L_{G,V}$ has a unique maximal submodule $J_{G,V}$ given by

$J_{G,V}(H)=\\{\sum_{i}x_{i}\otimes v_{i}\in L_{G,V}(H)|\forall y\in A(G\times
H):\sum_{i}(y\circ x_{i})v_{i}=0\\}.$

The simple head $S_{G,V}$, which is the quotient $L_{G,V}/J_{G,V}$ and a
simple object of $\mathcal{F}_{k}$, has the property that $S_{G,V}(G)\cong V$.

###### 2.5

Covering algebras. In general determination of the seeds and the equivalence
relation is not easy. The first step in this direction is to identify
essential algebras for $A$. Suppose $E^{\prime}_{G}$ is a subalgebra of
$E_{G}^{A}$ such that

$E_{G}^{A}=E^{\prime}_{G}+I_{G}^{A}.$

Then we get

$\hat{A}(G)=E^{\prime}_{G}/(E^{\prime}_{G}\cap I_{G}^{A}).$

In other words, one can consider $E^{\prime}_{G}$ as an approximation to the
essential algebra $\hat{A}(G)$. We call $E^{\prime}_{G}$ a covering algebra
for $E_{G}^{A}$.

Suppose $|G|$ is invertible in $k$. Then an example of covering algebra is the
truncated algebra $\tilde{e}_{G}^{G}A(G\times G)\tilde{e}_{G}^{G}$. Here
$\tilde{e}_{G}^{G}$ is the image of the primitive idempotent $e_{G}^{G}$ of
the Burnside ring $B(G)$ in the double Burnside ring $B(G\times G)$ as defined
in [4, Section 6.5]. We regard it as a morphism in $\mathcal{P}_{A}$ as
explained above. This is a covering algebra, by [4, Proposition 6.5.5]. The
algebra $E_{G}^{c}$ introduced in [2, Section 6] for fibered Burnside functors
is equal to $\tilde{e}_{G}^{G}E_{G}\tilde{e}_{G}^{G}$ by [4, Proposition
6.5.5]. It is possible to introduce other covering algebras. One aims to
choose a covering algebra for which the intersection with the ideal
$I_{G}^{A}$ is easier to determine.

###### 2.6

Idempotents in covering algebras. In [2] the covering algebra for the fibered
biset functors is studied through a set of central idempotents. This approach
can be abstracted to Green biset functors as follows. Note that it only holds
under an assumption and generalizes some very basic results.

We fix a covering algebra $E_{G}^{c}$ for $A$. Let $(X,\leqslant)$ be a finite
poset with minimum element $x_{0}$. Suppose there is a function

$e:X\to E_{G}^{c},x\mapsto e_{x}$

such that

  1. (a)

$e_{x_{0}}=1_{E_{G}^{c}}$

  2. (b)

$e_{x}\cdot e_{y}=\begin{cases}e_{x\vee y}&\mbox{\rm if}\quad x\vee
y\,\mbox{\rm exists}\\\ 0&\mbox{\rm otherwise}\end{cases}$

In particular $e_{x}$ is an idempotent in $E_{G}^{c}$ for each $x\in X$. We
define

$f_{x}=\sum_{x\leqslant y\in X}\mu_{x,y}e_{y}$

where $\mu_{x,y}$ is the Mobius function of the poset $X$. Then by the Mobius
inversion formula, we also get

$e_{x}=\sum_{x\leqslant y\in X}f_{y}\quad\mbox{\rm and}\quad
1_{E_{G}^{c}}=e_{x_{0}}=\sum_{x\in X}f_{x}.$

The following result is a generalization of Proposition 4.4 from [2] and
Theorem 10.1 from [6].

###### 2.7 Proposition

For all $x,y\in X$, we have

$e_{x}\cdot f_{y}=f_{y}\cdot e_{x}=\begin{cases}f_{y}&\mbox{\rm if}\quad x\leq
y\\\ 0&\mbox{\rm otherwise}\end{cases}$

and

$f_{x}\cdot f_{y}=\begin{cases}f_{x}&\mbox{\rm if}\quad x=y\\\ 0&\mbox{\rm
otherwise}\end{cases}$

###### Proof.

The proof is almost identical to the proof of [2, Proposition 4.4]. We repeat
it for completeness. For the first case, if $x\leqslant y$, then

$\displaystyle e_{x}\cdot f_{y}$ $\displaystyle=$
$\displaystyle\sum_{y\leqslant z}\mu_{y,z}e_{x}\cdot e_{z}$ $\displaystyle=$
$\displaystyle\sum_{y\leqslant z}\mu_{y,z}e_{z}=f_{y}.$

Here the second equality holds since $x\leqslant z$ implies $e_{x}\cdot
e_{z}=e_{z}$. Also it is clear that $e_{x}$ and $f_{y}$ commutes. For the rest
of the claims, we argue by induction on $d=d_{x}+d_{y}$ where for any $x\in
X$, the natural number $d_{x}$ is the largest $n\in\mathbb{N}_{0}$ such that
there exists a chain $x=a_{0}<a_{1}<\cdots<a_{n}$ in $X$. Now if $d=0$, then
both $x$ and $y$ are maximal in $X$, hence we have $f_{y}=e_{x}$. Therefore
either $x=y$ and $x\vee y=x$ and hence $e_{x}\cdot f_{y}=e_{x}=f_{y}$ or their
join does not exist and hence $e_{x}\cdot f_{y}=0$.

Next suppose $d>0$ and that the proposition holds for all smaller $d$. We
claim that if $f_{x}\cdot f_{y}$ is non-zero, then $x=y$. Indeed we have

$\displaystyle f_{x}\cdot f_{y}$ $\displaystyle=$
$\displaystyle\sum_{x\leqslant z,y\leqslant t}\mu_{x,z}\mu_{y,t}e_{z}\cdot
e_{t}.$

Here the product $e_{z}\cdot e_{t}$ is non-zero if and only if the join $z\vee
t$ exists, in which case, $x\vee y$ also exists and $x\vee y\leqslant z\vee
t$. Hence the above equality becomes

$\displaystyle f_{x}\cdot f_{y}$ $\displaystyle=$ $\displaystyle\sum_{x\vee
y\leqslant z}\mu_{x,z}\mu_{y,z}e_{z}.$

In particular $e_{x\vee y}\cdot f_{x}\cdot f_{y}=f_{x}\cdot f_{y}\not=0$.
Hence by the first part, either $x<x\vee y$ or $y<x\vee y$ and we get
$e_{x\vee y}\cdot f_{x}\cdot f_{y}=0$ by induction. This is a contradiction,
so we must have $x=y$.

Finally if $x=y$, then

$e_{x}=e_{x}^{2}=f_{x}^{2}+\sum_{x<y}f_{y}$

and hence $f_{x}^{2}=f_{x}$ as required. ∎

Note that the elements $e_{x},x\in X$ generates a commutative subalgebra
$\bigoplus_{x\in X}\mathbb{Z}e_{x}$ of $E_{G}^{c}$ and the set $\\{f_{x}|x\in
X\\}$ is a basis of this algebra consisting of orthogonal idempotents summing
up to the identity element.

###### 2.8

To resolve the structure of $E_{G}^{c}$ further, one needs more properties of
the idempotents introduced above. The properties in [2] depend on fibered
bisets and it is not clear to us how to make an abstraction to the above case.
In this paper we consider the special case of section Burnside functors [5].

##  3 Section Burnside rings

In this section we collect together definitions and basic results on section
Burnside rings as defined in [5]. We also prove a version of Goursat Theorem
for sections of direct products of groups.

###### 3.1

The categories $\hbox to0.0pt{\hss\phantom{|}}_{G}\mathsf{mor}$ and $\hbox
to0.0pt{\hss\phantom{|}}_{G}\mathsf{mor}^{\mathsf{Gal}}$. Let $G$ be a finite
group. We denote the category of finite $G$-sets and $G$-equivariant morphisms
by $\hbox to0.0pt{\hss\phantom{|}}_{G}\mathsf{set}$. For any pair of morphisms
$f:X\to Y$ and $f^{\prime}:X^{\prime}\to Y^{\prime}$, a morphism from $f$ to
$f^{\prime}$ is defined to be a pair $(\alpha:X\to X^{\prime},\beta:Y\to
Y^{\prime})$ of morphisms of $G$-sets such that the following diagram
commutes.

${X}$${Y}$${X^{\prime}}$${Y^{\prime}}$$\alpha$$f$$f^{\prime}$$\beta$

For $(\alpha,\beta):f\to f^{\prime}$ and $(\gamma,\delta):f^{\prime}\to
f^{\prime\prime}$, we define the composition
$(\gamma,\delta)\circ(\alpha,\beta)$ component-wise. Notice that the pair
$(\mathrm{id}_{X},\mathrm{id}_{Y})$ becomes the identity morphisms of $f:X\to
Y$. With this data, we form the category $\hbox
to0.0pt{\hss\phantom{|}}_{G}\mathsf{mor}$ of morphisms of $G$-sets. By [5,
Proposition 2.2], the disjoint union and the direct product of $G$-sets induce
a coproduct and a product in $\hbox to0.0pt{\hss\phantom{|}}_{G}\mathsf{mor}$,
respectively.

A morphism $f:X\to Y$ is called a Galois morphism if for any $x,x^{\prime}\in
X$ with $f(x)=f(x^{\prime})$, we have $f=f\circ\phi$ with $\phi(x)=x^{\prime}$
for some automorphism $\phi$ of $X$ in $\hbox
to0.0pt{\hss\phantom{|}}_{G}\mathsf{set}$. By [5, Section 9], the class of all
Galois morphisms of morphisms of $G$-sets forms a subcategory of $\hbox
to0.0pt{\hss\phantom{|}}_{G}\mathsf{mor}$ and the product and coproduct in
$\hbox to0.0pt{\hss\phantom{|}}_{G}\mathsf{mor}$ restricts to a product and a
coproduct in this subcategory. We denote the category of Galois morphisms by
$\hbox to0.0pt{\hss\phantom{|}}_{G}\mathsf{mor}^{\mathsf{Gal}}$.

Isomorphisms of morphisms of $G$-sets are defined in the usual way and we
denote the isomorphism class of the morphism $f:X\to Y$ by $[f:X\to Y]$. Given
a slice $S\leqslant T$ of $G$, we denote the isomorphism class of the
projection morphism $G/S\to G/T$ by $\langle G/T,G/S\rangle$. Note that by [5,
Remark 9.5], the morphism $G/S\to G/T$ is a Galois morphism if and only if
$S\unlhd T$.

###### 3.2

Slice and section Burnside rings. Following Bouc [5], we define the slice
Burnside ring $\Xi(G)$ of $G$ as the quotient of the free abelian group on the
isomorphism classes $[f:X\to Y]$ of morphisms of $G$-sets by the subgroup
generated by the relation

$[f_{1}\sqcup f_{2}:(X_{1}\sqcup
X_{2}){\rightarrow}Y]-[f_{1}:X_{1}{\rightarrow}f(X_{1})]-[f_{2}:X_{2}{\rightarrow}f(X_{2})]$

whenever $f:X\rightarrow Y$ is a morphism of finite $G$-sets with a
decomposition $X=X_{1}\sqcup X_{2}$ as a disjoint union of $G$-sets, where
$f_{1}=f_{|X_{1}}$ and $f_{2}=f_{|X_{2}}$. The product is induced by the
product of morphisms. The image of $f$ in $\Xi(G)$ is denoted by $[f]_{G}$ and
to simplify the notation further, we denote $\langle G/T,G/S\rangle$ by
$\langle T,S\rangle_{G}$.

The subring of $\Xi(G)$ generated by the isomorphism classes of Galois
morphisms of $G$-sets is called the section Burnside ring and is denoted by
$\Gamma(G)$.

With this notation, by [5, Lemma 3.3], the zero element in $\Xi(G)$ is
$[\emptyset\to\emptyset]_{G}$ and we have $[f:X\to Y]_{G}=[f:X\to f(X)]_{G}$
for any morphism $f:X\to Y$ of $G$-sets. Furthermore we have

$[f_{1}\sqcup f_{2}:X_{1}\sqcup X_{2}\to Y_{1}\sqcup
Y_{2}]_{G}=[f_{1}:X_{1}\to Y_{1}]_{G}+[f_{2}:X_{2}\to Y_{2}]_{G}$

In particular, by [5, Lemma 3.4], for any morphism $f:X\to Y$, there is a
decomposition

$[f]_{G}=\sum_{x\in[G\backslash X]}\langle G_{f(x)},G_{x}\rangle_{G}$

where the sum is over a complete set of representatives of $G$-orbits in $X$.
Therefore the slice Burnside ring $\Xi(G)$ has a basis $\langle
T,S\rangle_{G}$ as $(T,S)$ runs over a complete set of representatives of
$G$-conjugacy classes of pairs $(T,S)$ of subgroups $S\leqslant T\leqslant G$,
see [5, Theorem 4.6]. These pairs are called slices of $G$. Similarly, the
subset consisting of the pairs parameterized by the representatives of the
$G$-conjugacy classes of sections $(T,S)$ of $G$ forms a basis for the section
Burnside ring $\Gamma(G)$.

###### 3.3

The group $\Gamma(G,H)$. Let $H$ be a finite group. We write $\Gamma(G,H)$ for
the section Burnside ring of $G\times H$. It has a basis parameterized by the
representatives of $G\times H$-conjugacy classes of sections of the direct
product $G\times H$. As usual we shall consider the elements of $\Gamma(G,H)$
as morphisms from $H$ to $G$, and when considering as such, we denote the
Galois morphism $(X\xrightarrow{g}Y)$ of $G\times H$ and its image in
$\Gamma(G,H)$, respectively, by

$\Big{(}\frac{G\times H}{X\xrightarrow{g}Y}\Big{)}\quad\mbox{\rm
and}\quad\Big{[}\frac{G\times H}{X\xrightarrow{g}Y}\Big{]}$

In case we have a section $(T,S)$ of $G\times H$, we denote the projection
$\Big{(}\frac{G\times H}{S}\Big{)}\to\Big{(}\frac{G\times H}{T}\Big{)}$ and
its image in $\Gamma(G,H)$, respectively, by

$\Big{(}\frac{G\times H}{S\unlhd T}\Big{)}\quad\mbox{\rm
and}\quad\Big{[}\frac{G\times H}{S\unlhd T}\Big{]}$

###### 3.4

Digression on crossed modules. It turns out that crossed modules are useful to
express conditions on subgroups of direct products to be sections. We recall
basic definitions. See [10] for further details.

Let $G$ and $A$ be groups. Assume $G$ acts on $A$, written
$(g,a)\mapsto{{}^{g}}a$ for $g\in G,a\in A$ and also suppose that there is a
group homomorphism $\partial:A\rightarrow G$. The triple $(A,G,\partial)$ is
called a crossed module if the following two conditions hold.

  1. (i)

$\partial({{}^{g}}a)=g\partial(a)g{{}^{-1}},$

  2. (ii)

${{}^{\partial(b)}}a=bab^{-1}$

for all $g\in G$ and $a,b\in A$.

Let $(A,G,\partial)$ and $(A^{\prime},G^{\prime},\partial^{\prime})$ be
crossed modules. We define a morphism from $(A,G,\partial)$ to
$(A^{\prime},G^{\prime},\partial^{\prime})$ as a pair $(\alpha,\beta)$ of
group homomorphisms $\alpha:A\to A^{\prime}$ and $\beta:G\to G^{\prime}$
satisfying

  1. (a)

The diagram

${A}$${A^{\prime}}$${G}$${G^{\prime}}$$\scriptstyle{\alpha}$$\scriptstyle{\partial}$$\scriptstyle{\partial^{\prime}}$$\scriptstyle{\beta}$

commutes.

  2. (b)

For all $g\in G$ and $a\in A$, we have
$\alpha({{}^{g}}a)={{}^{\beta(g)}}\alpha(a)$.

The pair $(\alpha,\beta)$ is called an isomorphism, a monomorphism, an
epimorphism or an automorphism of crossed modules if the homomorphisms
$(\alpha,\beta)$ are both isomorphisms, monomorphisms, epimorphisms or
automorphisms, respectively. Hence the group of automorphisms of
$(A,G,\partial)$ is defined and denoted by $\mathrm{Aut}(A,G,\partial)$.

Let $\alpha:G\to\mathrm{Aut}(A),g\mapsto\alpha_{g}$ be the action of $G$ on
$A$ and $c:G\to\mathrm{Aut}(G),g\mapsto c_{g}$ be the conjugation action of
$G$ on itself. For any $g\in G$ the pair $(\alpha_{g},c_{g})$ is an
automorphism of $(A,G,\partial)$ and it is called an inner automorphism. The
induced function $\theta:G\to\mathrm{Aut}(A,G,\partial)$ is a homomorphism,
its image is a normal subgroup, denoted by $\mathrm{Inn}(A,G,\partial)$. As
usual the quotient
$\mathrm{Out}(A,G,\partial)=\mathrm{Aut}(A,G,\partial)/\mathrm{Inn}(A,G,\partial)$
is called the group of outer automorphisms of $(A,G,\partial)$.

###### 3.5

Goursat Theorem for sections. Since a basis for $\Gamma(G,H)$ is given in
terms of sections of $G\times H$, we first parametrize these sections in a way
as Goursat Theorem parametrizes subgroups of direct products. Let $U$ be a
subgroup of $G\times H$. We set

  1. (i)

$p_{1}(U)=\\{g\in G\mid(g,h)\in U\,\mbox{\rm for some}\,h\in H\\}$,
$p_{2}(U)=\\{h\in H\mid(g,h)\in U\,\mbox{\rm for some}\,g\in G\\}$,

  2. (ii)

$k_{1}(U)=\\{g\in G\mid(g,1)\in U\\}$, $k_{2}(U)=\\{h\in H\mid(1,h)\in U\\}$,

  3. (iii)

$\eta_{U}:p_{2}(U)/k_{2}(U)\to p_{1}(U)/k_{1}(U),h\cdot k_{2}(U)\mapsto g\cdot
k_{1}(U)$ provided $(g,h)\in U$.

By Goursat’ Theorem, the map $\eta_{U}$ above is a group isomorphism and the
correspondence mapping $U$ to the quintuple
$(p_{1}(U),k_{1}(U),\eta_{U},k_{2}(U),p_{2}(U))$ establishes a bijective
correspondence between the subgroups $U$ of $G\times H$ and the quintuples
$(P,K,\eta,L,Q)$ with $K\unlhd P\leqslant G$ and $L\unlhd Q\leqslant H$ and
$\eta:Q/L\to P/K$ a group isomorphism. We call
$(p_{1}(U),k_{1}(U),\eta_{U},k_{2}(U),p_{2}(U))$ the Goursat correspondent of
$U$. Also we call a quintuple $(P,K,\eta,L,Q)$ satisfying the above conditions
a Goursat quintuple.

To determine a similar correspondence for the sections of $G\times H$, let
$(T,S)$ be a section of $G\times H$ with the corresponding quintuples
$(P_{T},K_{T},\eta_{T},L_{T},Q_{T})$ and $(P_{S},K_{S},\eta_{S},L_{S},Q_{S})$.
Then by Goursat Theorem, we have

  1. (S1)

$1\leqslant K_{X}\unlhd P_{X}\leqslant G$ and $1\leqslant L_{X}\unlhd
Q_{X}\leqslant H$ for $X\in\\{T,S\\}$.

  2. (S2)

$\eta_{S}$ and $\eta_{T}$ induced respectively by $S$ and $T$ are
isomorphisms.

In addition the condition $S\unlhd T$ implies that the following statements
also hold:

  1. (S3)

$K_{S}\unlhd K_{T}$ and $L_{S}\unlhd L_{T}$.

  2. (S4)

$K_{S},P_{S}\unlhd P_{T}$ and $L_{S},Q_{S}\unlhd Q_{T}$,

Here (S3) is straightforward. To prove that $K_{S}\unlhd P_{T}$, let $x\in
K_{S}$ and $g\in P_{T}$ so that $(x,1)\in S$ and $(g,h)\in T$ for some $h\in
H$. Then since S is normal in T, we get $({}^{g}x,1)={}^{(g,h)}(x,1)\in S$ In
particular ${{}^{g}}x\in K_{S}$. Similarly if $x\in P_{S}$ with $(x,y)\in S$
then$({{}^{g}}x,{{}^{h}}y)=^{(g,h)}(x,y)\in S$, that is, ${{}^{g}}x\in P_{S}$.
The rest is proved in the same way.

  1. (S5)

$[K_{T},P_{S}]\leqslant K_{S}$ and $[L_{T},Q_{S}]\leqslant L_{S}$

Indeed let $x\in P_{S}$ with $(x,y)\in S$ and $g\in K_{T}$. Note that
$(g,1)\in T$. Then since S is normal in T, we get
${}^{(g,1)}(x,y)=({}^{g}x,y)\in S$. In particular $x^{-1}\cdot{}^{g}x\in
K_{S}$ which implies $xK_{S}={}^{g}xK_{S}$, as required. Note that this
condition is equivalent to the following condition.

  1. (S5′)

$K_{T}/K_{S}\leqslant C_{P_{T}/K_{S}}(P_{S}/K_{S})$ and $L_{T}/L_{S}\leqslant
C_{Q_{T}/L_{S}}(Q_{S}/L_{S})$.

By Condition (S4), we have that $P_{T}/K_{S}$ contains $P_{S}/K_{S}$ as a
subgroup and $P_{T}/K_{T}$ as a quotient. We have the following diagram.

${P_{T}/K_{S}}$${P_{S}/K_{S}}$${P_{T}/K_{T}}$$\scriptstyle{\pi}$$\scriptstyle{\iota}$

We denote the composition by $\partial=\pi\circ\iota$, it is given by
$\partial(xK_{S})=xK_{T}$. Together with (S5), the conjugation action of
$P_{T}/K_{S}$ on $P_{S}/K_{S}$ descents to a conjugation action of
$P_{T}/K_{T}$ on $P_{S}/K_{S}$. Now we get

  1. (S6)

$(P_{S}/K_{S},P_{T}/K_{T},\partial)$ and $(Q_{S}/L_{S},Q_{T}/L_{T},\partial)$
are crossed modules.

To prove (S6) one needs to justify the two conditions given in Section 3.4.
For the first condition, let $gK_{T}\in P_{T}/K_{T}$ and $xK_{S}\in
P_{S}/K_{S}$. Then

$\partial({}^{gK_{T}}xK_{S})=\partial({}^{g}xK_{S})={}^{g}xK_{T}={}^{gK_{T}}\partial(xK_{S}),$

hence the first condition of the definition holds. For the second condition,
let also $yK_{S}\in P_{S}/K_{S}$. Then

${}^{\partial(xK_{S})}yK_{S}={}^{xK_{T}}yK_{S}={}^{x}yK_{S}={}^{xK_{S}}yK_{S}$

which proves the second condition. Thus $(P_{S}/K_{S},P_{T}/K_{T},\partial)$
is a crossed module. Replacing $K$ with $L$ and $P$ with $Q$ in the above
arguments, one can also prove that $(Q_{S}/L_{S},Q_{T}/L_{T},\partial)$ is a
crossed module. With this identifications, the following also holds.

  1. (S7)

$(\eta_{S},\eta_{T}):(Q_{S}/L_{S},Q_{T}/L_{T},\partial)\to(P_{S}/K_{S},P_{T}/K_{T},\partial)$
is an isomorphism of crossed modules.

Since $\eta_{S}$ and $\eta_{T}$ are already isomorphisms, we only need to
check that the pair $(\eta_{S},\eta_{T})$ is a morphism of crossed modules,
see Section 3.4. Let $qL_{S}\in Q_{S}/L_{S}$. and let
$\eta_{S}(qL_{S})=pK_{S}$. Then

$\eta_{T}(\partial(qL_{S}))=\eta_{T}(qL_{T})=pK_{T}$

where the last equation holds since $(p,q)\in S$ implies it is in $T$. On the
other hand

$\partial(\eta_{S}(qL_{S}))=\partial(pK_{S})=pK_{T}$

and hence the diagram in Section 3.4 commutes. For the other condition, let
also $(g,h)\in T$ so that $\eta_{T}(hL_{T})=gK_{T}$. Then

$\eta_{S}({}^{hL_{T}}qL_{S})=\eta_{S}({}^{h}qL_{S})={}^{g}pK_{S}.$

Here the last equality holds since $S\unlhd T$. We clearly have

${}^{g}pK_{S}={}^{\eta_{T}(hL_{T})}\eta_{S}(qL_{S}).$

Hence $(\eta_{S},\eta_{T})$ is a morphism of crossed modules and hence we
proved (S7). Collecting the properties we obtain the following version of
Goursat’ Theorem for sections.

###### 3.6 Theorem

Let $G$ and $H$ be finite groups. Then there is a bijective correspondence
between

  1. (a)

the set of all sections $(T,S)$ of $G\times H$ and

  2. (b)

the set of all pairs
$((P_{1},K_{1},\eta_{1},L_{1},Q_{1}),(P_{2},K_{2},\eta_{2},L_{2},Q_{2}))$ of
Goursat quintuples satisfying the following conditions.

    1. (S3)

$K_{2}\unlhd K_{1}$ and $L_{2}\unlhd L_{1}$,

    2. (S4)

$K_{2},P_{2}\unlhd P_{1}$ and $L_{2},Q_{2}\unlhd Q_{1}$,

    3. (S5)

$[K_{1},P_{2}]\leqslant K_{2}$ and $[L_{1},Q_{2}]\leqslant L_{2}$.

    4. (S6)

$(P_{2}/K_{2},P_{1}/K_{1},\partial)$ and $(Q_{2}/L_{2},Q_{1}/L_{1},\partial)$
are crossed modules where $\partial$ is given by $\partial(xA_{2})=xA_{1}$ for
$A\in\\{K,L\\}$ and in both cases, the action is given by conjugation.

    5. (S7)

$(\eta_{2},\eta_{1}):(Q_{2}/L_{2},Q_{1}/L_{1},\partial)\to(P_{2}/K_{2},P_{1}/K_{1},\partial)$
is an isomorphism of crossed modules.

The correspondence is given by mapping a section $(T,S)$ of $G\times H$ to the
pair of Goursat correspondents of $T$ and $S$.

###### Proof.

By the arguments preceding the theorem, the mapping is well-defined and
injective. Hence it is sufficient to prove that any element of the later set
corresponds to a section in $G\times H$. Let
$((P_{1},K_{1},\eta_{1},L_{1},Q_{1}),(P_{2},K_{2},\eta_{2},L_{2},Q_{2}))$ be a
pair of Goursat quintuples satisfying the above conditions. Also let $(T,S)$
be the pair of subgroups of $G\times H$ corresponding to these Goursat
quintuples. Explicitly we have

$S=\\{(x,y)\in P_{2}\times Q_{2}\mid\eta_{2}(yL_{2})=xK_{2}\\}\,\,\mbox{\rm
and}\,\,T=\\{(g,h)\in P_{1}\times Q_{1}\mid\eta_{1}(hL_{1})=gK_{1}\\}$

We have to show that $S\unlhd T$. It is straightforward to show that
$S\subseteq T$ is equivalent to (S6) and $T\subset N_{G}(S)$ is equivalent to
(S7). Note that although the conditions (S3)-(S4)-(S5) are not used
explicitly, they are needed for the last two conditions to make sense. ∎

###### 3.7

Invariants of sections. Let $G$ and $H$ be finite groups. We denote the set of
all sections of $G$ by $\Sigma_{G}$. Let $(T,S)\in\Sigma_{G\times H}$. We
associate to $(T,S)$ its left and right invariants

$l(T,S):=(p_{1}(T),k_{1}(T),p_{1}(S),k_{1}(S))$

and

$r(T,S):=(p_{2}(T),k_{2}(T),p_{2}(S),k_{2}(S))$

respectively. In some cases, the middle invariants
$\;l_{0}(T,S):=(k_{1}(T),p_{1}(S))$ and $r_{0}(T,S):=(k_{2}(T),p_{2}(S))$ will
be more useful. The following result states further relations between Goursat
correspondents.

###### 3.8 Proposition

Let $G$ and $H$ be finite groups, $(T,S)\in\Sigma_{G\times H}$ with Goursat
correspondents $(P_{T},K_{T},\eta_{T},L_{T},Q_{T})$ and
$(P_{S},K_{S},\eta_{S},L_{S},Q_{S})$. Then there are group isomorphisms

  1. (a)

$\frac{P_{S}}{P_{S}\cap K_{T}}\cong\frac{Q_{S}}{Q_{S}\cap L_{T}},$

  2. (b)

$\frac{P_{T}}{P_{S}K_{T}}\cong\frac{Q_{T}}{Q_{S}L_{T}}$

and

  3. (c)

$\frac{P_{S}\cap K_{T}}{K_{S}}\cong\frac{Q_{S}\cap L_{T}}{L_{S}}.$

In particular if $P_{T}=G$, $K_{S}=1$ and $K_{T}\leqslant P_{S}$ then
$|G|\leqslant|H|$.

###### Proof.

(a) First note that since $K_{T}$ and $P_{S}$ are both normal in $P_{T}$, the
intersection $P_{S}\cap K_{T}$ is normal in $P_{S}$. Hence the isomorphism
$\eta_{S}:Q_{S}/L_{S}\to P_{S}/K_{S}$ pre-composed by $\pi:Q_{S}\to
Q_{S}/L_{S}$ and post-composed with $\pi:P_{S}/K_{S}\to P_{S}/(P_{S}\cap
K_{T})$ becomes a surjective homomorphism

$\tilde{\eta}_{S}:Q_{S}\to P_{S}/(P_{S}\cap K_{T}).$

The kernel of $\tilde{\eta}_{S}$ is clearly $Q_{S}\cap L_{T}$. Note that by
the second isomorphism theorem we also have the isomorphisms

$P_{S}K_{T}/K_{T}\cong P_{S}/(P_{S}\cap K_{T})\cong Q_{S}/(Q_{S}\cap
L_{T})\cong Q_{S}L_{T}/L_{T}.$

(b) As in the above case, post-compose $\eta_{T}$ with the canonical
homomorphism $\pi:P_{T}/K_{T}\to P_{T}/(P_{S}K_{T})$ to obtain

$\tilde{\eta}_{T}:Q_{T}/L_{T}\to P_{T}/(P_{S}K_{T})$

Let $(p,q)\in T$. Then we have $qL_{T}\in\ker\tilde{\eta}_{T}$ if and only if
$pK_{T}\in P_{S}K_{T}/K_{T}$. We claim that the last condition is equivalent
to $qL_{T}\in Q_{S}L_{T}/L_{T}$. There is nothing to prove if $p\in K_{T}$.
Suppose $p\in P_{S}$. It is sufficient to prove that $qL_{T}\in
Q_{S}L_{T}/L_{T}$ and this follows directly from the compatibility of
$\eta_{S}$ and $\eta_{T}$. Hence the kernel of $\tilde{\eta}_{T}$ is
$Q_{S}L_{T}/Q_{S}$ and the result follows from the first and the third
isomorphism theorems.

(c) Consider the restriction $\bar{\eta}_{S}$ of $\eta_{S}$ to the subgroup
$(Q_{S}\cap L_{T})/L_{S}$. Since $\eta_{S}$ is an isomorphism, the restriction
of $\bar{\eta}_{S}$ to its image is an isomorphism. To determine its image,
let $q\in Q_{S}\cap L_{T}$ and let $p\in G$ be such that $(p,q)\in S$. Then
since $(1,q)\in T$, we also get $(p,1)\in T$, that is, $p\in K_{T}$. Therefore
$pK_{S}$ is contained in $(P_{S}\cap K_{T})/K_{S}$, as required.

For the final statement, we write the order of $G$ as

$|G|=|G:P_{T}|\cdot|P_{T}:P_{S}K_{T}|\cdot|P_{S}K_{T}:K_{T}|\cdot|K_{T}:P_{S}\cap
K_{T}|\cdot|P_{S}\cap K_{T}:K_{S}|\cdot|K_{S}|$

and similarly the order of $H$ as

$|H|=|H:Q_{T}|\cdot|Q_{T}:Q_{S}L_{T}|\cdot|Q_{S}L_{T}:L_{T}|\cdot|L_{T}:Q_{S}\cap
L_{T}|\cdot|Q_{S}\cap L_{T}:L_{S}|\cdot|L_{S}|$

Then with the above isomorphisms, we have

$\frac{|G|}{|H|}=\frac{|G:P_{T}|\cdot|K_{T}:P_{S}\cap
K_{T}|\cdot|K_{S}|}{|H:Q_{T}|\cdot|L_{T}:Q_{S}\cap L_{T}|\cdot|L_{S}|}$

Since by the assumption $P_{T}=G$, $K_{S}=1$ and $K_{T}\leqslant P_{S}$ we
deduce that $\frac{|G|}{|H|}\leqslant 1$. ∎

###### 3.9

Composition product of sections.

Given finite groups $G$, $H$, $K$ and Galois morphisms $\Big{(}\frac{G\times
H}{X\xrightarrow{g}Y}\Big{)}$ and $\Big{(}\frac{H\times
K}{X^{\prime}\xrightarrow{f}Y^{\prime}}\Big{)}$, we define their composition
as the morphism

$\left(\frac{G\times
K}{X\times_{H}X^{\prime}\xrightarrow{g\times_{H}f}Y\times_{H}Y^{\prime}}\right)$

Here $X\times_{H}X^{\prime}$ and $Y\times_{H}Y^{\prime}$ are the usual
amalgamated products of bisets and
$(g\times_{H}f)(x,_{H}x^{\prime})=\big{(}g(x),_{H}f(x^{\prime})\big{)}$. It is
a Galois Morphism of $G\times K$-sets by Proposition 9.13 of [5]. Moreover the
above composition induces a bilinear associative product

$\Gamma(G,H)\times\Gamma(H,K)\to\Gamma(G,K).$

The following corollary gives an explicit formula for the composition of two
bases elements of the section Burnside ring.

###### 3.10 Corollary

(Mackey Formula) Let $(T,S)\in\Sigma_{G\times H}$ and $(V,U)\in\Sigma_{H\times
K}$. There exists an isomorphism of Galois morphisms

$\Big{(}\frac{G\times H}{S\unlhd T}\Big{)}\times_{H}\Big{(}\frac{H\times
K}{U\unlhd V}\Big{)}\cong\bigsqcup_{t\in p_{2}(S)\setminus
H/p_{1}(U)}\Big{(}\frac{G\times K}{S*{{}^{(t,1)}U}\unlhd
T*{{}^{(t,1)}V}}\Big{)}$

where $S*U:=\\{(g,k)\in G\times K\,|$ there exists $h\in H$ with $(g,h)\in S$
and $(h,k)\in U\\}.$

###### Proof.

This follows directly from the Mackey formula for bisets together with [5,
Lemma 3.3]. ∎

###### 3.11 Notation

Let $f:X\rightarrow Y$ be a Galois morphism in $\Gamma(G\times H)$. We define
the opposite $f^{op}\in\Gamma(H\times G)$ of $f$ as the Galois morphism
$f^{op}:X^{op}\rightarrow Y^{op}$ where $f^{op}(x)=f(x)$ for any $x\in X$.
Here the opposite biset $X^{op}$ is the $(H,G)$-biset equal to $X$ as a set
with the action defined in the following way. For all $h\in H$, $x\in X$,
$g\in G$, $h\cdot x\cdot g\;(in\;X^{op})=g^{-1}\cdot x\cdot h^{-1}(in\;X)$. In
particular, if $f=\Big{(}\frac{G\times H}{S\unlhd T}\Big{)}$ then
$f^{op}\cong\Big{(}\frac{H\times G}{S^{op}\unlhd T^{op}}\Big{)}$, where
$S^{op}:=\\{(h,g)\in H\times G\mid(g,h)\in S\\}$.

###### 3.12 Notation

Let $G$ and $H$ be finite groups. For any subgroups $A\leqslant B\leqslant H$
and any group homomorphism $\alpha:B\rightarrow G$, we set

${}_{\alpha}\Delta(A):=\\{(\alpha(a),a)\mid a\in A\\}\leqslant G\times H.$

For any subgroups $C\leqslant D\leqslant G$ and any group homomorphism
$\alpha:D\rightarrow H$, we set

$\Delta_{\alpha}(C):=\\{(c,\alpha(c))\mid c\in C\\}\leqslant G\times H.$

If $\alpha$ is the inclusion map of a subgroup, we write $\Delta(A)$ and
$\Delta(C)$ respectively.

###### 3.13

Basic bisets. Let $G$ be a finite group and $X$ be a $G$-set. The identity
morphism $X\to X$ is a Galois morphism and induces a homomorphism from the
Burnside ring $B(G)$ to $\Gamma(G)$. More explicitly if $X=G/H$ for some
subgroup $H$ of $G$, then its image in $\Gamma(G)$ is $[H,H]_{G}$. In
particular the basic bisets of induction, restriction, inflation and deflation
have images in $\Gamma(G,G)$ as listed below: Let $H\leqslant G$ and $N\unlhd
G$ and $\pi:G\to G/N$ be the natural homomorphism.

$\mathrm{Ind}_{H}^{G}:=\Big{(}\frac{G\times
H}{\Delta(H)\unlhd\Delta(H)}\Big{)}\in\Gamma(G,H),\quad\mathrm{Res}_{H}^{G}:=\Big{(}\frac{H\times
G}{\Delta(H)\unlhd\Delta(H)}\Big{)}\in\Gamma(H,G)$

$\mathrm{Inf}_{G/N}^{G}:=\Big{(}\frac{G\times
G/N}{\Delta_{\pi}(G)\unlhd\Delta_{\pi}(G)}\Big{)}\in\Gamma(G,G/N),\quad\mathrm{Def}_{G/N}^{G}:=\Big{(}\frac{G/N\times
G}{{}_{\pi}\Delta(G)\unlhd{{}_{\pi}\Delta(G)}}\Big{)}\in\Gamma(G/N,G)$

With these definitions we may decompose any section into a product as follows.
The statement is similar to [2, Proposition 2.8] and the proof is similar to
the proof of [4, Lemma 2.3.26]. We leave the straightforward calculations to
the reader.

###### 3.14 Proposition

Let G and H be finite groups and $(T,S)$ be a section of $G\times H$ with
$l(T,S)=(P_{T},K_{T},P_{S},K_{S})$ and $r(T,S)=(Q_{T},L_{T},Q_{S},L_{S})$.
Define $\bar{G}:={P_{T}}/{K_{S}}$, $\bar{H}:={Q_{T}}/{L_{S}}$,
$\bar{S}:=can(S/{({K_{S}}\times{L_{S}})})\leq\bar{G}\times\bar{H}$ and
$\bar{T}:=can(T/{({K_{S}}\times{L_{S}})})\leq\bar{G}\times\bar{H}$ where

$can:({P_{T}}\times{Q_{T}})/({K_{S}}\times{L_{S}})\rightarrow\bar{G}\times\bar{H}$

is the canonical homomorphism. Then

$\Big{(}\frac{G\times H}{S\unlhd
T}\Big{)}\cong\mathrm{Ind}_{P_{T}}^{G}\times_{P_{T}}\mathrm{Inf}_{\bar{G}}^{P_{T}}\times_{\bar{G}}\left(\frac{\bar{G}\times\bar{H}}{\bar{S}\unlhd\bar{T}}\right)\times_{\bar{H}}\mathrm{Def}_{\bar{H}}^{Q_{T}}\times_{Q_{T}}\mathrm{Res}_{Q_{T}}^{H}$

Furthermore we have

$k_{1}(\bar{S})=1$, $k_{2}(\bar{S})=1$, and $p_{1}(\bar{T})=\bar{G}$,
$p_{2}(\bar{T})=\bar{H}$.

##  4 The Green biset functor $\Gamma$

The functor $\Gamma$ associating $\Gamma(G)$ to the finite group $G$ has a
Green biset functor structure [5, Theorem 10.7]. Explicitly, we make $\Gamma$
a biset functor as follows. Let $G$ and $H$ be finite groups, $U$ be a
$(G,H)$-biset and $f:X\to Y$ be a Galois morphism of $H$-sets. Then the
morphism

$(f:X\to Y)\mapsto(U\times_{H}f:U\times_{H}X\to U\times_{H}Y)$

induces a functor from $\hbox
to0.0pt{\hss\phantom{|}}_{H}\mathsf{mor}^{\mathsf{Gal}}$ to $\hbox
to0.0pt{\hss\phantom{|}}_{G}\mathsf{mor}^{\mathsf{Gal}}$ and hence an abelian
group homomorphism from $\Gamma(H)$ to $\Gamma(G)$. Together with the product

$\Gamma(G)\times\Gamma(G^{\prime})\to\Gamma(G\times
G^{\prime}),(X\xrightarrow{f}Y,X^{\prime}\xrightarrow{f^{\prime}}Y^{\prime})\mapsto((X\times
X^{\prime})\xrightarrow{f\times f^{\prime}}(Y\times Y^{\prime}))$

the functor $G\mapsto\Gamma(G)$ becomes a Green biset functor.

As in Section 2.3, we associate the category $\mathcal{P}_{\Gamma}$ to the
Green biset functor $\Gamma$ where the objects are all finite groups, the
morphisms from $H$ to $G$ are given by $\Gamma(G\times H)$ and the composition
is defined as follows. Given $\alpha\in\Gamma(H\times K)$ and
$\beta\in\Gamma(G\times H)$, we have

$\beta\circ\alpha=\Gamma(\mathrm{Def}^{G\times\Delta(H)\times K}_{G\times
K}\mathrm{Res}^{G\times H\times H\times K}_{G\times\Delta(H)\times
K})(\beta\times\alpha).$

The next proposition shows that this composition is induced by the composition
product we defined in the previous section.

###### 4.1 Proposition

Let $\alpha:=\big{[}\frac{H\times K}{U\unlhd V}\big{]}$ and
$\beta:=\big{[}\frac{G\times H}{S\unlhd T}\big{]}$. Then

$\beta\circ\alpha=\beta\times_{H}\alpha.$

###### Proof.

First note that by definition $\beta\times\alpha=\big{[}\frac{G\times H\times
H\times K}{S\times U\unlhd T\times V}\big{]}$. Now by [5, Theorem 10.6],

$\beta\circ\alpha=\mathrm{Def}_{H\times
K}^{B}\mathrm{Res}_{B}^{A}(\beta\times\alpha)=\mathrm{Def}_{H\times
K}^{B}\big{(}\sum_{a\in[B\setminus A/S\times
U]}\big{\langle}(B\cap{{}^{a}}(T\times V)),(B\cap{{}^{a}}(S\times
U))\big{\rangle}_{B}\big{)}=$ $\sum_{a\in[B\setminus A/S\times
U]}\Big{\langle}\frac{(B\cap{{}^{a}}(T\times V))(1\times\Delta(G)\times
1)}{1\times\Delta(G)\times 1},\frac{(B\cap{{}^{a}}(S\times
U))(1\times\Delta(G)\times 1)}{1\times\Delta(G)\times 1}\Big{\rangle}_{H\times
K}$

where $A:=G\times H\times H\times K$ and $B:=G\times\Delta(H)\times K$.
Observe that $[B\setminus A/(S\times U)]=[p_{2}(S)\setminus G/p_{1}(U)]$ via
the bijection $Ba(S\times U)\mapsto p_{2}(S)g^{-1}\bar{g}p_{1}(U)$ for
$a=(h,g,\bar{g},k)$. Moreover we have a group isomorphism
$\frac{(B\cap(T\times V))(1\times\Delta(G)\times 1)}{1\times\Delta(G)\times
1}\cong T*V$ given by the homomorphism $(g,1,1,k)[1\times\Delta(G)\times
1]\mapsto(g,k)$.

Thus we obtain

$\beta\circ\alpha=\sum_{g\in[p_{2}(S)\setminus G/p_{1}(U)]}\langle
T*{{}^{(g,1)}}V,S*{{}^{(g,1)}}U\rangle_{H\times K}$

which is equal to $\beta\times_{H}\alpha$ by the Mackey formula given in the
previous section. ∎

Let $k$ be a commutative ring. We denote by $\mathcal{P}_{\Gamma}^{k}$ the
category obtained from $\mathcal{P}_{\Gamma}$ by extending the coefficients of
the morphisms to $k$, that is, by letting the morphisms from $H$ to $G$ to be
$k\Gamma(G\times H)$ and extending the composition linearly. Then a $k$-linear
functor $\mathcal{P}_{\Gamma}^{k}\to{}_{k}$mod is called a _section biset
functor (over $k$)_. We denote the functor category of all section biset
functors by $\mathcal{F}=\mathcal{F}_{\Gamma}^{k}$.

As explained in Section 2, simple section biset functors are determined by the
pairs $(G,V)$ where $G$ is a finite group and $V$ is a simple $E_{G}$-module.
To obtain a parametrizing set for simple section biset functors, we study the
structure of the endomorphism algebra $E_{G}$. Our results for the rest of the
paper follow [2] closely. We adapt the techniques from fibered biset functors
to show that the section biset functors has a similar theory.

##  5 The ring $E_{G}$ of endomorphisms

In this section we investigate structure of the endomorphism ring
$\Gamma(G\times G)$ in order to understand its module category. Throughout
this section, let $k$ be a field of characteristic zero. We write
$E_{G}=E_{G}^{k}=k\otimes\Gamma(G\times G)$. The main result in this section
is Theorem 5.18 which states that the covering algebra $E_{G}^{c}$ defined
below is isomorphic to a direct product of matrix rings over certain group
algebras. This is a version of Theorem 6.2 in [2]. Some preliminary proofs are
almost identical to the original versions stated and proved in [2]. Whenever
possible, we leave the verifications of the proofs to the reader. We present
certain proofs in the appendix.

###### 5.1

The poset $\mathcal{G}_{G}$. We write $\mathcal{G}_{G}$ for the set of all
pairs $(K,P)$ of normal subgroups of $G$ such that $[K,P]=1$. Note that a pair
$(K,P)$ of normal subgroups of $G$ is in $\mathcal{G}_{G}$ if and only if
$K\leqslant C_{G}(P)$ if and only if $P\leqslant C_{G}(K)$. There is a poset
structure on $\mathcal{G}_{G}$ given by $(K,P)\preceq(L,Q)$ if $K\leqslant L$
and $P\geqslant Q$. The pair $(1,G)$ is the minimum element and the pair
$(G,1)$ is the maximum element of this poset. In particular joins exists in
$\mathcal{G}_{G}$ and given $(K,P),(L,Q)\in\mathcal{G}_{G}$, one has
$(K,P)\vee(L,Q)=(KL,P\cap Q)$.

###### 5.2

To apply the results in Section 2.6, for each $(K,P)\in\mathcal{G}_{G}$, we
define

$E_{(K,P)}=\left(\frac{G\times G}{\Delta(P)\unlhd\Delta_{K}(G)}\right)$

where

$\Delta_{K}(G)=(K\times 1)\Delta(G)=\\{(g,h)\in G\times G|h^{-1}g\in K\\}.$

Clearly $\Delta(P)\leqslant\Delta_{K}(G)$ and normality follows since
$[K,P]=1$. Alternatively, if

$J_{K}^{G}:=\mathrm{Inf}_{G/K}^{G}\times_{G/K}\mathrm{Def}_{G/K}^{G}\in
B(G,G)$

and

$I_{P}^{G}:=\mathrm{Ind}_{P}^{G}\times_{P}\mathrm{Res}_{P}^{G}\in B(G,G)$

then the map $I_{P}^{G}\to J_{K}^{G}$ is a Galois morphism and its image in
$\Gamma(G\times G)$ is $[E_{(K,P)}]$. The left and the right invariants of the
section $(\Delta_{K}(G),\Delta(P))$ are both given by $(G,K,P,1)$.

###### 5.3 Definition

For each pair $(K,P)\in\mathcal{G}_{G}$ we set

$e_{(K,P)}:=\frac{1}{|G:P|}\big{[}E_{(K,P)}\big{]}\in E_{G}.$

The following proposition summarizes basic multiplication formulas for the
elements $E_{(K,P)}$. The proofs follow easily by direct calculations using
the Mackey product formula given in Corollary 3.10. We leave the
justifications to the reader.

###### 5.4 Proposition

Let $G$ and $H$ be finite groups. If
$(K,P),(K^{\prime},P^{\prime})\in\mathcal{G}_{G}$ and $(T,S)\in\Sigma_{G\times
H}$ with $l(T,S)=(G,K^{\prime},P^{\prime},1)$ then

  1. (a)

$E_{(K,P)}\times_{G}(\frac{G\times H}{S\unlhd T})\cong|P\backslash
G/P^{\prime}|(\frac{G\times H}{U\unlhd V})$ with $l(V,U)=(G,KK^{\prime},P\cap
P^{\prime},1)$, In particular one has

$E_{(K,P)}\times_{G}E_{(K^{\prime},P^{\prime})}\cong|P\backslash
G/P^{\prime}|E_{(KK^{\prime},P\cap P^{\prime})}.$

  2. (b)

Assume that $(K,P)\preceq(K^{\prime},P^{\prime})$. Then

$E_{(K,P)}\times_{G}\Big{(}\frac{G\times H}{S\unlhd
T}\Big{)}\cong|G:P|\Big{(}\frac{G\times H}{S\unlhd T}\Big{)}$

In particular

$E_{(K,P)}\times_{G}E_{(K^{\prime},P^{\prime})}\cong|G:P|E_{(K^{\prime},P^{\prime})}\quad
and\quad E_{(K,P)}\times_{G}E_{(K,P)}\cong|G:P|E_{(K,P)}.$

  3. (c)

Assume that $r(T,S)=(H,L,Q,1)$ for some $(L,Q)\in\mathcal{G}_{H}$. Then

$(\frac{G\times H}{S\unlhd T})\times_{H}(\frac{G\times H}{S\unlhd
T})^{op}\cong|H:Q|E_{(K^{\prime},P^{\prime})}$

###### 5.5

With this proposition the elements $e_{(K,P)},(K,P)\in\mathcal{G}_{G}$ are
idempotents in $E_{G}$ and we have

$e_{(K,P)}\cdot e_{(L,Q)}=e_{(KL,P\cap Q)}\quad\mbox{\rm and}\quad
e_{(1,G)}=1_{E_{G}}.$

In particular the conditions in Section 2.6 are satisfied. We have the
commutative subalgebra generated by $e_{(K,P)}$ as $(K,P)$ runs over
$\mathcal{G}_{G}$. Also setting

$f_{(K,P)}:=\sum_{(K,P)\preceq(L,Q)\in\mathcal{G}_{G}}\mu_{(K,P),(L,Q)}e_{(L,Q)}$
(1)

where $\mu_{?,?}$ is the Mobius function of the poset $\mathcal{G}_{G}$, we
get

$e_{(K,P)}=\sum_{(K,P)\preceq(L,Q)\in\mathcal{G}_{G}}f_{(L,Q)}$ (2)

and

$\sum_{(K,L)\in\mathcal{G}_{G}}f_{(K,L)}=e_{(1,G)}=1\in E_{G}$ (3)

and the following corollary of Proposition 2.7.

###### 5.6 Proposition

For all $(K,P),(L,Q)\in\mathcal{G}_{G}$ we have

$e_{(K,P)}\cdot f_{(L,Q)}=f_{(L,Q)}\cdot
e_{(K,P)}=\left\\{\begin{array}[]{l}f_{(L,Q)},\quad\mbox{if}\quad(K,P)\preceq(L,Q),\\\
0,\qquad\quad\mbox{otherwise},\end{array}\right.$

and

$f_{(K,P)}\cdot
f_{(L,Q)}=\left\\{\begin{array}[]{l}f_{(K,P)},\quad\mbox{if}\quad(K,P)=(L,Q),\\\
0,\qquad\quad\mbox{otherwise}.\end{array}\right.$

###### 5.7

Linked pairs. Although the idempotents $f_{(K,P)}$ for
$(K,P)\in\mathcal{G}_{G}$ are orthogonal, they are not necessarily central in
$E_{G}$. Next we define an equivalence relation on $\mathcal{G}_{G}$ so that
the class sums of $f_{(K,P)}$ becomes central idempotents in the covering
algebra that we construct below. Our first definition is more general.

Let $G$ and $H$ be finite groups and $(K,P)\in\mathcal{G}_{G}$ and
$(L,Q)\in\mathcal{G}_{H}$. We say that the quadruples $(G,K,P,1)$ and
$(H,L,Q,1)$ are linked if there exists $(T,S)\in\Sigma_{G\times H}$ with
$l(T,S)=(G,K,P,1)$ and $r(T,S)=(H,L,Q,1)$. In this case, we write
$(G,K,P,1)\underset{(T,S)}{\sim}(H,L,Q,1)$ or just $(G,K,P,1)\sim(H,L,Q,1)$.
In the case $H=G$, and $(K,P),(L,Q)\in\mathcal{G}_{G}$ are said to be
$G$-linked if $(G,K,P,1)\sim(G,L,Q,1)$. We write $(K,P)\sim_{G}(L,Q)$ or just
$(K,P)\sim(L,Q)$. Being linked is clearly an equivalence relation and we write
$\\{K,P\\}_{G}$ for the $G$-linkage class of $(K,P)$.

Note that when $(G,K,P,1)\underset{(T,S)}{\sim}(H,L,Q,1)$ then by Goursat
Theorem for sections, the corresponding crossed modules $(P,G/K,i_{P})$ and
$(Q,H/L,i_{Q})$ are isomorphic where $i_{P}$ is the restriction of the natural
homomorphism $G\to G/K$ to $P$. The converse is also true as shown below.

###### 5.8

Let $(K,P)\in\mathcal{G}_{G}$. There is an action of $G/K$ on $P$ via
conjugation since $P$ is normal in $G$ and $[K,P]=1$. It is straightforward to
check that $(P,G/K,i_{P})$ becomes a crossed module where $i_{P}$ is as above.
With this notation we get the following for the converse of the linkage
condition.

###### 5.9 Proposition

Let $G$ and $H$ be finite groups and $(K,P)\in\mathcal{G}_{G}$ and
$(L,Q)\in\mathcal{G}_{H}$. Then the following are equivalent:

  1. (a)

$(G,K,P,1)\sim(H,L,Q,1)$.

  2. (b)

$(P,G/K,i_{P})$ isomorphic to $(Q,H/L,i_{Q})$ as crossed modules.

###### Proof.

The first part, $(a)$ implies $(b)$, is Theorem 3.6(S7). For the converse, if
$(\alpha,\beta):(Q,H/L,i_{Q})\rightarrow(P,G/K,i_{P})$ is an isomorphism of
crossed modules, then the conditions (S3) - (S7) are satisfied and hence there
exists a section $(T,S)$ with left and right invariants given by $(G,K,P,1)$
and $(H,L,Q,1)$, respectively. In particular $(G,K,P,1)$ and $(H,L,Q,1)$ are
linked. ∎

###### 5.10

The partial order on the set $\mathcal{G}_{G}$ induces a partial order on the
set $\mathcal{G}_{G}/\sim$ of linkage classes: Given
$(K,P),(L,Q)\in\mathcal{G}_{G}$, we write $\\{K,P\\}_{G}\preceq\\{L,Q\\}_{G}$
if and only if there exists $(K^{\prime},P^{\prime})\in\\{K,P\\}_{G}$ and
$(L^{\prime},Q^{\prime})\in\\{L,Q\\}_{G}$ with
$(K^{\prime},P^{\prime})\preceq(L^{\prime},Q^{\prime})$. To see that the
induced relation is transitive, we show that
$\\{K,P\\}_{G}\preceq\\{L,Q\\}_{G}$ if and only if for each
$(K^{\prime},P^{\prime})\sim_{G}(K,P)$ there exists
$(L^{\prime},Q^{\prime})\sim(L,Q)$ such that
$(K^{\prime},P^{\prime})\preceq(L^{\prime},Q^{\prime})$. Clearly the converse
follows from the definition. To prove the forward implication, suppose
$(K,P)\preceq(L,Q)$ and $(K^{\prime},P^{\prime})\underset{(T,S)}{\sim}(L,Q)$
and consider

$E_{(L,Q)}\cdot\Big{(}\frac{G\times G}{S\trianglelefteq
T}\Big{)}=\Big{(}\frac{G\times G}{S^{\prime}\trianglelefteq
T^{\prime}}\Big{)}$

By Proposition 5.4(a), we have $l_{0}(T^{\prime},S^{\prime})=(L,Q)$. Putting
$r_{0}(T^{\prime},S^{\prime})=(L^{\prime},Q^{\prime})$, we get
$(L^{\prime},Q^{\prime})\sim(L,Q)$ and
$(K^{\prime},P^{\prime})\preceq(L^{\prime},Q^{\prime})$. With this definition,
we write

$e_{\\{K,P\\}_{G}}:=\sum_{(K^{\prime},P^{\prime})\in\\{K,P\\}_{G}}e_{(K^{\prime},P^{\prime})}\in\Gamma(G,G)$

and

$f_{\\{K,P\\}_{G}}:=\sum_{(K^{\prime},P^{\prime})\in\\{K,P\\}_{G}}f_{(K^{\prime},P^{\prime})}\in\Gamma(G,G)$

for $(K,P)\in\mathcal{G}_{G}$. The conditions of Proposition 2.7 are also
satisfied by the elements $e_{\\{K,P\\}_{G}}$ and $f_{\\{K,P\\}_{G}}$ as
$(K,P)$ runs over linkage classes in $\mathcal{G}_{G}$. Hence we obtain the
following corollary.

###### 5.11 Proposition

Let $(K,P),(L,Q)\in\mathcal{G}_{G}$. Then

$e_{\\{K,P\\}_{G}}f_{\\{L,Q\\}_{G}}=f_{\\{L,Q\\}_{G}}e_{\\{K,P\\}_{G}}=0\quad\mbox{unless}\quad\\{K,P\\}_{G}\preceq\\{L,Q\\}_{G}$

$e_{\\{K,P\\}_{G}}f_{\\{K,P\\}_{G}}=f_{\\{K,P\\}_{G}}e_{\\{K,P\\}_{G}}=f_{\\{K,P\\}_{G}}\quad\mbox{and}$

$f_{\\{K,P\\}_{G}}f_{\\{L,Q\\}_{G}}=\left\\{\begin{array}[]{l}f_{\\{K,P\\}_{G}},\quad\mbox{if}\quad\\{K,P\\}_{G}=\\{L,Q\\}_{G},\\\
0,\qquad\quad\mbox{otherwise}.\end{array}\right.$

###### 5.12

Covering algebra. As we have shown in Proposition 3.14, for any section
$(T,S)$ in $G\times G$, there is a factorization of $(\frac{G\times G}{S\unlhd
T})$ over $p_{1}(T)/k_{1}(S)$ and $p_{2}(T)/k_{2}(S)$. Hence while determining
the essential algebra $\bar{E}_{G}$, we can work with the smaller subalgebra
where this type of factorization is trivial. Following [2], we call the pair
$(T,S)\in\Sigma_{G\times H}$ covering if $p_{1}(T)=G,p_{2}(T)=H$ and
$k_{1}(S)=k_{2}(S)=1$, and denote the set of all such pairs by
$\Sigma_{G\times H}^{c}$. Notice that for a covering pair $(T,S)$, we have
$l_{0}(T,S)\in\mathcal{G}_{G}$ and $r_{0}(T,S)\in\mathcal{G}_{H}$. It is clear
that the elements $[\frac{G\times G}{S\unlhd T}]$ as $(T,S)$ runs over all
covering pairs generates a subalgebra of $E_{G}$. We denote it by $E_{G}^{c}$.
This is a covering algebra since by Proposition 3.14 any basis element outside
of $E_{G}^{c}$ factors through a group of smaller order.

###### 5.13

Next we aim to determine the covering algebra $E_{G}^{c}$ more explicitly so
that its intersection with the ideal $I_{G}$ is easier to unearth. For this
aim, we first decompose $E_{G}^{c}$ as a product of matrix algebras. Note that
for any $(K,P)\in\mathcal{G}_{G}$, the idempotents $e_{(K,P)}$ and $f_{(K,P)}$
are contained in $E_{G}^{c}$. The following theorem introduces certain groups
contained in $E_{G}^{c}$ together with special bisets. Later, as Theorem 5.18,
we show that the covering algebra is a direct sum of group algebras over these
groups. Some parts of this theorem are given as separate results in [2].

###### 5.14 Theorem

Let $G,H$ be finite groups and let $(K,P)\in\mathcal{G}_{G}$ and
$(L,Q)\in\mathcal{G}_{H}$ be linked pairs. Also let

$\Gamma_{(G,K,P)}=\left\\{\frac{1}{|G:P|}\left[\frac{G\times G}{S\unlhd
T}\right]\mid l(T,S)=(G,K,P,1)=r(T,S)\right\\}$

and

${}_{(G,K,P)}\Gamma_{(H,L,Q)}=\left\\{\left[\frac{G\times H}{U\unlhd
V}\right]\mid l(V,U)=(G,K,P,1),r(V,U)=(H,L,Q,1)\right\\}.$

Then

  1. (a)

The set $\Gamma_{(G,K,P)}$ is a finite group under the multiplication induced
by the product in $E_{G}$. The identity is $e_{(K,P)}$ and inverses are given
by taking opposites.

  2. (b)

The set ${{}_{(G,K,P)}}\Gamma_{(H,L,Q)}$ is a
$(\Gamma_{(G,K,P)},\Gamma_{(H,L,Q)})$-biset which is both left and right
transitive and left and right free.

  3. (c)

Any $\gamma\in{{}_{(G,K,P)}}\Gamma_{(H,L,Q)}$ induces a group isomorphism

$\gamma:\Gamma_{(H,L,Q)}\xrightarrow{\sim}\Gamma_{(G,K,P)}.$

  4. (d)

The functor

$k[_{(G,K,P)}\Gamma_{(H,L,Q)}]\otimes_{k\Gamma_{(H,L,Q)}}-:{}_{k\Gamma_{(H,L,Q)}}\rm{mod}\to{}_{k\Gamma_{(G,K,P)}}\rm{mod}$

is an equivalence of categories. It induces a canonical bijection

$\mathrm{Irr}(k\Gamma_{(H,L,Q)})\xrightarrow{\sim}\mathrm{Irr}(k\Gamma_{(G,K,P)}).$
(4)

  5. (e)

There is an isomorphism of groups

$\Gamma_{(G,K,P)}\cong\mathrm{Out}(P,G/K,i_{P})$

where the right hand side is the group of outer automorphisms of the crossed
module $(P,G/K,i_{P})$.

###### Proof.

All except the last claim are easy justifications, we leave the details to the
reader. We only prove the last part by constructing an isomorphism. Define

$\Theta:\mathrm{Aut}(P,G/K,i_{P})\to\Gamma_{(G,K,P)}$

associating $(\alpha,\beta)$ to the element in $\Gamma_{(G,K,P)}$
corresponding to the section $(T,S)$ where

$T=\\{(x,y)\mid xK=\beta(yK)\,\mbox{for}\,x,y\in G\\}$

and

$S=\\{(\alpha(p),p)\mid p\in P\\}$

Note that $(T,S)$ is indeed a section by Goursat’s theorem for sections. It is
easy to check that $\Theta$ is a group homomorphism. Moreover if $(T,S)$ is a
section of $G\times G$ with $l_{0}(T,S)=(K,P)=r_{0}(T,S)$ then by Goursat’s
Theorem for sections, we obtain an automorphism of the crossed module
$(P,G/K,\iota_{P})$. It remains to show that the kernel of $\Theta$ is the
group of inner automorphisms. ∎

As a preparation for the proof of Theorem 5.18, we need the following
technical results. Statements and proofs are very similar to the ones proved
in [2, Section 6]. We state our versions in this section without proofs and
collect all the proofs in the appendix. The following proposition is the
version of Lemma 6.3 and its corollary from [2].

###### 5.15 Proposition

With the above notation

  1. (a)

Let $(T,S)$ be covering and $(K,P),(L,Q)\in\mathcal{G}_{G}$. If
$f_{\\{K,P\\}_{G}}[\frac{G\times G}{S\unlhd T}]f_{\\{L,Q\\}_{G}}$ is non zero,
then $\\{K,P\\}_{G}=\\{L,Q\\}_{G}$.

  2. (b)

The set $\\{f_{\\{K,P\\}_{G}}\mid\\{K,P\\}_{G}\in\mathcal{G}_{G}/\sim\\}$ is a
set of mutually orthogonal central idempotents of $E_{G}^{c}$ and their sum is
1.

  3. (c)

The covering algebra $E_{G}^{c}$ decomposes into its two-sided ideals as

$E_{G}^{c}=\bigoplus_{\\{K,P\\}_{G}\in\mathcal{G}_{G}/\sim}f_{\\{K,P\\}_{G}}E_{G}^{c}.$
(5)

###### 5.16

We can also decompose $E_{G}^{c}$ into $k$-submodules via linkage classes. For
any $\\{K,P\\}_{G}\in\mathcal{G}_{G}/\sim$, let $E_{G}^{c,\\{K,P\\}}$ be the
submodule of $E_{G}^{c}$ generated by all $\big{[}\frac{G\times G}{S\unlhd
T}\big{]}$ with $(T,S)\in\Sigma_{G\times G}^{c}$ and $l_{0}(T,S)\sim_{G}(K,P)$
(or equivalently $r_{0}(T,S)\sim_{G}(K,P)$). Then

$E_{G}^{c}=\bigoplus_{\\{K,P\\}_{G}\in\mathcal{G}_{G}/\sim}E_{G}^{c,\\{K,P\\}}$
(6)

This decomposition is related to the ideal decomposition in the following way.
The following version of [2, Lemma 6.6] hides some parts of the result that
are needed for the proof. The proof is presented in Appendix A.

###### 5.17 Lemma

Let $(K,P)\in\mathcal{G}_{G}$.

  1. (a)

The following equality holds.

$\bigoplus_{\\{K,P\\}_{G}\preceq\\{L,Q\\}_{G}\in\mathcal{G}_{G}/\sim}E_{G}^{c}f_{\\{L,Q\\}_{G}}=\bigoplus_{\\{K,P\\}_{G}\preceq\\{L,Q\\}_{G}\in\mathcal{G}_{G}/\sim}E_{G}^{c,\\{L,Q\\}}$

  2. (b)

The projection

$\omega:E_{G}^{c,\\{K,P\\}}\to E_{G}^{c}f_{\\{K,P\\}_{G}},\;b\mapsto
bf_{\\{K,P\\}_{G}}$

with respect to the ideal decomposition of $E_{G}^{c}$ is an isomorphism of
$k$-modules. Its inverse is given by the projection with respect to the
submodule decomposition of $E_{G}^{c}$.

  3. (c)

If $\\{K,P\\}_{G}=\\{(K_{1},P_{1}),(K_{2},P_{2}),\ldots,(K_{n},P_{n})\\}$ then
$\omega$ becomes the direct sum, over $i$ and $j$, of the $k$-module
isomorphisms

$\omega_{ij}:k[{}_{(G,K_{i},P_{i})}\Gamma_{(G,K_{j},P_{j})}]\to
f_{(K_{i},P_{i})}E_{G}^{c}f_{(K_{j},P_{j})}$

given by $b_{ij}\mapsto f_{(K_{i},P_{i})}bf_{(K_{j},P_{j})}$.

  4. (d)

The isomorphism $\omega_{ii}$ as defined above induces an isomorphism of
$k$-algebras

$k[\Gamma_{(G,K,P)}]\to f_{(K,P)}E_{G}^{c}f_{(K,P)}$

given by $a\mapsto f_{(K,P)}af_{(K,P)}$.

Now we are ready to state the main result of this section. The proof is given
in Appendix A.

###### 5.18 Theorem

There exists a $k$-algebra isomorphism

$E_{G}^{c}\xrightarrow{\sim}\underset{\\{K,P\\}_{G}\in\mathcal{G}_{G}/\sim}{\bigoplus}$
Mat ${}_{|\\{K,P\\}_{G}|}k\Gamma_{(G,K,P)}$

with the following property:

For every
$\\{K,P\\}_{G}=\\{(K_{1},P_{1}),\cdots,(K_{n},P_{n})\\}\in\mathcal{G}_{G}/\sim$,
the isomorphism maps $f_{(K_{i},P_{i})}\in E_{G}^{c}$ to the idempotent matrix
$e_{i}=\mbox{diag}(0,\cdots,0,1,0,\cdots,0)\in\mbox{Mat}_{|\\{K,P\\}_{G}|}(k\Gamma_{(G,K,P)})$,
for $i=1,\cdots,|\\{K,P\\}_{G}|$, in the $\\{K,P\\}_{G}$-component.

##  6 The essential algebra ${\bar{E}}_{G}$

Let $k$ be a field of characteristic zero, and fix a finite group $G$. In this
section we determine the essential algebra ${\bar{E}}_{G}$ and its simple
modules. Once more the results in this section are similar to the results in
[2, Section 8].

###### 6.1

As in the case of fibered biset functors, the essential algebra is isomorphic
to a subalgebra of the covering algebra. Generators of this subalgebra are
characterized as follows. Let $(K,P)\in\mathcal{G}_{G}$. We call $(K,P)$ a
reduced pair if $e_{K,P}$ is not contained in the essential ideal $I_{G}$. We
denote the subset of $\mathcal{G}_{G}$ consisting of the reduced pairs by
$\mathcal{R}_{G}=\mathcal{R}_{k}(G)$. It is easy to prove that being reduced
is compatible with being linked, that is, if
$(K,P),(K^{\prime},P^{\prime})\in\mathcal{G}_{G}$ are $G$-linked then $(K,P)$
is reduced if and only if $(K^{\prime},P^{\prime})$ is reduced. (cf. [2,
Notation 8.1]) We write $\widetilde{\mathcal{R}}_{G}$ for the set of linkage
classes of reduced pairs for $G$. Note also that $\mathcal{R}_{G}$ is a lower
set, that is, if $(K,P)$ is reduced and $(K^{\prime},P^{\prime})\preceq(K,P)$,
then $(K^{\prime},P^{\prime})$ is also reduced.

The following lemma determines a basis for the essential ideal. Its proof is
similar to the proof of [2, Lemma 8.2] and our version is given in Appendix A.

###### 6.2 Lemma

The ideal $I_{G}$ of $E_{G}$ is generated as a $k$-module by the standard
basis elements $\big{[}\frac{G\times G}{S\unlhd T}\big{]}$ with
$(T,S)\in\Sigma_{G\times G}$ satisfying

  1. ($i$)

$p_{1}(T)\neq G$ and $k_{1}(S)\neq 1$ or

  2. ($ii$)

$p_{1}(T)=G$, $k_{1}(S)=1$ and $l_{0}(T,S)\not\in\mathcal{R}_{G}$

Equivalently, it is generated by the standard basis elements
$\big{[}\frac{G\times G}{S\unlhd T}\big{]}$ with $(T,S)\in\Sigma_{G\times G}$
satisfying

  1. ($i^{\prime}$)

$p_{2}(T)\neq G$ and $k_{2}(S)\neq 1$ or

  2. ($ii^{\prime}$)

$p_{2}(T)=G$, $k_{2}(S)=1$ and $r_{0}(T,S)\not\in\mathcal{R}_{G}$

Now we are ready to determine the intersection of the covering algebra and the
ideal $I_{G}$. This also reveals the structure of the essential algebra. The
proof is given in Appendix A.

###### 6.3 Proposition

  1. (a)

The algebra $E_{G}^{c}$ is a covering algebra for $E_{G}$, in the sense of
Section 2.5.

  2. (b)

The equality

$E_{G}^{c}\cap
I_{G}=\bigoplus_{\begin{smallmatrix}\\{K,P\\}_{G}\in\mathcal{G}_{G}/\sim&\\\
(K,P)\not\in\mathcal{R}_{G}\end{smallmatrix}}f_{\\{K,P\\}_{G}}E_{G}^{c}$ (7)

holds.

  3. (c)

The canonical epimorphism $E_{G}\rightarrow\bar{E}_{G}$ maps the subalgebra

$\bigoplus_{\\{K,P\\}_{G}\in\widetilde{\mathcal{R}}_{G}}f_{\\{K,P\\}_{G}}E_{G}^{c}$

of $E_{G}^{c}$ isomorphically onto $\bar{E}_{G}$.

  4. (d)

For each $(K,P)\in\mathcal{R}_{G}$, the map

$k\Gamma_{(G,K,P)}\rightarrow\bar{f}_{(K,P)}\bar{E}_{G}\bar{f}_{(K,P)},\quad
a\mapsto\bar{f}_{(K,P)}\bar{a}\bar{f}_{(K,P)}$

is a $k$-algebra isomorphism.

  5. (e)

There is a bijective correspondence between the isomorphism classes of simple
$\bar{E}_{G}$-modules and the set of triples $(K,P,[V])$ where $(K,P)$ runs
over linkage classes of reduced pairs for $G$ and for a representative $(K,P)$
of the linkage class of $(K,P)$, $[V]$ runs over irreducible
$k\Gamma_{(G,K,P)}$-modules. The correspondence is given by associating the
triple $(K,P,[V])$ to the $\bar{E}_{G}$-module
$\bar{E}_{G}\bar{f}_{(K,P)}\otimes_{k\Gamma_{(G,K,P)}}V$.

With this result it is important to know more explicit results about reduced
pairs. Unfortunately we do not have a complete description of these pairs. The
following theorem includes some partial results.

###### 6.4 Theorem

Let $(K,P)\in\mathcal{G}_{G}$.

  1. (a)

The pair $(K,P)$ is reduced if $K\leqslant P$.

  2. (b)

If $(K,P)$ is reduced, then for any non-trivial $N\unlhd G$ with $N\leqslant
K$, we have $P\cap N\neq 1$.

  3. (c)

The pair $(K,P)$ is not reduced if there is a group $H$ and a pair
$(L,Q)\in\mathcal{G}_{H}$ such that $|H|<|G|$ and $(G,K,P,1)\sim(H,L,Q,1)$.

###### Proof.

  1. (a)

Assume for contradiction that $(K,P)$ is not reduced. Hence $e_{(K,P)}\in
I_{G}$, that is there exist a finite group $H$ with order strictly less than
the order of $G$ and sections $(T,S)$ and $(V,U)$ in $G\times H$ and $H\times
G$ respectively such that $e_{(K,P)}$ occurs as a summand in
$\Big{[}\frac{G\times H}{S\unlhd
T}\Big{]}\underset{H}{\cdot}\Big{[}\frac{H\times G}{U\unlhd V}\Big{]}$. Then
by the Mackey formula there is a section of $H\times G$, say
$(V^{\prime},U^{\prime})$, such that
$(S*U^{\prime},T*V^{\prime})=(\Delta_{K}(G),\Delta(P))$. Thus it follows that
$l(T,S)=(G,K^{\prime},P^{\prime},1)$ for some
$(K^{\prime},P^{\prime})\preceq(K,P)$. Now by Proposition 5.4(b), we have
$e_{(K,P)}\underset{G}{\cdot}\Big{[}\frac{G\times H}{S\unlhd
T}\Big{]}=\Big{[}\frac{G\times H}{S^{\prime}\unlhd T^{\prime}}\Big{]}$ with
$(T^{\prime},S^{\prime})\in\Sigma_{G\times H}$ satisfying
$l(T^{\prime},S^{\prime})=(G,K,P,1)$. Finally Proposition 3.8(c) applied to
$(T^{\prime},S^{\prime})$ implies that $|G|\leqslant|H|$, a contradiction.

  2. (b)

Let $(K,P)\in\mathcal{R}_{G}.$ Assume, for contradiction, that there exist
$N\unlhd G$ with $N\leqslant K$ such that $P\cap N=1$. Let
$x=\Big{[}\mathrm{Inf}_{G/K}^{G}\mathrm{Iso}(\eta_{T})\mathrm{Def}_{\frac{G/N}{K/N}}^{G/N},\mathrm{Ind}_{P}^{G}\mathrm{Iso}(\eta_{S})\mathrm{Res}^{G/N}_{PN/N}\Big{]}$
and
$y=\Big{[}\mathrm{Inf}_{\frac{G/N}{K/N}}^{G/N}\mathrm{Iso}(\eta_{T}^{-1})\mathrm{Def}_{G/K}^{G},\mathrm{Ind}^{G/N}_{PN/N}\mathrm{Iso}(\eta_{S}^{-1})\mathrm{Res}_{P}^{G}\Big{]}$,
with the canonical isomorphisms
$\eta_{T}:\frac{G/N}{K/N}\xrightarrow{\sim}G/K$ and
$\eta_{S}:PN/N\xrightarrow{\sim}P$. Then
$e_{(K,P)}=\frac{1}{|G:P|}x\cdot_{G/N}\frac{1}{|G:PN|}y\in I_{G}$, which is a
contradiction since $|G/N|<|G|$.

  3. (c)

This part is straightforward by the definition of linkage.

∎

###### 6.5 Corollary

The pair $(K,P)\in\mathcal{G}_{G}$ is reduced in each of the following cases:

  1. (a)

$K=1$.

  2. (b)

$P=G$

In particular the essential algebra $\bar{E}_{G}$ is non-zero for any finite
group $G$.

###### 6.6 Corollary

The pair $(K,P)\in\mathcal{G}_{G}$ is not reduced in each of the following
cases:

  1. (a)

$P<K$.

  2. (b)

$PK=G$ and $K\not\leqslant P$.

###### Proof.

In both cases it is sufficient to find a triple $(H,L,Q)$ with $|H|<|G|$ which
is linked to $(G,K,P)$.

  1. (a)

Let $G/K$ act on $P$ by conjugation and construct the semi-direct product
$\bar{G}=P\rtimes G/K$. Write $\bar{P}$ for the image (under
$\alpha^{-1}:x\mapsto(x,1)$) of $P$ in $\bar{G}$ so that we have an
isomorphism $\beta:\bar{G}/\bar{P}\xrightarrow{\sim}G/K$.

We define the subgroups $S=\\{(x,(x,1)):x\in P\\}$ and
$T=\\{(g,(x,yK)):gK=\beta((x,yK)\bar{P})\\}$ of $G\times\bar{G}$ and claim
that $S\unlhd T$. Indeed we clearly have $S\leqslant T$ and the normality
follows by direct calculations. Also note that $P<K$ and $[K,P]=1$, hence $P$
is abelian. The Goursat correspondents of $T$ and $S$ are
$(G,K,\beta,\bar{P},\bar{G})$ and $(P,1,\alpha,1,\bar{P})$ respectively. Hence
the quadruples $(G,K,P,1)$ and $(\bar{G},\bar{P},\bar{P},1)$ are linked, as
required.

  2. (b)

Let $S=\Delta(P)\leqslant G\times P$ and $T=\\{(g,h)\in G\times P|g^{-1}h\in
K\\}$. Clearly $S$ is a normal subgroup of $T$. Since $G=PK$, the pair $(T,S)$
is covering and hence $(G,K,P)$ is linked to $(P,K\cap P,P)$. Note that
$K\not\leqslant P$ is also necessary since otherwise $G=PK$ implies $G=P$ and
the factorization is not over a group of smaller order.

∎

###### 6.7

Finally we parameterize simple modules over $\bar{E}_{G}$. Let

$\mathcal{S}_{G}=\mathcal{S}_{k}(G):=\\{((K,P),[V])\mid(K,P)\in\mathcal{R}_{G},[V]\in\mathrm{Irr}(k\Gamma_{(G,K,P)})\\}.$

Then define an equivalence relation on $\mathcal{S}_{G}$ by
$((K,P),[V])\sim((K^{\prime},P^{\prime}),[V^{\prime}])$ if $(K,P)$ and
$(K^{\prime},P^{\prime})$ are $G$-linked and the canonical bijection
$\mbox{\rm
Irr}(k\Gamma_{(G,K^{\prime},P^{\prime})})\xrightarrow{\sim}\mbox{\rm
Irr}(k\Gamma_{(G,K,P)})$ from Theorem 5.14 maps $[V^{\prime}]$ to $[V]$. Also
let $\tilde{\mathcal{S}}_{G}$ be a set of representatives of the equivalence
classes of $\mathcal{S}_{G}$, that is,

$\tilde{\mathcal{S}}_{G}:=\\{((K,P),[V])\mid(K,P)\in\tilde{\mathcal{R}}_{G},[V]\in\mbox{\rm
Irr}(k\Gamma_{(G,K,P)})\\}.$

By the canonical isomorphism from Proposition 6.3(c), we can view each simple
$k\Gamma_{(G,K,P)}$-module as a simple
$\bar{f}_{(K,P)}\bar{E}_{G}\bar{f}_{(K,P)}$-module, and we can view
$\bar{E}_{G}\bar{f}_{(K,P)}$ as $(\bar{E}_{G},k\Gamma_{(G,K,P)})$-bimodule.
Hence $\bar{E}_{G}\bar{f}_{(K,P)}\otimes_{k\Gamma_{(G,K,P)}}V$ is a simple
$\bar{E}_{G}$-module for each simple $k\Gamma_{(G,K,P)}$-module. This induces
a bijection between the set $\tilde{\mathcal{S}}_{G}$ and Irr$(\bar{E}_{G})$.

##  7 Simples functors over $k\Gamma$

In this final section we parameterize simple section biset functors. Once more
we follow [2]. In the previous section we have classified simple modules over
the endomorphism ring $E_{G}$ for any finite group $G$. This gives us the map
from the set of seeds to the set of isomorphism classes of simple section
biset functors. In this section we introduce an equivalence relation on seeds
to make this map bijective.

###### 7.1

By the parametrization in Section 6.7, we write the set of all seeds for
$k\Gamma$ as

${\rm{Seeds}}(k\Gamma)=\\{(G,K,P,[V])|G\in
Ob(\mathcal{P}_{\Gamma}),(K,P,[V])\in\tilde{\mathcal{S}}_{G}\\}.$

Given a seed $(G,K,P,[V])$, we construct a simple $E_{G}$-module
$\widetilde{V}$ by

$\widetilde{V}=\bar{E}_{G}\bar{f}_{(K,P)}\otimes_{k\Gamma_{(G,K,P)}}V.$

Hence as explained in Section 2.4, we associate a simple section functor
$S_{G,\widetilde{V}}=S_{G,K,P,[V]}$ to the seed $(G,K,P,[V])$. Clearly
replacing $V$ with another isomorphic copy of $V$ does not change the
isomorphism type of the corresponding simple section biset functor. In
particular we obtain a function

$\omega:{\rm Seeds}(k\Gamma)\to{\rm Irr}(k\Gamma)$

where Irr$(k\Gamma)$ denotes the set of isomorphism classes of simple section
biset functors. Note that $\omega$ is surjective. Indeed if $S$ is a simple
section biset functor, we let $G$ be a minimal group for $S$. Then $S(G)$ is a
simple module for the essential algebra $\bar{E}_{G}$, and hence there is a
triple $(K,P,[V])\in\tilde{\mathcal{S}}_{G}$ such that
$S(G)\cong\bar{E}_{G}\bar{f}_{(K,P)}\otimes_{k\Gamma_{(G,K,P)}}V$. Therefore
$S\cong S_{G,K,P,[V]}$.

###### 7.2

As in [2], the relation on Seeds$(k\Gamma)$ given by

$(G,K,P,[V])\sim(H,L,Q,[W])$

if and only if $(G,K,P,1)\sim(H,L,Q,1)$ and $V\cong
k[{}_{(G,K,P)}\Gamma_{(H,L,Q)}]\otimes_{k\Gamma_{(H,L,Q)}}W$ is an equivalence
relation. If these conditions are satisfied, we say that the quadruples
$(G,K,P,[V])$ and $(H,L,Q,[W])$ are _linked_. We claim that the function
$\widetilde{\omega}$ induced by $\omega$ on the set of linkage classes
Seeds$(k\Gamma)/\sim$ of seeds is bijective. To prove this claim, we follow
the techniques from [2]. Ideas used in the proof below can be found in [2,
Section 9].

###### 7.3

Let $M$ be a section biset functor and $G$ and $H$ be finite groups. We have
decompositions

$M(G)=\bigoplus_{(K,P)\in\mathcal{G}_{G}}f_{(K,P)}M(G),\quad
M(H)=\bigoplus_{(L,Q)\in\mathcal{G}_{H}}f_{(L,Q)}M(H).$

Although these decompositions may not be related to each other, in general,
the terms corresponding to linked quadruples are isomorphic. Indeed suppose
there are pairs $(K,P)\in\mathcal{G}_{G}$ and $(L,Q)\in\mathcal{G}_{H}$ such
that the triples $(G,K,P)$ and $(H,L,Q)$ are linked so that the set
${}_{(G,K,P)}\Gamma_{(H,L,Q)}$ is non-empty. Then, for each
$x\in{}_{(G,K,P)}\Gamma_{(H,L,Q)}$, the map

$f_{(K,P)}M(G)\to f_{(L,Q)}M(H),m\mapsto|G:P|^{-1}x\cdot m$

is an isomorphism of $k$-modules with the inverse given by multiplication by
$x^{\rm op}$ (cf. [2, Lemma 9.3]). This follows easily since $xx^{\rm
op}=e_{(K,P)}$ and $x^{\rm op}x=e_{(L,Q)}$.

###### 7.4

Let $(H,L,Q,[W])$ be a seed and $S_{H,\widetilde{W}}$ be the corresponding
simple section biset functor so that

$S_{H,\widetilde{W}}(H)=\widetilde{W}=\bar{E}_{G}\bar{f}_{(L,Q)}\otimes_{k\Gamma_{(H,L,Q)}}W.$

Note that since there is an isomorphism of $k$-algebras
$k\Gamma_{(H,L,Q)}\cong f_{(L,Q)}E_{H}^{c}f_{(L,Q)}$, by Lemma 5.17, we may
regard $f_{(L,Q)}S_{H,\widetilde{W}}(H)$ as a $k\Gamma_{(H,L,Q)}$-module, and
as such, it is isomorphic to $W$.

In order to determine other seeds which corresponds to the functor
$S_{(H,\widetilde{W})}$, we need to know the evaluations
$S_{H,\widetilde{W}}(G)$ for groups with $|G|=|H|$. Hence let $G$ be a group
of order $|H|$. Clearly there is a seed of the form $(G,K,P,[V])$
corresponding to $S_{H,\widetilde{W}}$ only if $S_{H,\widetilde{W}}(G)$ is
non-zero, which we assume from now on. Then since $H$ is a minimal group for
$S_{H,\widetilde{W}}$, $G$ is also minimal. In particular
$S_{H,\widetilde{W}}(G)$ is annihilated by $I_{G}$, and hence it is a simple
$\bar{E}_{G}$-module. Put $\widetilde{V}=S_{H,\widetilde{W}}(G)$. By Section
6.7, there is a triple $(K,P,[V])$ such that
$\widetilde{V}\cong\bar{E}_{G}\bar{f}_{(K,P)}\otimes_{k\Gamma_{(G,K,P)}}V$.
Hence $\bar{f}_{(K,P)}S_{H,\widetilde{W}}(G)\cong V$ and
$S_{H,\widetilde{W}}\cong S_{G,\widetilde{V}}$.

Now by Lemma A.7, we must have a surjective homomorphism

$k{}_{(G,K,P)}\Gamma_{(H,L,Q)}\otimes_{k\Gamma_{(H,L,Q)}}W\to V$

of $k\Gamma_{(G,K,P)}$-modules. In particular the set
${}_{(G,K,P)}\Gamma_{(H,L,Q)}$ is non-empty, which implies that the quadruples
$(G,K,P,1)$ and $(H,L,Q,1)$ are linked. Moreover by Theorem 5.14, the
$k\Gamma_{(G,K,P)}$-module
$k{}_{(G,K,P)}\Gamma_{(H,L,Q)}\otimes_{k\Gamma_{(H,L,Q)}}W$ is simple. Hence
we must have an isomorphism

$k{}_{(G,K,P)}\Gamma_{(H,L,Q)}\otimes_{k\Gamma_{(H,L,Q)}}W\cong V$

of $k\Gamma_{(G,K,P)}$-modules. As a result the seeds $(G,K,P,[V])$ and
$(H,L,Q,[W])$ are linked, that is, seeds corresponding to the simple functor
$S_{H,\widetilde{W}}$ are all linked to $(H,L,Q,[W])$.

###### 7.5

Finally we show that if $(G,K,P,[V])$ and $(H,L,Q,[W])$ are linked then
$S_{G,\widetilde{V}}\cong S_{H,\widetilde{W}}$. First note that by Lemma A.6,
the groups $G$ and $H$ are of the same order. As discussed above, the
evaluation $S_{H,\widetilde{W}}(G)$ is a simple $\bar{E}_{G}$-module. Also
since $(G,K,P,1)\sim(H,L,Q,1)$, we get
$\bar{f}_{(K,P)}S_{H,\widetilde{W}}(G)\cong\bar{f}_{(L,Q)}S_{H,\widetilde{W}}(H)$
as $k$-modules, by Section 7.3. In particular
$\bar{f}_{(K,P)}S_{H,\widetilde{W}}(G)$ is non-zero. It remains to show that
there is an isomorphism $S_{H,\widetilde{W}}(G)\cong\widetilde{V}$ of
$\bar{E}_{G}$-modules. We already know that

$V\cong k{}_{(G,K,P)}\Gamma_{(H,L,Q)}\otimes_{k\Gamma_{(H,L,Q)}}W.$

Hence by Lemma A.7, there is a homomorphism
$V\to\bar{f}_{(K,P)}S_{H,\widetilde{W}}(G)$ of $k\Gamma_{(G,K,P)}$-modules.
Now we have the following isomorphisms.

$\displaystyle\\{0\\}\not=\mathrm{Hom}_{k\Gamma_{(G,K,P)}}(V,\bar{f}_{(K,P)}S_{H,\widetilde{W}}(G))\cong\mathrm{Hom}_{k\Gamma_{(G,K,P)}}(V,\mathrm{Hom}_{\bar{E}_{G}}(\bar{E}_{G}\bar{f}_{(K,P)},S_{H,\widetilde{W}}(G)))$
$\displaystyle\cong\mathrm{Hom}_{\bar{E}_{G}}(\bar{E}_{G}\bar{f}_{(K,P)}\otimes_{k\Gamma_{(G,K,P)}}V,S_{H,\widetilde{W}}(G))\cong\mathrm{Hom}_{\bar{E}_{G}}(\widetilde{V},S_{H,\widetilde{W}}(G)).$

In particular there is a non-zero homomorphism $\widetilde{V}\to
S_{H,\widetilde{W}}(G)$. Since both of these modules are simple, they must be
isomorphic. With this step, we have completed the proof of following theorem.

###### 7.6 Theorem

There is a bijective correspondence between

  1. (a)

the set Irr$(k\Gamma)$ of isomorphism classes of simple section biset functors

  2. (b)

the set Seeds$(k\Gamma)/\sim$ of linkage classes of quadruples $(G,K,P,[V])$.

###### 7.7 Remark

Let $G$ and $H$ be non-isomorphic finite groups. For reduced pairs
$(K,P)\in\mathcal{R}_{G}$ and $(L,Q)\in\mathcal{R}_{H}$ if there exists a
section $(T,S)$ such that $(G,K,P,1)\underset{(T,S)}{\sim}(H,L,Q,1)$ then we
can take $V:=k{}_{(G,K,P)}\Gamma_{(H,L,Q)}\otimes_{k\Gamma_{(H,L,Q)}}W$ to be
the irreducible $k\Gamma_{(G,K,P)}$-module corresponding to
$W\in\mathrm{Irr}(k\Gamma_{(H,L,Q)})$. Then $S_{G,\widetilde{V}}\cong
S_{H,\widetilde{W}}$, which shows that there exists simple section biset
functors with non-isomorphic minimal groups.

For example let $G$ be the group of quaternions, $\langle
x,y\,|\,x^{4}=1,\,yxy^{-1}=x^{-1},\,x^{2}=y^{2}\rangle$ and $H:=\langle
a,b\,|\,a^{4}=b^{2}=1,\,bab^{-1}=a^{-1}\rangle$ the dihedral group of order 8.
Consider the section $(T,S)$ of $G\times H$ given by

$T:=\langle(x,a),(y,b)\rangle\,\rm{and}\,S:=\langle(x,a)\rangle$

Clearly $S$ is a subgroup of $T$ and since ${}^{(y,b)}(x,a)=(x^{-1},a^{-1})$
moreover $S$ is normal in $T$ and hence $(T,S)$ is a section of $G\times H$.
The left and right invariant of $(T,S)$ are as follows

$l(T,S)=(G,\langle x^{2}\rangle,\langle x\rangle,1)\quad r(T,S)=(H,\langle
a^{2}\rangle,\langle a\rangle,1).$

So $(K,P):=l_{0}(T,S)=(\langle x^{2}\rangle,\langle
x\rangle)\in\mathcal{G}_{G}$ and $(L,Q):=r_{0}(T,S)=(\langle
a^{2}\rangle,\langle a\rangle)\in\mathcal{G}_{H}$ are linked pairs. Moreover
by Theorem 6.4 (a) both $(K,P)$ and $(L,Q)$ are reduced pairs.

##  Appendix A Proofs

###### A.1

Proof of 5.15.

  1. (a)

By definition $f_{\\{K,P\\}}$ is a sum of $f_{(K^{\prime},P^{\prime})}$’s
where $(K^{\prime},P^{\prime})$ runs through the linkage class of
$\\{K,P\\}_{G}$. So for the product given in (a) being non-zero there should
be at least a pair say $(K^{\prime},P^{\prime})$ in the linkage class
$\\{K,P\\}_{G}$ and a pair $(L^{\prime},Q^{\prime})$ in the linkage class of
$\\{L,Q\\}_{G}$ such that the following product is non-zero
$f_{(K^{\prime},P^{\prime})}[\frac{G\times G}{S\unlhd
T}]f_{(L^{\prime},Q^{\prime})}$. Moreover, $f_{(L^{\prime},Q^{\prime})}$ is
the sum of $e_{(L^{\prime\prime},Q^{\prime\prime})}$’s where
$(L^{\prime\prime},Q^{\prime\prime})$ runs through all poset elements of
$\mathcal{G}_{G}$ which are bigger than $(L^{\prime},Q^{\prime})$. Then by the
similar reasoning $f_{(K^{\prime},P^{\prime})}[\frac{G\times G}{S\unlhd
T}]e_{(L^{\prime\prime},Q^{\prime\prime})}$ must be non-zero at least for some
$(L^{\prime},Q^{\prime})\preceq(L^{\prime\prime},Q^{\prime\prime})$. Hence we
have $[\frac{G\times G}{S\unlhd
T}]e_{(L^{\prime\prime},Q^{\prime\prime})}\not=0$, and by the Proposition 5.4
its in the form $[\frac{G\times G}{U\unlhd V}]$ for some
$(V,U)\in\Sigma_{G}^{c}$. Also by the same proposition we know that the right
middle invariant of section $(V,U)$ is bigger than
$(L^{\prime\prime},Q^{\prime\prime})$. Let indicate by
$(K^{\prime\prime},P^{\prime\prime})$ the left middle invariant of the section
$(V,U)$. Then
$f_{(K^{\prime},P^{\prime})}e_{(K^{\prime\prime},P^{\prime\prime})}[\frac{G\times
G}{S\unlhd
T}]e_{(L^{\prime\prime},Q^{\prime\prime})}=f_{(K^{\prime},P^{\prime})}[\frac{G\times
G}{S\unlhd T}]e_{(L^{\prime\prime},Q^{\prime\prime})}\not=0$ and so
$f_{(K^{\prime},P^{\prime})}e_{(K^{\prime\prime},P^{\prime\prime})}\not=0$,
which implies
$(K^{\prime\prime},P^{\prime\prime})\preceq(K^{\prime},P^{\prime})$ by
Proposition 5.6. If we gather it all together,

$\\{L,Q\\}_{G}=\\{L^{\prime},Q^{\prime}\\}_{G}\preceq\\{L^{\prime\prime},Q^{\prime\prime}\\}_{G}\preceq\\{r_{0}(V,U)\\}_{G}=\\{l_{0}(V,U)\\}_{G}=\\{K^{\prime\prime},P^{\prime\prime}\\}_{G}\preceq\\{K^{\prime},P^{\prime}\\}_{G}=\\{K,P\\}_{G}.$

Note that by the similar reasoning we also can derive
$\\{K,P\\}_{G}\preceq\\{L,Q\\}_{G}$ and hence there must be an equality.

  2. (b)

By the Proposition 5.11, for $\\{K,P\\}_{G}\in\mathcal{G}_{G}/\sim$, the
elements $f_{\\{K,P\\}_{G}}$ are the mutually orthogonal idempotents of the
covering algebra. Also their sum is equal to the sum of all $f_{(K,P)}$ where
$(K,P)$ run through all the poset elements of $\mathcal{G}_{G}$ and this sum
is equal to $1$ by the equation (3). Hence for any $b\in E_{G}^{c}$

$\sum_{\\{K,P\\}_{G}\in\mathcal{G}_{G}/\sim}f_{\\{K,P\\}_{G}}b=b=\sum_{\\{K,P\\}_{G}\in\mathcal{G}_{G}/\sim}bf_{\\{K,P\\}_{G}}$

and by the previous case we obtain $f_{\\{K,P\\}_{G}}b=bf_{\\{K,P\\}_{G}}$ for
every $\\{K,P\\}_{G}\in\mathcal{G}_{G}/\sim$.

  3. (c)

This is immediate from case $(b)$.

###### A.2

Proof of 5.17. Using Proposition 5.6 one can derive that the product
$[\frac{G\times H}{S\unlhd T}]f_{(L,Q)}$ is zero for any section
$(T,S)\in\Sigma_{G\times G}^{c}$ with $r_{0}(T,S)=(K,P)$ and
$(K,P)\npreceq(L,Q)\in\mathcal{G}_{G}$. Similarly one can get its left-sided
version too.

  1. (a)

To prove the equality first note that by the Proposition 5.15 the left hand
side equals the annihilator of the set of all
$f_{\\{K^{\prime},P^{\prime}\\}_{G}}$ with
$\\{K^{\prime},P^{\prime}\\}_{G}\nsucceq\\{K,P\\}_{G}$. On the other hand by
the observation above the right hand side also annihilates
$f_{\\{K^{\prime},P^{\prime}\\}_{G}}$ for all
$\\{K^{\prime},P^{\prime}\\}_{G}\nsucceq\\{K,P\\}_{G}$, which shows that the
right hand side is contained in the left hand side. But right hand side
obviously contains the left hand side since it contains $f_{\\{L,Q\\}_{G}}$
for every $\\{L,Q\\}_{G}\preceq\\{K,P\\}_{G}$.

  2. (b)

By the same way we proved previous case one can also show that

$\bigoplus_{\\{K,P\\}_{G}\prec\\{L,Q\\}_{G}\in\mathcal{G}_{G}/\sim}E_{G}^{c}f_{\\{L,Q\\}_{G}}=\bigoplus_{\\{K,P\\}_{G}\prec\\{L,Q\\}_{G}\in\mathcal{G}_{G}/\sim}E_{G}^{c,\\{L,Q\\}}.$

Hence $E_{G}^{c,\\{K,P\\}}$ and $E_{G}^{c}f_{\\{K,P\\}_{G}}$ are both
complements to the same submodule given above. So $\omega$ is an isomorphism
of $k$-modules.

  3. (c)

By definition, and since $f_{\\{K,P\\}_{G}}$ is central in $E_{G}^{c}$ we have
a direct sum decomposition

$E_{G}^{c,\\{K,P\\}}=\bigoplus_{i,j=1}^{n}k[_{(G,K_{i},P_{i})}\Gamma_{(G,K_{j},P_{j})}]$
(8)

and

$E_{G}^{c}f_{\\{K,P\\}_{G}}=f_{\\{K,P\\}_{G}}E_{G}^{c}f_{\\{K,P\\}_{G}}=\bigoplus_{i,j=1}^{n}f_{\\{K_{i},P_{i}\\}_{G}}E_{G}^{c}f_{\\{K_{j},P_{j}\\}_{G}}.$

into $k$-submodules. Moreover for $b_{ij}\in
k[_{(G,K_{i},P_{i})}\Gamma_{(G,K_{j},P_{j})}]$ using the fact we observed in
the beginning of the proof its clear that

$\omega(b_{ij})=f_{\\{K,P\\}_{G}}b_{ij}f_{\\{K,P\\}_{G}}=f_{(K_{i},P_{i})}b_{ij}f_{(K_{j},P_{j})}\in
f_{(K_{i},P_{i})}E_{G}^{c}f_{(K_{j},P_{j})}.$

Then each $\omega_{ij}$ is an isomorphism of $k$-modules since $\omega$ is an
isomorphism of $k$-modules.

  4. (d)

Let $a$ and $b$ be arbitrarily chosen elements in the $k$-algebra
$k[\Gamma_{(G,K,P)}]$. Also enumerate $\\{K,P\\}_{G}$ as in the previous case,
and choose $(K,P)=(K_{i},P_{i})$. Using the fact that $f_{\\{K,P\\}_{G}}$ is
central and $f_{(K_{i},P_{i})}$’s are orthogonal idempotents in $E_{G}^{c}$ we
have

$\omega_{ii}(a)\omega_{ii}(b)=f_{(K,P)}af_{(K,P)}bf_{(K,P)}=f_{(K,P)}af_{\\{K,P\\}_{G}}bf_{(K,P)}=f_{(K,P)}f_{\\{K,P\\}_{G}}abf_{(K,P)}=f_{(K,P)}abf_{(K,P)}=\omega_{ii}(ab).$

###### A.3

Proof of 5.18. In the Proposition 5.15 we have given a decomposition of
$E_{G}^{c}$ into two-sided ideals. So it suffices to show that there exists a
$k$-algebra isomorphism between $E_{G}^{c}f_{\\{K,P\\}_{G}}$ and
Mat${}_{n}(k\Gamma_{(G,K,P)})$. Aiming this we construct a map from
Mat${}_{n}(k\Gamma_{(G,K,P)})$ to $E_{G}^{c,\\{K,P\\}}$ and since we already
have a $k$-algebra isomorphisms $\omega$, composing this two maps we get
desired isomorphism.

Enumerate the elements of $\\{K,P\\}_{G}$ as in the Lemma 5.17. Then chose an
element $(T_{i},S_{i})\in\Sigma_{G\times G}^{c}$ with
$l_{0}(T_{i},S_{i})=(K,P)$ and $r_{0}(T_{i},S_{i})=(K_{i},P_{i})$, and set
$x_{i}:=[\frac{G\times G}{S\unlhd T}]$. Denote by
$y_{i}=\frac{1}{|G:P_{i}|}x_{i}$ and by
$\bar{y_{i}}=\frac{1}{|G:P|}x_{i}^{op}$. Then its an easy calculation to show
that $y_{i}\underset{G}{\cdot}\bar{y_{i}}=e_{(K,P)}$ and
$\bar{y_{i}}\underset{G}{\cdot}y_{i}=e_{(K_{i},P_{i})}$. Hence the maps
$\sigma_{ij}:k\Gamma_{(G,K,P)}\rightarrow
k[_{(G,K_{i},P_{i})}\Gamma_{(G,K_{j},P_{j})}],\;a\mapsto\bar{y_{i}}ay_{j}$ are
$k$-module isomorphisms with the inverse maps $b\mapsto y_{i}b\bar{y_{j}}$.
Taking the direct sum of the maps $\sigma_{ij}$ and using the direct sum
decomposition in (8) we obtain a $k$-module isomorphism
$\sigma:Mat_{n}(k\Gamma_{(G,K,P)})\rightarrow E_{G}^{c,\\{K,P\\}}$. Infact it
is a $k$-algebra isomorphism. So we have a desired isomorphism
$\omega\circ\sigma$ of $k$-algebras, which maps diagonal idempotent matrix
$e_{i}$ to $f_{(K_{i},P_{i})}$.

###### A.4

Proof of 6.2 Let $(T,S)\in\Sigma_{G\times G}$. If $(T,S)$ satisfies $(i)$ then
by the decomposition in Proposition 3.14 we have $[\frac{G\times G}{S\unlhd
T}]\in I_{G}$. If it satisfies $(ii)$ then
$(K,P):=l_{0}{(T,S)}\not\in\mathcal{R}_{G}$, which by definition means that
$e_{(K,P)}\in I_{G}$, but then since $I_{G}$ is an ideal $[\frac{G\times
G}{S\unlhd T}]=e_{(K,P)}[\frac{G\times G}{S\unlhd T}]\in I_{G}$.

Conversely, to prove that every element in $I_{G}$ can be written as a
$k$-linear combination of elements as given in the lemma it suffices to show
that $[\frac{G\times H}{S\unlhd T}]\underset{H}{\cdot}[\frac{H\times
G}{U\unlhd V}]$ can be written as such a linear combination for any group
$|H|<|G|$ and $(T,S)\in\Sigma_{G\times H}$, $(V,U)\in\Sigma_{H\times G}$. By
the Mackey formula this product consist of the elements of the form
$[\frac{G\times G}{S*U^{\prime}\unlhd T*V^{\prime}}]$ for some
$(V^{\prime},U^{\prime})\in\Sigma_{H\times G}$. Assume that
$(T*V^{\prime},S*U^{\prime})$ does not satisfy conditions $(i)$ and $(ii)$. So
we have that the left invariant of the section $(T*V^{\prime},S*U^{\prime})$
equal to $(G,K,P,1)$ for some $(K,P)\in\mathcal{G}_{G}$ and
$(K,P)\in\mathcal{R}_{G}$. Proving the similar version of Proposition [2, 2.6]
for the elements of the poset $\mathcal{G}_{G}$ one can show that
$l(T,S)=(G,K^{\prime},P^{\prime},1)$ with
$(K^{\prime},P^{\prime})\preceq(K,P)$. But then since $\mathcal{R}_{G}$ is a
lower set the pair $(K^{\prime},P^{\prime})$ also is reduced and we obtain the
contradiction

$e_{(K^{\prime},P^{\prime})}=\Big{[}\frac{G\times H}{S\unlhd
T}\Big{]}\underset{H}{\cdot}\Big{[}\frac{G\times H}{S\unlhd T}\Big{]}^{op}\in
I_{G}.$

###### A.5

Proof of 6.3

  1. (a)

By the Lemma 6.2 a standart basis element $\big{[}\frac{G\times G}{S\unlhd
T}\big{]}$ of $E_{G}$ belongs to covering algebra if $(T,S)\in\Sigma_{G\times
G}^{c}$ and it belongs to $I_{G}$ if $(T,S)\not\in\Sigma_{G\times G}^{c}$ so
their sum is a whole endomorphism ring. Thus $E_{G}^{c}$ is a covering algebra
for $E_{G}$.

  2. (b)

The standart basis elements $\big{[}\frac{G\times G}{S\unlhd T}\big{]}$ with
$(T,S)\in\Sigma_{G\times G}^{c}$ such that $l_{0}(T,S)\not\in\mathcal{R}_{G}$
generate the intersection of covering algebra and ideal $I_{G}$ as $k$-module.
Hence we have an equality

$E_{G}^{c}\cap
I_{G}=\bigoplus_{\begin{smallmatrix}\\{K,P\\}_{G}\in\mathcal{G}_{G}/\sim&\\\
(K,P)\not\in\mathcal{R}_{G}\end{smallmatrix}}E_{G}^{c,\\{K,P\\}}$

Since the set of reduced elements is a lower set and due to the Lemma 5.17 we
have the equality

$\bigoplus_{\begin{smallmatrix}\\{K,P\\}_{G}\in\mathcal{G}_{G}/\sim&\\\
(K,P)\not\in\mathcal{R}_{G}\end{smallmatrix}}E_{G}^{c,\\{K,P\\}}=\bigoplus_{\begin{smallmatrix}\\{K,P\\}_{G}\in\mathcal{G}_{G}/\sim&\\\
(K,P)\not\in\mathcal{R}_{G}\end{smallmatrix}}f_{\\{K,P\\}_{G}}E_{G}^{c}.$

Thus we get the desired result.

  3. (c)

It is a consequence of the Proposition 5.15 and Part (b).

  4. (d)

It is a consequence of the Proposition 5.17(d) and Part (c).

  5. (e)

Let $[V]$ be the class of simple $k\Gamma_{(G,K,P)}$-module and
$(K,P)\in\widetilde{\mathcal{R}}_{G}$ such that $(K,P)=(K_{i},P_{i})$ where
the elements of $\\{K,P\\}_{G}$ are enumerated as in the Lemma 5.17. Then by
the Morita equivalence of algebras Mat${}_{n}(k\Gamma_{(G,K,P)})$ and
$k\Gamma_{(G,K,P)}$, $[V]$ corresponds to the class of simple module
Mat${}_{n}(k\Gamma_{(G,K,P)})e_{i}\otimes_{k\Gamma_{(G,K,P)}}V$. Then for
$y_{i}=e_{i}=e_{(K,P)}$ the isomorphism $\omega\circ\sigma$ (from the proof of
Theorem 5.18) transports this simple module to the irreducible module
$f_{\\{K,P\\}_{G}}E_{G}^{c}f_{(K,P)}\otimes_{k\Gamma_{(G,K,P)}}V=E_{G}^{c}f_{(K,P)}\otimes_{k\Gamma_{(G,K,P)}}V$.
The last equality due to the fact that $f_{\\{K,P\\}_{G}}$ is commutative in
the covering algebra, and $f_{(K,P)}$ are orthogonal idempotents. So by the
isomorphism in the Part (c) we are done.

###### A.6 Lemma

Let $G$ and $H$ be finite groups, $(K,P)\in\mathcal{R}_{G}$ and
$(L,Q)\in\mathcal{R}_{H}$ be such that $(G,K,P,1)\sim(H,L,Q,1)$. Then $G$ and
$H$ have the same order.

###### Proof.

Since $(G,K,P,1)$ and $(H,L,Q,1)$ are linked, there is a section
$[\frac{G\times H}{S\unlhd T}]\in{}_{(G,K,P)}\Gamma_{(H,L,Q)}$. Assume
$|G|>|H|$. Then there is a factorization $|H:Q||G:P|e_{(K,P)}=[\frac{G\times
H}{S\unlhd T}]\times_{H}[\frac{G\times H}{S\unlhd T}]^{op}$. This is
impossible since $e_{(K,P)}$ is reduced. ∎

###### A.7 Lemma

Let $G$ and $H$ be finite groups of the same order, let $\Gamma_{k}^{c}(G,H)$
be the submodule of $\Gamma_{k}(G,H)$ spanned by all sections corresponding to
covering pairs. Also let $(K,P)\in\mathcal{G}_{G}$ and
$(L,Q)\in\mathcal{G}_{H}$. Then

  1. (a)

There is an isomorphism of $(k\Gamma_{(G,K,P)},k\Gamma_{(H,L,Q)})$-bimodules

$k[{}_{(G,K,P)}\Gamma_{(H,L,Q)}]\to f_{(K,P)}\Gamma_{k}^{c}(G,H)f_{(L,Q)}$

given by mapping $b$ to $f_{(K,P)}bf_{(L,Q)}$. Here the actions on the left
and on the right of $f_{(K,P)}\Gamma_{k}^{c}(G,H)f_{(L,Q)}$ are given through
the isomorphism in Theorem 5.14.

  2. (b)

Suppose $(L,Q)\in\mathcal{R}_{H}$ and $W$ is a simple
$k\Gamma_{(H,L,Q)}$-module. Then there is an epimorphism

$k[{}_{(G,K,P)}\Gamma_{(H,L,Q)}]\otimes_{k\Gamma_{(H,L,Q)}}W\to
f_{(K,P)}S_{H,\widetilde{W}}(G)$

of $k\Gamma_{(G,K,P)}$-modules.

The proof of this technical lemma is almost identical to the proof of [2,
Lemma 9.5]. We leave the justification to the reader.

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