Contents

group theory

# Contents

## Definition

###### Definition

(Burnside marks and Burnside character)

Let $G$ be a finite group and let $X \in G Set^{fin}$ any finite G-set. For $[H]$ the conjugacy class of a subgroup $H \subset G$, the $[H]$-mark on $X$ is the cardinality of the set of $H$-fixed points, hence the natural number

$\phi_{[H]}(X) \;\coloneqq\; \left\vert X^{H} \right\vert$

of elements $x \in X$ such that for each $h \in H \subset G$ we have $h(x) = x$.

This construction extends to a ring homomorphism

(1)$\array{ \phi &\colon& A(G) &\overset{\phi}{\longrightarrow}& \mathbb{Z}^{Conj(G)} \\ && [X] &\mapsto& \big( [H] \mapsto \left\vert X^H\right\vert\big) }$

from the Burnside ring to the ring of tuples of integers of length the number $Conj(G)$ of conjugacy classes of subgroups of $G$.

This morphism is also called the Burnside character or mark homomorphism.

###### Remark

(marks in terms of homs)

Equivalent the set of $[H]$-marks of a G-set $X$ (Def. ) is the hom-set in GSet from $G/H$ to $X$.

###### Proof

This follows from a basic standard argument. For completeness, we make it explicit:

By transitivity of the action on $G/H$ a $G$-equivariant function $f \colon G/H \to X$ is fully specified by its image $f([e]) \in X$ of the equivalence class $[e] \in G/H$ of the neutral element. Since this $[e] \in G/H$ is fixed precisely by the elements in $H \subset G$ it may, again by $G$-equivariance, be mapped to any $H$-fixed point $f([e]) \in X^H \subset X$.

Of particular interest are the marks of the transitive G-sets, i.e. those isomorphic to sets $G/H$ of coset, for $H\subset G$ a subgroup. These arrange into a table of marks:

###### Definition

(table of marks)

The table of Burnside marks (or table of marks, for short) of a finite group $G$ is the matrix indexed by conjugacy classes $[H]$ of subgroups $H \subset G$ whose $([H_i], [H_j])$-entry is the $[H_j]$-marks of $G/H_i$ (Def. ), hence the number of fixed points of the action of $H_j$ on the coset space $G/H_i$:

$M_{i j} \;\coloneqq\; \left\vert \left(G/H_i\right)^{H_j} \right\vert \,.$

(e.g. Pfeiffer 97, chapter The Burnside Ring and the Table of Marks)

###### Remark

(table of marks in terms of homs)

The expression of marks in terms of homs (Remark ) means here that the table of marks (Def. ) is equivalently given by

$M_{i j} \;\coloneqq\; \left\vert \left(G/H_i\right)^{H_j} \right\vert \;=\; \left\vert Hom_{G Set}(G/H_j, G/H_i)\right\vert \,.$

## Properties

### Relation to characters

The following Prop. that the Burnside character plays the same role for finite G-sets as characters of representations play for finite-dimensional linear representations, in that it faithfully reflects $G$-sets. In fact the marks of a $G$-set over cyclic subgroups coincides with the character of its permutation representation over any ground field (Prop. ) below.

###### Proposition

(Burnside character is injective)

The Burnside character (1) is injective. Hence any two finite G-sets are isomorphic precisely if they have the same Burnside marks (Def. ).

###### Proposition

(mark homomorphism on cyclic groups agrees with characters of corresponding permutation representations)

For $S \in G Set_{fin}$ a finite G-set, for $k$ any field and $k[S] \in Rep_k(G)$ the corresponding permutation representation, the character $\chi_{k[S]}$ of the permutation representation at any $g \in G$ equals the Burnside marks (Def. ) of $S$ under the cyclic group $\langle g\rangle \subset G$ generated by $g$:

$\chi_{k[S]}\big( g \big) \;=\; \left\vert X^{\langle g \rangle} \right\vert \;\in\; \mathbb{Z} \longrightarrow k \,.$

Hence the mark homomorphism (Def. ) of $G$-sets restricted to cyclic subgroups coincides with the characters of their permutation representations.

This statement immediately generalizes from plain representations to virtual representations, hence to the Burnside ring.

(e.g. tom Dieck 09, (2.15))

###### Proof

By definition of character of a linear representation, we have that

$\chi_{k[S]}(g) = tr_{k[S]}(g)$

is the trace of the linear endomorphism $k[S] \overset{g}{\to} k[S]$ of the given permutation representation.

Now the canonical $k$-linear basis for $k[S]$ is of course the set $S$ itself, and so

\begin{aligned} \chi_{k[S]}(g) & = \underset{ s \in S }{\sum} \left\{ \array{ 1 &\vert& g(s) = s \\ 0 &\vert& \text{otherwise} } \right. \\ & = \left\vert S^g \right\vert \\ & = \left\vert S^{\langle g \rangle} \right\vert \end{aligned}

Here in the first step we spelled out the definition of trace in the canonical basis, and in the second step we observed that the fixed point set of a cyclic group equals that of any one of its generating elements.

### Relation to Burnside product

We discuss, in Prop. below, how the table of marks encodes the product in the Burnside ring of the given finite group $G$. For this purpose we first consider two Lemma: Lemma and Lemma .

###### Lemma

(linear order on set of conjugacy classes of subgroups)

There exists a linear order $\leq_{lin}$ on the set of conjugacy classes $[H]$ of subgroups of $G$ such that a subgroup inclusion $H \subset H'$ implies that $[H] \leq_{lin} [H']$.

###### Proof

This follows by the general existence of linear extensions of partial orders applied to the subgroup lattice of $G$.

###### Lemma

(table of marks is lower triangular invertible matrix)

With respect to any linear order on the conjugacy classes of subgroups as in Lemma , the table of marks (Def. ) becomes a lower triangular matrix over the integers with non-zero entries on the diagonal. In particular, it is an invertible matrix.

###### Proof

That the subgroup $H_j \subset G$ has any fixed points in $G/H_i$ means that there is a $g \in G$ such that $h g H_i = g H_i$ for all $h \in H_j$, and thus that $g^{-1}H_{j}g$ is a subgroup of $H_{i}$, or in other words, that $H_{j}$ is conjugate to a subgroup of $H_{i}$. Hence

$\big( M_{i j} \gt 0 \big) \;\Rightarrow\; \big( [H_j] \leq_{lin} [H_i] \big) \,,$

and thus the matrix $M$ is lower triangular.

Since at least $H = 1_{G/H}$ is fixed by $H$, we moreover have that the diagonal entries are non-zero.

In the following, given a G-set $G/H_i$ we write $[G/H_i] \in A(G)$ for its isomorphism class, regarded as an element in the Burnside ring.

###### Definition

(Burnside multiplicities)

Given a choice of linear order on the conjugacy classes of subgroups of $G$ (for instance as in Lemma ), we say that the corresponding structure constants of the Burnside ring (or Burnside multiplicities) are the natural numbers

$n_{i j}^\ell \;\in\; \mathbb{N}$

uniquely defined by the equation

(2)$[G/H_i] \times [G/H_j] \;=\; \underset{ \ell }{\sum} n_{i j}^\ell [G/H_\ell] \,.$
###### Proposition

(Burnside ring product in terms of table of marks)

The Burnside ring structure constants $\left( n_{i j}^\ell\right)$ (Def. ) are equal to the following algebraic expression in the table of marks $\left( M_{i j}\right)$ and its inverse matrix $\left( \left(M^{-1}\right)_{r s} \right)$ (which exists by Lemma ):

$n_{i j}^\ell \;=\; \underset{m}{\sum} M_{i m} \cdot M_{j m} \cdot (M^{-1})_{m \ell}$
###### Proof

Let $t$ be the dimension of $M$, i.e. $M$ is a $t \times t$ matrix. For any $1 \leq m \leq t$, we compute as follows:

\begin{aligned} \underset{1 \leq \ell \leq t}{\sum} n_{i j}^\ell \cdot M_{\ell m} & = \underset{1 \leq \ell \leq t}{\sum} n_{i j}^\ell \cdot \left\vert Hom_{GSet} \big( G/H_m , G/H_\ell \big) \right\vert \\ & = \left\vert Hom_{GSet} \big( G/H_m , \underset{\ell}{\sum} n_{i j}^\ell \cdot G/H_\ell \big) \right\vert \\ & = \left\vert Hom_{GSet}\big( G/H_m, \; G/H_i \times G/H_j \big) \right\vert \\ & = \left\vert Hom_{GSet}\big( G/H_m, \; G/H_i \big) \right\vert \cdot \left\vert Hom_{GSet}\big( G/H_m, \; G/H_j \big) \right\vert \\ & = M_{i m} \cdot M_{j m} \end{aligned}

Here the third step uses the defining equation (2) of the structure constants $n_{i j}^\ell$, while all other steps use that the mark homomorphism is a ring homomorphism, which we made manifest by expressing the marks via hom-sets (Remark ).

Thus we have that $\left( n_{i j}^{1}, n_{i j}^{2}, \ldots, n_{i j}^{t} \right)$ is a solution to the following system of equations.

\begin{aligned} M_{i 1} \cdot M_{j 1} &= M_{1 1}x_{1} + \cdots + M_{t 1}x_{t} \\ M_{i 2} \cdot M_{j 2} &= M_{1 2}x_{1} + \cdots + M_{t 2} x_{t} \\ &\vdots \\ M_{i t} \cdot M_{j t} &= M_{1 t}x_{1} + \cdots + M_{t t} x_{t} \end{aligned}

But, since $M$ is invertible, the unique solution to this system of equations is given by the product of $M^{-1}$ and the transposition of $\left( M_{i 1} \cdot M_{j 1}, \ldots, M_{i t} \cdot M_{j t} \right)$. The claim follows immediately.

###### Corollary

The table of marks of a finite group determines its Burnside ring. That is to say, if the tables of marks of a pair of groups $G_{1}$ and $G_{2}$ are equal, then the Burnside ring of $G_{1}$ is isomorphic to the Burnside ring of $G_{2}$.

###### Proof

The Burnside ring of a finite group is a free abelian group on the set $G / H_1, \ldots, G / H_t$, where $H_1, \ldots, H_t$ are representatives of the conjugacy classes of that group, equipped with a certain multiplication. Thus it suffices to check that the structure constants of the Burnside rings coincide, which is established by the previous proposition.

## References

The concept was introduced in

Textbook accounts and lecture notes include

Computer implementation is discussed in

• Götz Pfeiffer, The Subgroups of $M_{24}$, or How to Compute the Table of Marks of a Finite Group, Experiment. Math. 6 (1997), no. 3, 247–270 (doi:10.1080/10586458.1997.10504613, web)