Computational aspects of Burnside rings, part I: the ring structure

Abstract

The Burnside ring $\mathcal {B}(G)$ of a finite group G, a classical tool in group theory and representation theory, is studied from the point of view of computational commutative algebra. Starting from a table of marks, we describe efficient algorithms for computing a presentation, the image of the mark homomorphism, the prime ideals and the prime ideal graph, the singular locus, the conductor in its integral closure, the connected components of its spectrum, and its idempotents. On the way, we provide methods for identifying p-residual subgroups, direct products of subgroups of coprime order, commutator subgroups, and perfect subgroups.

1 Introduction

The Burnside ring is a classical tool in the theory of finite groups. It was introduced for the first time by Burnside (1911) in his book. Given a finite group G, it is usually defined as the Grothendieck ring of isomorphism classes of finite G-sets. It can be viewed as the representation ring of permutation representations of G.

Many papers have been written about the Burnside ring $\mathcal {B}(G)$, most of them from a group theoretical point of view. In this series of papers we change the perspective and consider the structure of $\mathcal {B}(G)$ as a commutative ring. As such, we can use the methods of computational commutative algebra to deal with it algorithmically. Since many properties of groups are reflected in the structure of this ring, we set ourselves the goal to find efficient algorithms for checking or computing them. For this endeavour, our starting point, i.e., the input of all algorithms, is the table of marks of a finite group. Although it is known that different groups can have the same table of marks (see for instance Thévenaz 1988), there is still a lot of information about the group contained in it. In particular, the table of marks determines a presentation of the Burnside ring and thus makes it possible to perform effective computations with this commutative ring.

Let us describe the contents in more detail. In Sect. 2 we recall the definition and the basic properties of the Burnside ring $\mathcal {B}(G)$. Given representatives $H_1,\dots ,H_s$ for the conjugacy classes of subgroups of G, it is a free $\mathbb {Z}$-module with basis $[G/H_1],\dots ,[G/H_s]$. Then, in Sect. 3, we provide the definition and the basic properties of the table of marks $\mathcal {T}(G)$. After we order $H_1,\dots ,H_s$ such that $\# H_1 \le \cdots \le \# H_s$, the matrix $\mathcal {T}(G)=(m_{ij}) \in \mathrm{Mat}_s(\mathbb {Z})$ is defined by $m_{ij}= \# (G{/}H_i)^{H_j}$, where the action is by left multiplication. We also observe that using only the knowledge of $\mathcal {T}(G)$, we cannot check whether one subgroup is conjugate to a normal subgroup of another, or identify the conjugacy class of the normalizer of a subgroup.

In Sect. 4 we introduce the mark homomorphism $\Phi _G: \mathcal {B}(G) \longrightarrow \mathbb {Z}^s$ and spell out a direct method for checking whether a tuple is in the image of $\Phi _G$ which avoids the use of possibly hard-to-compute standard congruences. Moreover, using this mark homomorphism, we write down an algorithm to compute a “canonical presentation” of $\mathcal {B}(G)$ of the form $\mathcal {B}(G)\cong \mathbb {Z}[x_1,\dots ,x_{s-1}]/I_G$. The defining ideal $I_G$ is then identified in Sect. 5 as a vanishing ideal of a finite point set, called the set of mark points, and its primary decomposition allows us to write down all prime ideals of $\mathcal {B}(G)$ explicitly. In particular, we conclude that $\mathcal {B}(G)$ is a 1-dimensional, reduced, Cohen–Macaulay ring.

In Sect. 6 we begin a study of the Zariski topology on $\mathrm{Spec}(\mathcal {B}(G))$. The prime ideal graph (first introduced in Nicolson 1978) of $\mathcal {B}(G)$ has the s minimal primes as its vertices and a connecting edge between two vertices if and only if some maximal ideal contains both minimal primes. Among others, we provide algorithms for computing the prime graph as a labeled graph, and for computing the prime graph of a finite nilpotent group. In particular, we show that the prime graph of a finite nilpotent group is a direct product of complete graphs, and we get an alternative proof of Dress’ well-known connectedness theorem (cf. Dress 1969, Prop. 2) in this case.

The next section contains a characterization of the singular locus of $\mathcal {B}(G)$, an algorithm for its computation, as well as an algorithm for identifying the conjugacy class of the p-residual subgroup of a subgroup of G. Finally, in Sect. 8, we compute idempotents and quasi-idempotents of $\mathcal {B}(G)$ and determine the connected components of $\mathrm{Spec}(\mathcal {B}(G)$ effectively. The quasi-idempotents are the least multiples $r_ie_i$ of the idempotents $e_i$ of the ghost ring $\mathbb {Z}^s$ such that $r_ie_i\in \mathrm{Im}(\Phi _G)$. Since we identify $\mathbb {Z}^s$ with the integral closure of $\mathcal {B}(G)$ in its full ring of quotients, they are the generators of the conductor of $\mathcal {B}(G)$. To compute them efficiently, we do not use the usual formula involving commutator subgroups (which cannot be identified from $\mathcal {T}(G)$) but a straightforward diagonalization algorithm for $\mathcal {T}(G)$.

To find the connected components of $\mathrm{Spec}(\mathcal {B}(G))$, we apply the naive breadth-first search algorithm. Then we use the characterization of Dress (1969) which characterizes perfect subgroups of G via the connected components of $\mathrm{Spec}(\mathcal {B}(G))$ to compute the conjugacy classes of perfect subgroups of G. For the last task, namely to compute the primitive idempotents of $\mathcal {B}(G)$, we can use the information about the connected components of $\mathcal {B}(G)$ and represent their characteristic functions as linear combinations of the rows of $\mathcal {T}(G)$. Again this results in an algorithm which relies only on the table of marks, in contrast to an implementation of the usual formula for the primitive idempotents (cf. Yoshida 1983) involving the Möbius function of the subgroup lattice.

All algorithms in this paper were implemented in the computer algebra system ApCoCoA (see The ApCoCoA Team 2013), using the tables of marks coming from GAP (see The GAP Group 2014). Our definitions and notation follow the books by Kreuzer and Robbiano (2000, 2005), unless explicitly noted otherwise. Good proofs of some standard results about Burnside rings that we need can be found in Karpilovsky (1995).

2 Definition and basic properties

In the following we let G be a finite group. A finite set X is called a G -set if there is an action of G from the left on X, i.e., a map $\varphi :\, G\times X \longrightarrow X$ such that $\varphi (gh,a)=\varphi (g,\varphi (h,a))$ and $\varphi (e,a)=a$ for $g,h\in G$, $a\in X$, and the neutral element $e\in G$. As usual, we write $g\cdot a =\varphi (g,a)$. The subsets $G\cdot a$ for $a\in X$ are called the orbits of the action. Choosing representatives $a_1,\dots ,a_n$ for the orbits, we have a disjoint union . For every $a\in X$, the subgroup $G_a=\{g\in G\mid g\cdot a = a\}$ is called the stabilizer of a. There is a bijection between the set of left cosets $G{/}G_{a_i}$ and $G\cdot a_i$ defined by $gG_{a_i} \mapsto g\cdot a_i$.

If we choose two representatives $a_i,b_i\in X$ of the same orbit $G\cdot a_i = G\cdot b_i$, the subgroups $G_{a_i}$ and $G_{b_i}$ are conjugate to each other, since $b_i=g\cdot a_i$ implies $G_{b_i}\cdot g\cdot a_i = g\cdot a_i$, and therefore $g^{-1}G_{b_i}g\cdot a_i =a_i$. This shows $g^{-1}G_{b_i}g \subseteq G_{a_i}$, and the reverse containment follows similarly. Thus, for $i\in \{1,\dots ,n\}$, the orbit $G\cdot a_i$ of $\varphi$ may be identified with the set of all G / H where H ranges over the conjugacy class of the subgroup $G_{a_i}$.

Definition 2.1

The Burnside ring $\mathcal {B}(G)$ of G is defined to be the set of all formal differences of isomorphism classes of finite G-sets. Addition is given by disjoint union of G-sets, and multiplication by their cartesian product equipped with the diagonal action.

For a G-set X, let [X] denote its isomorphism class. Since every finite G-set is the disjoint union of its orbits, the Burnside ring is generated as a $\mathbb {Z}$-module by the isomorphism classes of orbit types. As we saw above, these isomorphism classes can be identified with the sets [G / H] where H ranges over the conjugacy class of a subgroup of G. For G-sets X, Y, the addition in $\mathcal {B}(G)$ is defined by , and the multiplication is defined by $[X] \cdot [Y] = [X\times Y]$. Our first goal is to make these operations explicit in terms of the G-sets [G / H]. The following proposition describes the additive structure of $\mathcal {B}(G)$ explicitly.

Proposition 2.2

(Burnside) Let G be a finite group.

• (a) Two G-sets X, Y are isomorphic as G-sets if and only if $[X] = [Y]$ in $\mathcal {B}(G)$.

• (b) Let $H_1,\dots ,H_s$ be representatives of the conjugacy classes of subgroups of G. Then $\mathcal {B}(G)$ is a free $\mathbb {Z}$-module with basis $\{[G{/}H_1],\dots ,[G{/}H_s]\}$.

Proof

See for instance Karpilovsky (1995, Thm. 2.1). $\square$

To get also the multiplicative structure of $\mathcal {B}(G)$ under control, we introduce the table of marks of a finite group next.

3 The table of marks

Let G be a finite group and X a G-set. For a subgroup H of G, we denote by $X^H=\{x\in X \mid h\cdot x = x$ for all $h\in H\}$ the set of fixed points of X under the action of H. The number $\# X^H$ is called the mark of H on X. If K is a further subgroup of G, we consider the action of K on G / H defined by $k\cdot gH = kgH$. Then the mark of K on G / H is denoted by

$$\begin{aligned} \mathrm{m}(H,K) \;=\; \# \{gH \in G{/}H \mid KgH=gH\} \;=\; \#(G{/}H)^K. \end{aligned}$$

Notice that $KgH=gH$ is equivalent to $K\subseteq gHg^{-1}$, and also to $g^{-1}Kg\subseteq H$. This shows $\mathrm{m}(H,K)=0$ if K is not conjugate to a subgroup of H.

Definition 3.1

Let s be the number of conjugacy classes of subgroups of G, and let $H_1,\dots ,H_s$ be subgroups of G which represent these conjugacy classes. We assume that these subgroups are numbered such that $\# H_i\le \# H_j$ for $i\le j$. Then the matrix $\mathcal {T}(G)\in \mathrm{Mat}_s(\mathbb {Z})$ whose entry in position (i, j) is $\mathrm{m}(H_i,H_j)$ is called the table of marks of G.

Let us collect some easy properties of the table of marks of G.

Remark 3.2

• (a) Since $H_1=\{e\}$ is the trivial subgroup of G, we have $\mathrm{m}(H_1,H_1)=\mathrm{ord}(G)$ and $\mathrm{m}(H_1,H_j)=0$ for $j>1$. Thus the first row of $\mathcal {T}(G)$ is always $(\# G,0,\dots ,0)$.

• (b) Since $H_s=G$, we have $G{/}H_s=\{e\}$, and therefore $\mathrm{m}(H_s,H_j) = 1$ for $j=1,\dots ,s$. Thus the last row of $\mathcal {T}(G)$ is always $(1,\dots ,1)$.

• (c) The first column of $\mathcal {T}(G)$ corresponds to the conjugacy class $[\{e\}]$ of the trivial subgroup which fixes all elements of $[G{/}H_i]$. Hence the first entry of the i-th row if $\mathcal {T}(G)$ is $\#(G{/}H_i)$.

• (d) For $j> i$, the group $H_j$ satisfies $\# H_j\ge \# H_i$. Thus it is not conjugate to a subgroup of $H_i$ and we have $\mathrm{m}(H_i,H_j)=0$. Hence the matrix $\mathcal {T}(G)$ is a lower triangular matrix.

The table of marks of a finite group can be made unique in the following way.

Remark 3.3

In Definition 3.1, we required that the conjugacy classes of subgroups of G and their representatives $H_1,\dots ,H_s$ are ordered such that $\# H_i \le \# H_j$ for $i\le j$. In fact, it would have been sufficient to require that $i\le j$ if $H_i$ is conjugate to a subgroup of $H_j$. But in both cases, the table of marks $\mathcal {T}(G)$ is not uniquely determined by G, since there may be several conjugacy classes of subgroups of the same order. To remedy this situation, we fix the following ordering of the rows and columns of $\mathcal {T}(G)$.

• (a) First order $H_1,\dots ,H_s$ in increasing cardinality. Equivalently, the first column of $\mathcal {T}(G)$ will be ordered decreasingly.

• (b) If $H_i,\dots ,H_{j}$ have the same number of elements, order them such that the diagonal entries of $\mathcal {T}(G)$ satisfy $m_{ii}\ge \cdots \ge m_{jj}$.

• (c) If the rows of $\mathcal {T}(G)$ corresponding to $[G{/}H_i],\dots , [G{/}H_j]$ have the same first element and the same diagonal element, reorder them descendingly with respect to the lexicographic order. Process these blocks from top to bottom in $\mathcal {T}(G)$.

In this way, we arrive at a table which we call the canonical table of marks of G. In the following, unless we explicitly mention otherwise, we shall always use the canonical table of marks of a finite group.

The table of marks of a finite group can be calculated using methods of computational group theory (see for instance Holt et al. 2005, Section 10.5 or Pfeiffer 1997). In this paper, it is considered as the input for all further algorithms. Whenever necessary, we used the computer algebra system GAP (see The GAP Group 2014) to determine it. Let us consider some examples.

Example 3.4

The table of marks of a finite group G has size $2\times 2$ if and only if $G=C_p$ is a cyclic group of prime order p. In this case, we have $\mathcal {T}(C_p)= \left( {\begin{array}{c}p\;0\\ 1\;1\end{array}}\right)$.

Example 3.5

The table of marks of the symmetric group $S_3$ is

$$\begin{aligned} \mathcal {T}(S_3)= \begin{pmatrix} 6 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 3 &{}\quad 1 &{}\quad 0 &{}\quad 0\\ 2 &{}\quad 0 &{}\quad 2 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 \end{pmatrix} \end{aligned}$$

Example 3.6

The table of marks of the cyclic group $C_4$ of order four is

$$\begin{aligned} \mathcal {T}(C_4)= \begin{pmatrix} 4 &{}\quad 0 &{}\quad 0\\ 2 &{}\quad 2 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 1 \end{pmatrix} \end{aligned}$$

Example 3.7

The table of marks of the Klein four-group $V_4$ is

$$\begin{aligned} \mathcal {T}(V_4)= \begin{pmatrix} 4 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 2 &{}\quad 2 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 2 &{}\quad 0 &{}\quad 2 &{}\quad 0 &{}\quad 0\\ 2 &{}\quad 0 &{}\quad 0 &{}\quad 2 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 \end{pmatrix} \end{aligned}$$

The following proposition contains a small observation about the distribution of zeros in $\mathcal {T}(G)$.

Proposition 3.8

Let r be the number of distinct prime factors of $\# G$. Then every column of $\mathcal {T}(G)$, except the first one, contains at least r zeros.

Proof

Let $p_1,\dots ,p_r$ be the distinct primes dividing $\# G$. Let $H_1,\dots ,H_r$ be subgroups of G such that $H_i$ represents the conjugacy class of the $p_i$-Sylow subgroups of G. For every non-trivial subgroup U of G, there exists a prime $p_i$ dividing $\# U$. By Cauchy’s theorem, the subgroup U contains an element of order $p_i$. Now consider the action of U on the sets $G{/}H_j$ for $j\ne i$. Since $p_i$ does not divide the order of $H_j$, the group U is not conjugate to a subgroup of $H_j$. Hence the action has no fixed points, i.e., we have $\mathrm{m}(H_j,U)=0$. Together with the zero in the first row of $\mathcal {T}(G)$, this provides the claimed zeros. $\square$

Examples 3.6 and 3.7 show that for a p-group G, the second column of T(G) may contain one or more zeros, so the bound given in the proposition is sharp. We end this section with a collection of further properties of the table of marks of a finite group.

Proposition 3.9

Let $\mathcal {T}(G)=(m_{ij}) \in \mathrm{Mat}_s(\mathbb {Z})$ be the table of marks of a finite group G, and let $H_1,\dots ,H_s$ be subgroups of G such that $[G{/}H_i]$ corresponds to the i-th row of $\mathcal {T}(G)$.

• (a) For $i=1,\dots ,s$, we have $m_{ii} = [N_G(H_i) : H_i]$, where $N_G(H_i)$ is the normalizer of $H_i$ in G, i.e., the largest subgroup of G containing $H_i$ as a normal subgroup.

• (b) For $i=1,\dots ,s$, the subgroup $H_i$ of G is a normal subgroup if and only if $m_{i1}= m_{ii}$. More generally, the number $m_{i1}/m_{ii}$ is the size of the conjugacy class of $H_i$ in G.

• (c) Let $i\in \{1,\dots ,s\}$. Every entry in the i-th row of $\mathcal {T}(G)$ is a multiple of $m_{ii}$. More precisely, the entry $m_{ij}$ equals the product of $m_{i1}$ by the number of conjugates of $H_i$ containing $H_j$.

• (d) The number of distinct conjugates of $H_j$ contained in $H_i$ is given by $(m_{ij}\, m_{j1})/(m_{jj}\, m_{i1})$.

Proof

Claim (a) follows from Solomon (1967, Lemma 2). The first part of claim (b) follows immediately from (a). The second part of (b) and (c) follow from the facts that $H_jgH_i=gH_i$ is equivalent to $H_j \subseteq gH_ig^{-1}$ and $\# N_G(H_i)$ elements g give the same conjugate $gH_i g^{-1}$. Claim (d) follows from a similar counting argument (see Pfeiffer 1997, Prop. 1.4). $\square$

Finally, we note that there exist certain types of information which cannot be derived from the table of marks.

Remark 3.10

Let $\mathcal {T}(G)$ be the table of marks of a finite group G.

• (a) Using only the knowledge of $\mathcal {T}(G)$, it is not possible to check whether one subgroup of G is conjugate to a normal subgroup of another one (see Huerta-Aparicio et al. 2009, Theorem 4.2.5).

• (b) It is also not possible to identify the normalizer of a subgroup (see Huerta-Aparicio et al. 2009, Theorem 4.2.6).

4 Computing a presentation

In the following we let G be a finite group and $\mathcal {T}(G)$ its table of marks. Our goal in this section is to compute a natural presentation of the Burnside ring $\mathcal {B}(G)$. The additive structure of $\mathcal {B}(G)$ is determined by the fact that the sets [G / H], where H ranges over a set of representatives of the conjugacy classes of subgroups of G, form a $\mathbb {Z}$-basis of $\mathcal {B}(G)$. The multiplicative structure is obtained from the following observation.

Proposition 4.1

Let $H_1,\dots ,H_s$ be subgroups of G which represent the distinct equivalence classes of subgroups of G under conjugation. Then the map

$$\begin{aligned} \Phi _G:\ \mathcal {B}(G) \longrightarrow \prod _{i=1}^s \;\mathbb {Z} \quad \hbox {given by }[G{/}U] \mapsto (m(U,H_i))_{i=1,\dots ,s} \end{aligned}$$

is an injective ring homomorphism. It is called the mark homomorphism.

For a proof, see for instance Bouc (2000, Sec. 3.2). Since the image of $\Phi _G$ is a free $\mathbb {Z}$-submodule of $\mathbb {Z}^s$, we have the following easy Algorithm 1 to check whether or not a given tuple is in $\mathrm{Im}(\Phi _G)$.

Remark 4.2

In the classical theory of Burnside rings, containment in the image of $\Phi _G$ is usually checked via the so-called standard congruences. Typically, these congruences require the computation of a sum which extends over all cyclic subgroups of a certain group. Although it is possible to determine the cyclic subgroups of a given group from the table of marks, a more efficient version of this method could be based on the computation of the syzygies of the row vectors of $\mathcal {T}(G)$.

On the right-hand side of $\Phi _G$, the $\mathbb {Z}$-module $\mathbb {Z}^s$ is a ring by component-wise multiplication. It is called the ghost ring of G. In order to have a more invariant representation of the ghost ring, we can use the set $\mathcal {C}(G)$ of equivalence classes of subgroups of G under conjugation and write the ghost ring as $\mathbb {Z}^{\mathcal {C}(G)}$.

The fact that $\Phi _G$ is a ring homomorphism implies that we can compute the product of two classes $[G{/}H_i],[G{/}H_j]\in \mathcal {B}(G)$ by multiplying the corresponding rows of $\mathcal {T}(G)$ component-wise and representing the result as a $\mathbb {Z}$-linear combination of the rows of $\mathcal {T}(G)$. Thus we get equalities

$$\begin{aligned}{}[G{/}H_i] \cdot [G{/}H_j] \;=\; \sum _{k=1}^s \; c_{ijk} \, [G{/}H_k] \end{aligned}$$

with $c_{ijk}\in \mathbb {Z}$. In fact, since this product corresponds to the decomposition of the natural action of G on $G{/}H_i \times G{/}H_j$, the numbers $c_{ijk}$ will be non-negative. Thus we arrive at the following algorithm for computing a presentation of $\mathcal {B}(G)$ using generators and relations.

Corollary 4.3

Let $\mathcal {T}(G)\in \mathrm{Mat}_s(\mathbb {Z})$ be the table of marks of G. Then Algorithm 2 computes a tuple $F=(f_1,\dots ,f_m)$ of polynomials in $\mathbb {Z}[x_1,\dots ,x_{s-1}]$ such that $\mathcal {B}(G)\cong \mathbb {Z}[x_1,\dots ,x_{s-1}]/\langle f_1,\dots ,f_m\rangle$.

Under this isomorphism, the set $[G{/}H_i]$ corresponds to the residue class $\bar{x}_{s-i}$ if $H_i$ is a subgroup of G in the conjugacy class corresponding to the i-th row of $\mathcal {T}(G)$.

Proof

Let $H_1,\dots ,H_s$ be subgroups of G such that the conjugacy class $[G{/}H_i]$ corresponds to the i-th row of $\mathcal {T}(G)$ for $i=1,\dots ,s$. From the description of $\Phi _G$ via component-wise multiplication and the fact that the last row of $\mathcal {T}(G)$ is $(1,\dots ,1)$ it is clear that $[G{/}H_s]$ represents the multiplicative identity element of $\mathcal {B}(G)$. Since the sets $[G{/}H_i]$ are a $\mathbb {Z}$-basis of $\mathcal {B}(G)$, it follows that $[G{/}H_1],\dots ,[G{/}H_{s-1}]$ is a system of $\mathbb {Z}$-algebra generators. We represent $[G{/}H_i]$ by an indeterminate $x_{s-i}$. Then the multiplicative structure of $\mathcal {B}(G)$ is determined by the relations resulting from the multiplication table of the algebra generators. These relations are precisely the polynomials computed in Steps 4 and 5 of the algorithm. $\square$

Let us compute this algebra structure in some examples.

Example 4.4

Let $G=S_3$ be the symmetric group on three symbols. Using the table of marks given in Example 3.5 and the algorithm, we compute:

$$\begin{aligned} (6,0,0,0) \cdot (6,0,0,0)&= 6\cdot (6,0,0,0)\\ (6,0,0,0) \cdot (3,1,0,0)&= 3\cdot (6,0,0,0)\\ (6,0,0,0) \cdot (2,0,2,0)&= 2 \cdot (6,0,0,0)\\ (3,1,0,0) \cdot (3,1,0,0)&= (6,0,0,0) + (3,1,0,0)\\ (3,1,0,0) \cdot (2,0,2,0)&= (6,0,0,0)\\ (2,0,2,0) \cdot (2,0,2,0)&= 2\cdot (2,0,2,0) \end{aligned}$$

Therefore the algorithm computes the following presentation:

$$\begin{aligned} \mathcal {B}(S_3) \;\cong \; \mathbb {Z}[x_1,x_2,x_3]/\langle&x_3^2-6x_3,\, x_3x_2-3x_3,\, x_3x_1-2x_3,\\&x_2^2-x_3-x_2,\, x_2x_1-x_3,\, x_1^2-2x_1\rangle \end{aligned}$$

Example 4.5

For a prime p, the Burnside ring of the cyclic group $C_p$ with p elements satisfies $\mathcal {B}(C_p)\cong \mathbb {Z}[x]/\langle x^2-p\,x\rangle$.

Example 4.6

The Burnside ring of the Klein four-group (see Example 3.7) has the presentation

$$\begin{aligned} \mathcal {B}(V_4) \;\cong \; \mathbb {Z}[x_1,x_2,x_3,x_4]/\langle&x_4^2-4x_4,\, x_4x_3-2x_3,\, x_4x_2-2x_2,\, x_4x_1-2x_1,\, x_3^2-2x_3\\&x_3x_2-x_4,\, x_3x_1-x_4,\, x_2^2-2x_2,\, x_2x_1-x_4,\, x_1^2-2x_1 \rangle . \end{aligned}$$

The presentation of $\mathcal {B}(G)$ computed by Algorithm 2 will be called the canonical presentation and the ideal will be called the (canonical) defining ideal of $\mathcal {B}(G)$ and denoted by I(G). In the next section, we begin our study of the algebraic structure of $\mathcal {B}(G)$ based on the canoncial presentation. But before, let us end this section with a criterion which tells us when we can simplify the canonical presentation of $\mathcal {B}(G)$ somewhat.

Proposition 4.7

Let $H_1,\dots ,H_s$ be subgroups of G which represent the distinct conjugacy classes of subgroups of G.

• (a) For $i,j\in \{1,\dots ,s\}$, let $[G{/}H_i] \cdot [G{/}H_j] = \sum _{k=1}^s\, c_{ijk} [G{/}H_k]$ with $c_{ijk}\in \mathbb {N}$. Then $c_{ijk}\ne 0$ implies that $H_k$ is conjugate to a subgroup of $H_i$ and to a subgroup of $H_j$.

• (b) If there exits a prime p dividing $\mathrm{ord}(G)$ such that G has a p -complement, i.e., a subgroup of complementary order to a p-Sylow subgroup of G, then the first row of $\mathcal {T}(G)$ is the component-wise product of two other rows of $\mathcal {T}(G)$.

Proof

Claim (a) follows from Solomon (1967, Lemma 1). To prove (b), we let $H_i$ be a p-Sylow subgroup of G and $H_j$ a subgroup of complementary order. Since no prime divides both the order of $H_i$ and the order of $H_j$, no subgroup of G is conjugate to a subgroup of both $H_i$ and $H_j$. Hence the claim follows from (a) by multiplying the rows corresponding to $[G{/}H_i]$ and $[G{/}H_j]$ in $\mathcal {T}(G)$. $\square$

In particular, part (b) of this proposition says that the canonical defining ideal of $\mathcal {B}(G)$ contains a polynomial of the form $x_{s-1}-x_i\,x_j$ with $i,j\in \{1,\dots ,s-2\}$. Therefore we can substitute $x_{s-1}$ by $x_i\,x_j$ in I(G) and get a simplified presentation $\mathcal {B}(G)\cong \mathbb {Z}[x_1,\dots ,x_{s-2}]/J(G)$ of $\mathcal {B}(G)$. We call J(G) the simplified defining ideal of $\mathcal {B}(G)$. The following example shows that not every canonical presentation of a Burnside ring can be simplified.

Example 4.8

Consider the table of marks of the alternating group $A_6$. The second column corresponds to the conjugacy class of a subgroup generated by a double transposition. It contains zero entries in the following rows:

$$\begin{aligned} \begin{array}{ccccccccccccccc} 120 &{}\quad 0 &{}\quad 6 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad \cdots &{}\quad 0\\ 120 &{}\quad 0 &{}\quad 0 &{}\quad 6 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad \cdots &{}\quad 0\\ 72 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 2 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad \cdots &{}\quad 0\\ 40 &{}\quad 0 &{}\quad 4 &{}\quad 4 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 4 &{}\quad 0 &{}\quad \cdots &{}\quad 0 \end{array} \end{aligned}$$

Since $A_6$ has no subgroups having 40, 72, or 120 elements, and therefore no row in $\mathcal {T}(A_6)$ starts with 9, 5, or 3, the first row of $\mathcal {T}(A_6)$ is not the product of other rows. Thus the canonical presentation of $\mathcal {B}(A_6)$ cannot be simplified.

5 Computing the prime spectrum

From the description of the additive and the multiplicative structure of $\mathcal {B}(G)$ it follows immediately that $\mathcal {B}(G)$ is a 1-dimensional reduced Cohen–Macaulay ring. The next proposition yields more precise information. It provides an important connection between the entries of the table of marks T(G) of G and the defining ideal I(G).

Proposition 5.1

For $i=1,\dots ,s$, let $p_i=(m_{s{-}1\,i},\, m_{s{-}2\,i},\dots ,m_{1i})$, and let $\mathbb {X}_G=\{p_1,\dots ,p_s\} \subset \mathbb {Z}^{s-1}$. Then the following claims hold.

• (a) The vanishing ideal

  $$\begin{aligned} \mathcal {I}(\mathbb {X}) \;=\; \{ f\in \mathbb {Z}[x_1,\dots ,x_{s-1}] \mid f(p_1) = \cdots = f(p_s) = 0\} \end{aligned}$$

  of $\mathbb {X}_G$ equals the ideal I(G).

• (b) The ideal I(G) is reduced. Its minimal primes are the linear ideals $\mathfrak {p}_i = \langle x_1-m_{s{-}1\,i},\dots , x_{s-1} - m_{1i} \rangle$, where $i\in \{1,\dots ,s\}$.

Proof

To prove (a), we note that $\mathbb {X}_G$ is a set of s points in an $(s-1)$-dimensional affine space. Since the table of marks is lower triangular and invertible, these points are in linearly general position, i.e., any subset of t points of $\mathbb {X}_G$ spans a $(t-1)$-dimensional affine space. Consequently, the vanishing ideal of $\mathbb {X}_G$ does not contain a linear polynomial and its leading term ideal with respect to a degree compatible term ordering is $\langle x_i x_j \mid i,j=1,\dots ,s\rangle$, the monomial ideal generated by all terms of degree two. Now it is clear that the generators of I(G) computed by Algorithm 2 are a Gröbner basis, and thus a system of generators, of I(G) for such a term ordering.

Now we prove (b). The vanishing ideal of a set of points is the intersection of the vanishing ideals of the individual points, in our case of the ideals $\mathfrak {p}_i$. Since these ideals are clearly prime ideals of height $s-1$, it follows that I(G) is reduced and the representation $I(G)=\mathfrak {p}_1\cap \cdots \cap \mathfrak {p}_s$ is its primary decomposition. $\square$

Part (a) of this proposition says that the ideal I(G) is an ideal of points, i.e., the vanishing ideal of a finite set of points. Let us give this point set a name.

Definition 5.2

Let $\mathcal {T}(G)=(m_{ij}) \in \mathrm{Mat}_s(\mathbb {Z})$ be the table of marks of a finite group G. For $i=1,\dots ,s$, let $p_i=(m_{s{-}1\,i},\, m_{s{-}2\,i},\dots ,m_{1i})$, and let $\mathbb {X}_G=\{p_1,\dots ,p_s\} \subset \mathbb {Z}^{s-1}$. Then $\mathbb {X}_G$ is called the set of mark points of G.

One question one may ask is whether and how the geometry of the point set $\mathbb {X}_G$ reflects group-theoretic properties of G. In view of the proposition and the fact that $\mathbb {Z}[x_1,\dots ,x_s]/\mathfrak {p}_i \cong \mathbb {Z}$, it is clear that $\mathcal {B}(G)$ is a one-dimensional reduced ring and that its prime ideals can be described as follows.

Corollary 5.3

In the setting of the proposition, the spectrum $\mathrm{Spec}(\mathcal {B}(G))$ can be described as follows.

• (a) There are s minimal primes $\bar{\mathfrak {p}}_1,\dots , \bar{\mathfrak {p}}_s$ which can be computed from the table of marks as the residue class ideals of the ideals $\mathfrak {p}_i= \langle x_1-m_{s{-}1\,i},\dots , x_{s-1} - m_{1i} \rangle$.

• (b) The maximal ideals of $\mathcal {B}(G)$ are of the form $\bar{\mathfrak {m}}_{ip}$ where $\mathfrak {m}_{ip} = \mathfrak {p}_i + \langle p\rangle$ and $p\in \mathbb {N}$ is a prime number.

• (c) A maximal ideal $\bar{\mathfrak {m}}_{ip}$ contains two minimal primes $\bar{\mathfrak {p_i}}, \bar{\mathfrak {p}}_j$ if and only if the reductions $\bar{p}_i, \bar{p}_j \in \mathbb {F}_p^{s-1}$ of the mark points $p_i$ and $p_j$ agree.

Proof

Claim (a) was already shown in the proposition and (b) follows from (a) and the isomorphism $\mathbb {Z}[x_1,\dots ,x_{s-1}]/\mathfrak {p}_i \cong \mathbb {Z}$ since every maximal ideal contains a minimal prime.

To prove (c), we note that $\mathfrak {p}_j \subset \mathfrak {m}_{ip}$ is equivalent to $x_k - m_{s{-}k\, j}\in \mathfrak {m}_{ip}$ for $k=1,\dots ,s-1$. By reducing modulo p, we see that this is equivalent to

$$\begin{aligned} x_k - \bar{m}_{s{-}k\, j} \;\in \; \langle x_1- \bar{m}_{s{-}1\, i}, \dots , x_{s-1} - \bar{m}_{1i} \rangle \end{aligned}$$

in $\mathbb {F}_p[x_1,\dots ,x_{s-1}]$, and therefore to $\bar{m}_{s{-}k\, j} = \bar{m}_{s{-}k\, i}$ for $k=1,\dots ,s-1$. $\square$

Let us determine the prime spectrum of $\mathcal {B}(G)$ in a concrete case.

Example 5.4

Let us consider a group $G=\mathbb {Z}/p^n\mathbb {Z}$ where p is a prime number and $n\ge 1$. The subgroups $H_i= \langle \bar{p}^{\,n-i}\rangle$ with $i\in \{0,\dots ,n\}$ are normal and represent all conjucagy classes of subgroups of G. Notice that $\# H_i=p^i$ for $i=0,\dots ,n$. The table of marks of G is given by

$$\begin{aligned} \mathcal {T}(G) \;=\; \begin{pmatrix} p^n &{}\quad 0 &{}\quad 0 \;\cdots \;0 &{}\quad 0 &{}\quad 0\\ p^{n-1} &{}\quad p^{n-1} &{}\quad 0 \;\cdots \;0 &{}\quad 0 &{}\quad 0\\ \vdots &{}\quad \vdots &{}\quad \vdots &{}\quad \vdots &{}\quad \vdots \\ p^2 &{}\quad p^2 &{}\quad p^2 \;\cdots \; p^2 &{}\quad 0 &{}\quad 0\\ p &{}\quad p &{}\quad p \;\cdots \; p &{}\quad p &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 1 \;\cdots \; 1 &{}\quad 1 &{}\quad 1 \end{pmatrix} \end{aligned}$$

By the corollary, the minimal primes of $\mathcal {B}(G)$ are $\bar{\mathfrak {p}}_1, \dots , \bar{\mathfrak {p}}_{n+1}$, where $\mathfrak {p}_{n+1-i} = \langle x_1-p, x_2-p^2,\dots ,x_i-p^i, x_{i+1}, \dots , x_n \rangle$ for $i=0,\dots ,n$. (Here we let $\mathfrak {p}_{n+1}= \langle x_1,\dots ,x_n\rangle$.)

Thus we see that $\mathrm{Spec}(\mathcal {B}(G))$ has $n+1$ irreducible components, and using the prime p in part (c) of the corollary, it follows that $\mathrm{Spec}(\mathcal {B}(G))$ is connected.

Via Corollary 5.3.c, we can study the connected components of $\mathrm{Spec}(\mathcal {B}(G))$. The first result in this direction was shown by A. Dress who characterized them and showed that $\mathrm{Spec}(\mathcal {B}(G))$ is connected if and only if G is solvable (see Dress 1969). In the following section we want to examine and refine this result.

6 The prime ideal graph

Let G be a finite group, let $\mathcal {T}(G)=(m_{ij})\in \mathrm{Mat}_s(\mathbb {Z})$ be its table of marks, let $\mathcal {B}(G)$ be its Burnside ring, let I(G) be its canonical defining ideal, and let $\mathbb {X}_G$ be its set of mark points. Corollary 5.3(c) allows us to interpret the connected components of $\mathrm{Spec}(\mathcal {B}(G))$ as the connected components of the following graph.

Definition 6.1

Let $\mathbb {X}_G=\{p_1,\dots ,p_s\}$ be the set of mark points of a finite group G. Using $\mathbb {X}_G$ as the set of vertices, we define a graph $\Gamma _G$ by letting $(p_i,p_j)$ be an edge of $\Gamma _G$ if and only if there exists a prime number p such that we have $\bar{p}_i = \bar{p}_j$ in $\mathbb {F}_p^{s-1}$. The graph $\Gamma _G$ is called the prime ideal graph of G.

Thus two minimal primes $\bar{\mathfrak {p}}_i,\bar{\mathfrak {p}}_j$ of the Burnside ring $\mathcal {B}(G)$ have a common point in $\mathrm{Spec}(\mathcal {B}(G))$ if and only $p_i$ and $p_j$ are joined by an edge in $\Gamma _G$. The prime ideal graph of G was first considered in Nicolson (1978). To study it further, we use the following subgroups.

Lemma 6.2

For every subgroup H of G and every prime number p, there exists a unique smallest normal subgroup $O^p(H)$ such that $H/O^p(H)$ is a p-group. It is called the p -residual subgroup of H.

Proof

This follows from the fact that, given two normal p-subgroups $K_1,K_2$ of H, the subgroup $K_1\cap K_2$ is also normal in H and we have an injective group homomorphism $H/(K_1\cap K_2) \hookrightarrow H/K_1 \times H/K_2$ given by $h(K_1\cap K_2) \mapsto (hK_1,hK_2)$. $\square$

Notice that, in general, the conjugacy class of the p-residual subgroup $O^p(H)$ of H cannot be identified using the defining property directly, since one cannot check whether a subgroup of G is conjugate to a normal subgroup of H based on $\mathcal {T}(G)$ alone (see Remark 3.10.a). However, in Sect. 7 we provide an algorithm for performing this task using the prime ideal graph $\Gamma _G$. The following proposition says that $\Gamma _G$ can be considered as a labelled graph.

Proposition 6.3

Let $\mathbb {X}_G$ be the set of mark points and $\Gamma _G$ the prime ideal graph of a finite group G. Let $p_i,p_j\in \mathbb {X}_G$ be two distinct points, and let $H_i,H_j$ be subgroups of G corresponding to the i-th and j-th column of $\mathcal {T}(G)$, respectively.

• (a) The points $p_i,p_j$ are joined by an edge in $\Gamma _G$ if and only if we have $O^p(H_i)=O^p(H_j)$.

• (b) If the two points $p_i,p_j$ are joined by an edge in $\Gamma _G$ then there exists an integer $\alpha \in \mathbb {Z}$ such that $\# H_i = p^\alpha \cdot \# H_j$.

• (c) There exists at most one prime number p such that $\mathfrak {m}_{i,p}=\mathfrak {m}_{j,p}$, i.e., there exists at most one edge in $\Gamma _G$ joining $p_i$ and $p_j$ in $\Gamma _G$.

Proof

Claim (a) was shown in Dress (1969, Prop. 1.d). Claim (b) follows immediately from (a), and (c) is a consequence of (b). $\square$

By attaching the corresponding prime p to every edge of $\Gamma _G$, we get the labelled prime ideal graph of G. Although it is straightforward, let us spell out the algorithm to compute the (labelled) prime ideal graph of G.

To get a better feeling for prime ideal graphs, let us have a look at some examples.

Example 6.4

For the cyclic group $C_8=\mathbb {Z}/8\mathbb {Z}$, there is a unique normal subgroup of each order in $\{1,2,4,8\}$. Hence the diagonal elements in the table of marks are equal to the elements in the first column and henceforth divisible by two, except for $m_{44}$, of course. Consequently, the mark points of $C_8$ all reduce to $(0,\dots ,0)$ modulo two, and $\Gamma _G$ is the following complete graph on four vertices.

As we shall see later, prime ideal graphs which are complete graphs are typical for p-groups. Also the following case of a cyclic group is easy to understand.

Example 6.5

Consider the cyclic group $C_{10} \cong C_2\times C_5$. From Proposition 6.8 below, we could determine its prime ideal graph as a direct product of graphs. Instead, let us compute it in a more direct manner. Since $C_{10}$ has unique normal subgroups of orders 1, 2, 5, and 10, its table of marks is given by

$$\begin{aligned} \mathcal {T}(C_{10}) \;=\; \begin{pmatrix} 10 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 5 &{}\quad 5 &{}\quad 0 &{}\quad 0\\ 2 &{}\quad 0 &{}\quad 2 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 \end{pmatrix} \end{aligned}$$

Hence the prime ideal graph of $C_{10}$ is the following labelled square.

For non-commutative groups, the prime ideal graph has a more intricate structure. Let us look at the simplest case first.

Example 6.6

The symmetric group $S_3$ has four conjugacy classes of subgroups: $\{e\}$, $C_2$, $A_3$, and $S_3$. It table of marks was computed in Example 3.5. If we reduce it modulo two and modulo three, we see that $\Gamma _{S_3}$ is the following labelled graph.

Our last introductory example has already some features of more general cases: complete subgraphs representing Sylow subgroups which are attached to the vertex “1” corresponding to the trivial subgroup, and “tentacles” attached to these subgraphs.

Example 6.7

The table of marks of the alternating group $A_4$ is given by

$$\begin{aligned} \mathcal {T}(A_4) \;=\; \begin{pmatrix} 12 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 6 &{}\quad 2 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 4 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 0\\ 3 &{}\quad 3 &{}\quad 0 &{}\quad 3 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 \end{pmatrix} \end{aligned}$$

Thus its prime ideal graph has the following shape.

Notice that there are two complete subgraphs attached to node “1”: one of them, labelled by “3”, is a complete graph on two vertices, and the other one, labelled by “2”, is the complete graph on three vertices. As we shall see later, they correspond to a 3-Sylow subgroup generated by a 3-cycle and the 2-Sylow subgroup $V_4$ generated by the double transpositions. It is interesting to observe that the three 2-element subgroups in $V_2$ are conjugate to each other in $A_4$ and thus contribute only one vertex to the complete subgraph. When we connect $\mathcal {B}(A_4)$ and $\mathcal {B}(V_4)$ using the restriction and induction maps, this will come to create tricky intricacies.

The next proposition explains the behaviour of $\mathcal {B}(G)$ and $\Gamma _G$ under direct products of groups.

Proposition 6.8

Let $G_1,G_2$ be two finite groups such that $\mathrm{ord}(G_1)$ and $\mathrm{ord}(G_2)$ are relatively prime.

• (a) There is a canonical isomorphism of rings $\mathcal {B}(G_1) \otimes \mathcal {B}(G_2) \longrightarrow \mathcal {B}(G_1\times G_2)$ given by $[G_1/H_1] \otimes [G_2/H_2] \mapsto [G_1\times G_2/H_1\times H_2]$.

• (b) The table of marks of $G_1\times G_2$ can be obtained by taking the Kronecker product $\mathcal {T}(G_1)\otimes \mathcal {T}(G_2)$ and reordering the rows and columns such that the subgroups representing the conjugacy classes of subgroups of $G_1\times G_2$ are listed in increasing order.

• (c) The graph $\Gamma _{G_1\times G_2}$ is the Cartesian product of the graphs $\Gamma _{G_1}$ and $\Gamma _{G_2}$, i.e., its set of vertices is the Cartesian product of the set of vertices $\{v_1,\dots ,v_s\}$ of $\Gamma _{G_1}$ and the set of vertices $\{w_1,\dots ,w_t\}$ of $\Gamma _{G_2}$, and there is an edge from $(v_i,w_j)$ to $(v_k,w_l)$ if and only if $v_i=v_k$ and there is an edge in $\Gamma _{G_2}$ from $w_j$ to $w_l$ or if $w_j=w_l$ and there is an edge in $\Gamma _{G_1}$ from $v_1$ to $v_k$.

Proof

Part (a) is contained in Karpilovsky (1995, Thm. 2.9). It follows from the fact that every subgroup of $G_1\times G_2$ is of the form $H_1\times H_2$ with subgroups $H_1\subseteq G_1$ and $H_2\subseteq G_2$ (see Karpilovsky 1995, Lemma 2.8).

To prove part (b), we first note that subgroups $H_1\times H_2$ and $K_1\times K_2$ are conjugate in $G_1\times G_2$ if and only if $H_1$, $K_1$ are conjugate in $G_1$ and $H_2$, $K_2$ are conjugate in $G_2$. Thus the size of $\mathcal {T}(G_1\times G_2)$ is the product of the sizes of $\mathcal {T}(G_1)$ and $\mathcal {T}(G_2)$. Furthermore, we have $\mathrm{m}(H_1\times H_2, K_1\times K_2)= \mathrm{m}(H_1,K_1)\cdot \mathrm{m}(H_2,K_2)$. This implies the claim.

Finally we show (c). Let p be a prime dividing $\mathrm{ord}(G_2)$. Since it does not affect the prime ideal graph, we use the reordered form $\mathcal {T}(G_1)\otimes \mathcal {T}(G_2)$ of the table of marks of $G_1\times G_2$. If we look at the last row of blocks of this matrix, we see that two columns can be equal modulo p only if the two corresponding columns of the last block are equal modulo p. Hence, if there is an edge from $(v_i, w_j)$ to $(v_k,w_l)$, then $\bar{w}_j = \bar{w}_l$.

Now there are two cases. If $i=k$, i.e., if we choose the same block column both times, the entries of $\bar{w}_j = \bar{w}_l$ are multiplied by the same numbers in each row. Hence the two columns in $\mathcal {T}(G_1)\otimes \mathcal {T}(G_2)$ are equal. But if we have $i\ne j$, i.e., if we choose columns from two distinct block columns, we can use the fact that each row of blocks is obtained by multiplying $\mathcal {T}(G_2)$ by a divisor of $\mathrm{ord}(G_1)$. Since p does not divide $\mathrm{ord}(G_1)$, the mod p reductions of the two columns of $\mathcal {T}(G_1)\otimes \mathcal {T}(G_2)$ cannot be equal.

For a prime p dividing $\mathrm{ord}(G_1)$, we argue in the same way, but using the matrix $\mathcal {T}(G_2)\otimes \mathcal {T}(G_1)$. Altogether, the claim follows. $\square$

The following example is an application of this proposition.

Example 6.9

Given the group $C_{30} \cong C_2\times C_3\times C_5$, we can apply the preceding proposition two times and get the following prime ideal graph.

For a finite abelian group, the preceding proposition allows us to reduce the description of the prime ideal graph to the case of a finite abelian p-group for some prime p. More generally, we can characterize arbitrary finite p-groups using their labelled prime ideal graph. The next lemma is a crucial ingredient for this characterization.

Lemma 6.10

Let p be a prime, let G be a finite p-group, and let $H,K\subset G$ be two non-trivial, proper subgroups of G. Then the prime p divides $m(H,K)=\# (G{/}H)^K$.

Proof

Let us write down the class equation for the operation of K on G / H:

$$\begin{aligned} \# (G{/}H) \;=\; \# (G{/}H)^K + \sum _{\{x\in G{/}H \mid K\ne K_x\}} [K : K_x] \end{aligned}$$

Since H is a proper subgroup of G, the prime p divides $\#(G{/}H)$. Since K is a non-trival p-group and $K_x \ne K$, the prime p divides all summands $[K:K_x]$. Hence p divides $\# (G{/}H)^K$. $\square$

Now we are ready to characterize p-groups as follows.

Proposition 6.11

Let G be a finite group and p a prime number. Then the following conditions are equivalent.

• (a) The group G is a p-group.

• (b) The prime ideal graph is a complete graph and all of its edges have the label p.

Proof

Given a finite p-group G, the lemma shows that p divides all elements of $\mathcal {T}(G)$ except the ones in the last row. Hence the reductions mod p of all mark points are equal to $(0,\dots ,0)$ and the prime ideal graph is a complete graph, all of whose edges are labelled by p.

Conversely, if all mod p reductions of the mark points are equal, they have to be equal to the last mark point, i.e., to zero. Thus all entries of the first $s-1$ rows of $\mathcal {T}(G)$ are powers of p. In particular, the entry $m_{11}=\mathrm{ord}(G)$ is a power of p. $\square$

Using the last two propositions, we can describe the prime ideal graph of finite abelian and, more generally, of finite nilpotent groups.

Remark 6.12

Let G be a finite nilpotent group.

• (a) It is well-known that G is the direct product of its p-Sylow subgroups. Using Proposition 6.8, it follows that the prime ideal graph of G is the direct product of the prime ideal graphs of these p-Sylow subgroups. Every p-Sylow subgroup S is a p-group. Hence its prime ideal graph is the complete graph on s nodes, where s is the number of conjugacy classes of subgroups of S.

• (b) For an abelian p-group G, the number s equals the number N(G) of its subgroups. This number has been studied classically, although it is difficult to give a closed formula for N(G). For an attempt in this direction, see Călugăreanu (2004). In simple cases it is, however, possible to evaluate N(G) in closed form. For instance, we have $N(C_p\times C_p)=p+3$ and $N(C_p\times C_p\times C_p) = 2p^2+2p+4$.

One immediate consequence of this remark is that the spectrum of the Burnside ring of a finite nilpotent group is connected. To continue with a more refined study of the prime ideal graph and the connected components of the spectrum of $\mathcal {B}(G)$, we need to introduce additional tools first.

7 The singular locus

In this section we determine the regular and the singular points in the spectrum of $\mathcal {B}(G)$. Recall that the Burnside ring is 1-dimensional and reduced. Hence its localizations at the minimal primes $\bar{\mathfrak {p}}_i$ are fields and thus regular. The following proposition classifies the maximal ideals which correspond to regular points of $\mathrm{Spec}(\mathcal {B}(G))$. We use the notation introduced in the previous sections.

Proposition 7.1

Let $i\in \{1,\dots ,s\}$, and let $p\in \mathbb {N}$ be a prime number. For the maximal ideal $\bar{\mathfrak {m}}_{ip}$ of $\mathcal {B}(G)$, the following conditions are equivalent.

• (a) The localization $\mathcal {B}(G)_{\bar{\mathfrak {m}}_{ip}}$ is a regular local ring.

• (b) The minimal prime $\bar{\mathfrak {p}}_i$ is the only minimal prime of $\mathcal {B}(G)$ contained in $\bar{\mathfrak {m}}_{ip}$.

Proof

If $\mathcal {B}(G)_{\bar{\mathfrak {m}}_{ip}}$ is a regular local ring, it is an integral domain, and therefore the zero ideal is its only minimal prime. Conversely, if the localization $\mathcal {B}(G)_{\bar{\mathfrak {m}}_{ip}}$ contains only one minimal prime, the fact that it is a reduced ring implies that it is an integral domain and that $\bar{\mathfrak {p}}_i\,\mathcal {B}(G)_{\bar{\mathfrak {m}}_{ip}}$ is the zero ideal. From the fact that $\bar{\mathfrak {m}}_{ip} = \bar{\mathfrak {p}}_{i}+\langle p\rangle$ is follows that the maximal ideal of $\mathcal {B}(G)_{\bar{\mathfrak {m}}_{ip}}$ is generated by one element, namely by $\bar{p}$, and hence that $\mathcal {B}(G)_{\bar{\mathfrak {m}}_{ip}}$ is a regular local ring. $\square$

As a consequence, we can characterize the singular locus $\mathrm{Sing}(\mathcal {B}(G))$ of the spectrum of $\mathcal {B}(G)$ as follows.

Proposition 7.2

Let $i\in \{1,\dots ,s\}$, and let $p\in \mathbb {N}$ be a prime number. For the maximal ideal $\bar{\mathfrak {m}}_{ip}$ of $\mathcal {B}(G)$, the following conditions are equivalent.

• (a) We have $\bar{\mathfrak {m}}_{ip} \in \mathrm{Sing}(\mathcal {B}(G))$.

• (b) The ring $\mathcal {B}(G)_{\bar{\mathfrak {m}}_{ip}}$ has a non-trivial zerodivisor.

• (c) There is an edge in the prime ideal graph $\Gamma _G$ starting at the mark point $p_i$ which is labeled by the prime p.

• (d) The prime p divides $[N_G(O^p(H_i)) : O^p(H_i)]$.

Proof

The equivalence of (a) and (b) was shown in the proof of the preceding proposition. The equivalence of (a) and (c) follows from Corollary 5.3.c and the preceding proposition.

Next we show that (c) implies (d). Let $j\in \{1,\dots ,s\}$ be such that $O^p(H_i)$ is conjugate to $H_j$. By Dress (1969, Prop. 1.d), there exists an edge $(p_i,p_j)$ in $\Gamma _G$ which is labeled by p. In particular, the reductions $\bar{p}_i,\bar{p}_j \in \mathbb {F}_p^{s-1}$ are the same. Notice that, by (c), we may assume that $i\ne j$. Hence the first non-zero entry $m_{jj}=[N_G(O^p(H_i)) : O^p(H_i)]$ in the j-th column of $\mathcal {T}(G)$ has to be congruent to zero modulo p.

Finally, we prove that (d) implies (c). Let $j\in \{1,\dots ,s\}$ be such that $O^p(H_i)$ is conjugate to $H_j$. Since p divides $m_{jj}=[N_G(H_j) : H_j]$, the p-Sylow subgroup of $N_G(H_j)/H_j$ is non-trivial. Hence $H_j$ is a normal subgroup of a subgroup U of G such that $U{/}H_j$ is a non-trivial p-group, and therefore we have $H_j=O^p(U)$. Letting $k\in \{1,\dots ,s\}$ be the index such that U is conjugate to $H_k$, we see that $(p_j,p_k)$ is an edge in $\Gamma _p$ which is labeled by p, and thus also $(p_i,p_k)$ is such an edge. $\square$

From this description of the set of edges of $\Gamma _G$ labeled by a fixed prime p we obtain that the corresponding graph is a disjoint union of complete graphs. Each of these complete graphs corresponds to exactly one singular maximal ideal of B(G). This yields Algorithm 4 for computing $\mathrm{Sing}(\mathcal {B}(G))$.

Given one of the complete subgraphs of $\Gamma _G$ labeled p, the vertex with the lowest index j, i.e. the conjugacy class $[H_j]$ corresponding to the smallest group $H_j$, is the one such that $O^p(H_i)$ is conjugate to $H_j$ for all $p_i$ in this subgraph. Hence we get Algorithm 5 for identifying the conjugacy class of $O^p(H_i)$.

8 Idempotents and quasi-idempotents

As in the previous sections, we let G be a finite group and we assume that we are given its table of marks $\mathcal {T}(G)=(m_{ij})\in \mathrm{Mat}_s(\mathbb {Z})$.

Definition 8.1

The $\mathbb {Q}$-algebra $\mathcal {B}_{\mathbb {Q}}(G) = \mathbb {Q}\otimes \mathcal {B}(G)$ is called the Burnside algebra of G.

By Proposition 5.1, the Burnside algebra of G is given by

$$\begin{aligned} \mathcal {B}_{\mathbb {Q}}(G) = \mathbb {Q}[x_1,\dots ,x_{s-1}]/ (\mathfrak {p}_1\cap \cdots \cap \mathfrak {p}_s) \end{aligned}$$

where $\mathfrak {p}_i = \langle x_1-m_{s{-}1\,i},\dots , x_{s-1} - m_{1i} \rangle$, for $i\in \{1,\dots ,s\}$. Thus the Chinese Remainder Theorem implies that $\mathcal {B}_{\mathbb {Q}}(G)$ is isomorphic to $\mathbb {Q}^s$. Since this ring consists only of zero-divisors and units, it is the full ring of quotients of $\mathcal {B}(G)$.

Proposition 8.2

In the above setting, let $\gamma _G:\; \mathcal {B}_{\mathbb {Q}}(G) \longrightarrow \mathbb {Q}^s$ be the composition

$$\begin{aligned} \mathcal {B}_{\mathbb {Q}}(G) \cong \mathbb {Q}[x_1,\dots ,x_{s-1}/ (\mathfrak {p}_1 \cap \cdots \cap \mathfrak {p}_s) \cong \prod _{j=1}^s \mathbb {Q}[x_1,\dots ,x_{s-1}]/\mathfrak {p}_j \cong \mathbb {Q}^s \end{aligned}$$

• (a) The map $\gamma _G$ is an isomorphism of $\mathbb {Q}$-algebras and the canonical diagram

  

  is commutative.

• (b) The ghost ring $\mathbb {Z}^{\mathcal {C}(G)}$ is isomorphic to the integral closure of the Burnside ring $\mathcal {B}(G)$ in its full ring of quotients.

Proof

First we show (a). The map $\Phi _G$ is given by $\Phi _G([G{/}H_i])= (m_{i1},\dots ,m_{is})$ for $i=1,\dots ,s$. For $i=s$, i.e., for $[G{/}H_s]=[G{/}G]$, the diagram commutes since this is the unit element in $\mathcal {B}(G)$ and all maps are algebra homomorphisms. Now let $i\in \{1,\dots ,s-1\}$ and $j=s-i$. Then $[G{/}H_i]$ is the residue class of $x_{s-i}=x_j$ under the canonical presentation. The Chinese Remainder Theorem maps $\bar{x}_j$ to $(\bar{x}_j,\dots ,\bar{x}_j)$ where the residue classes have to be taken appropriately. The isomorphism $\mathbb {Q}[x_1,\dots ,x_{s-1}] /\mathfrak {p}_k \rightarrow \mathbb {Q}$ maps $\bar{x}_j$ to $\bar{m}_{s{-}j\,k} = \bar{m}_{ik}$. Therefore the image of $[G{/}H_i]$ under $\gamma _G$ is $(m_{i1},\dots ,m_{is})$, and the diagram commutes.

Claim (b) follows from (a) since $\mathbb {Z}^s$ is integrally closed im $\mathbb {Q}^s$. $\square$

After identifying the mark homomorphism $\Phi _G:\; \mathcal {B}(G) \longrightarrow \mathbb {Z}^s$ as the embedding of the Burnside ring into its integral closure, we recall the notion of the conductor of $\mathcal {B}(G)$. This is defined as the ideal

$$\begin{aligned} \mathfrak {f}(\mathcal {B}(G)) \;=\; \{ a \in \mathcal {B}(G) \mid a\cdot \mathbb {Z}^s \subset \mathrm{Im}(\Phi _G)\} \end{aligned}$$

Equivalently, $\mathfrak {f}(\mathcal {B}(G))$ can be described as the annihilator of $\mathbb {Z}^s/\mathrm{Im}(\Phi _G)$ or as the largest ideal of $\mathcal {B}(G)$ which is also an ideal of $\mathbb {Z}^s$.

Clearly, an element $a\in \mathcal {B}(G)$ is contained in the conductor if and only if $a\ e_i \in \mathrm{Im}(\Phi _G)$ for $i=1,\dots ,s$. If we write $\Phi _G(a)=(a_1,\dots ,a_s)$, then this is equivalent to $a_i\,e_i\in \mathrm{Im}(\Phi _G)$ for $i=1,\dots ,s$. Keeping in mind that $\mathrm{Im}(\Phi _G)$ is a free $\mathbb {Z}$-module of rank s and that, consequently, $\mathbb {Z}^s/\mathrm{Im}(\Phi _G)$ is a torsion group, we are led to the following definition.

Definition 8.3

For $i=1,\dots ,s$, let $r_i$ be the smallest positive integer such that $r_i e_i \in \mathrm{Im}(\Phi )$. Then the elements $\Phi _G^{-1}(r_1 e_1), \dots , \Phi _G^{-1}(r_s e_s)$ are called the quasi-idempotents of $\mathcal {B}(G)$ and the numbers $r_i$ are called its quasi-idempotent indices.

By what we remarked above, the quasi-idempotents of $\mathcal {B}(G)$ generate the conductor $\mathfrak {f}(\mathcal {B}(G))$. Algorithm 6 computes the quasi-idempotent indices of $\mathcal {B}(G)$.

It is easy to see that this algorithm is correct and performs nothing but a stepwise diagonalization of $\mathcal {T}(G)$. To understand its performance better, we apply it to a concrete example.

Example 8.4

The table of marks of the alternating group $A_5$ is given by

$$\begin{aligned} \mathcal {T}(A_5) \;=\; \begin{pmatrix} 60 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 30 &{}\quad 2 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 20 &{}\quad 0 &{}\quad 2 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 15 &{}\quad 3 &{}\quad 0 &{}\quad 3 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 12 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 2 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 10 &{}\quad 2 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 6 &{}\quad 2 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 0\\ 5 &{}\quad 1 &{}\quad 2 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0\\ 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 \end{pmatrix} \end{aligned}$$

Now let us use Algorithm 6 to compute the quasi-idempotents of $\mathcal {B}(A_5)$.

• (1) For $i=1$, we have $(v_1,\dots ,v_s)=60 e_1$, an therefore $r_1=60$.

• (2) For $i=2$, we have to multiply $(30,2,0,\dots ,0)$ by 2 and subtract the first row of $\mathcal {T}(A_5)$. Hence we get $r_2=4$.

• (3) For $i=3$, we have to multiply $(20,0,2,0,\dots ,0)$ by 3 and subtract the first row of $\mathcal {T}(A_5)$. We obtain $r_3=6$.

• (4) For $i=4$, we first have to multiply $(15,3,0,3,0,\dots ,0)$ by two and subtract three times the second row of $\mathcal {T}(A_5)$. The intermediate result is $(-60,0,0,6,0,\dots ,0)$. Now we add the first row and conclude that $r_4=6$.

• (5) For $i=5$, we multiply $(12,0,0,0,2,0,\dots ,0)$ by 5, subtract the first row of $\mathcal {T}(A_5)$ and get $r_5=10$.

• (6) Continuing in this way, we also find $r_6=r_7=2$, $r_8=3$, and $r_9=1$.

In particular, the result $r_9=1$ says that $\mathcal {B}(A_5)$ contains the non-trivial idempotent element $\Phi _G^{-1}(e_9)$, and hence $\mathrm{Spec}(\mathcal {B}(A_5))$ is not connected.

The quasi-idempotent indices could also have been computed using the following formula.

Remark 8.5

For a positive integer n, let $\mathrm{sqfr}(n)$ be the squarefree part of n, i.e., the product of the distinct prime numbers dividing n. Nicolson (1978) proved the formula $r_i=\mathrm{sqfr}([H_i : H_i'])\cdot m_{ii}$, where $H_i'$ is the commutator subgroup of $H_i$. Since it is possible (although a bit tedious) to identify the row of $\mathcal {T}(G)$ corresponding to $H_i'$, we could have computed $r_i$ directly using this formula. However, we think that Algorithm 6 is much more efficient.

From the quasi-idempotent indices, we can compute the actual quasi-idempotents of $\mathcal {B}(G)$ as follows.

Remark 8.6

To compute the quasi-idempotents of $\mathcal {B}(G)$ from the table of marks, we can, for instance, use either of the following two ways.

• (a) In Algorithm 6, start with the tuple $e_i$ (representing the fact that we initialize v to contain the i-th row of $\mathcal {T}(G)$) and keep track of the operations performed by Steps 5 and 6 of this algorithm, so that the final tuple $(c_{i1},\dots ,c_{is})$ contains the coefficients of the representation of $r_ie_i$ in terms of the rows of $\mathcal {T}(G)$.

• (b) After running Algorithm 6, write $r_ie_i$ as a linear combination of the rows of $\mathcal {T}(G)$, i.e., write $r_ie_i = \sum _{j=1}^s c_{ij} (m_{j1},\dots ,m_{js})$ with $c_{ij}\in \mathbb {N}$.

In both cases, the elements $q_i=\sum _{j=1}^s c_{ij} [G{/}H_j]$ with $1\le i\le s$ are the quasi-idempotents of $\mathcal {B}(G)$.

Now let us turn to the actual idempotents of $\mathcal {B}(G)$. Recall that an idempotent e of $\mathcal {B}(G)$ is called a primitive idempotent if it is not of the form $e=e'+e''$ with idempotents $e',e''$ such that $e'e''=0$. The following proposition was shown in Dress (1969, Prop. 2). It characterizes the primitive idempotents of $\mathcal {B}(G)$.

Proposition 8.7

The primitive idempotents of $\mathcal {B}(G)$ are in 1–1 correspondence with the connected components of $\mathrm{Spec}(\mathcal {B}(G))$. These, in turn, are in 1–1 correspondence with the indices $i\in \{1,\dots ,s\}$ such that $H_i$ is a perfect subgroup of G.

This proposition suggests that we should first look for a way to compute the connected components of $\mathrm{Spec}(\mathcal {B}(G))$. Based on the algorithm for computing the prime ideal graph (see Algorithm 3), this can be achieved by a simple breadth-first search. Of course, for large graphs, many more efficient algorithms are well-known.

Let us apply this algorithm in the case of the group $A_5$ discussed above.

Example 8.8

For the group $A_5$, the table of marks given in Example 8.4 and Algorithm 3 indicate that there are two connected components of $\mathrm{Spec}(\mathcal {B}(G))$. To check this, we follow the steps of Algorithm 7.

• (1) First compute the prime ideal graph and get $\Gamma _G=\{(1,2), (1,3), (1,5), (2,4),(3,6), (4,8), (5,7)\}$.

• (2) Let $L=[1,2,\dots ,9]$ and $M=[\;]$.

• (4), (5) Choose $i=1$, let $L=[2,\dots ,9]$, and let $C=[1]$.

• (6)–(9) Then iteration stops when $C=[1,2,\dots ,8]$ and $L=[9]$.

• (10) Let $M=[C]$.

• (4), (5) Choose $i=9$, let $L=[\;]$, and let $C'=[9]$.

• (10) Let $M=[C,C']$.

The algorithms returns the pair $M=[[1,\dots ,8],[9]]$. Hence $\mathrm{Spec}(\mathcal {B}(G))$ has two connected components, namely the set of prime ideals containing one of $\bar{\mathfrak {p}}_1, \dots , \bar{\mathfrak {p}}_8$, and the set of prime ideals containing $\bar{\mathfrak {p}}_9$.

This example and the following one illustrate the observation in Dress (1969, p. 216), that $\mathrm{Spec}(\mathcal {B}(G))$ has two connected components, one of which is a single point, if and only if the group G is minimally simple.

Example 8.9

For the group $A_6$, Algorithm 7 finds four connected components of $\mathrm{Spec}(\mathcal {B}(G))$, namely $C=\{\bar{\mathfrak {p}}_1, \dots , \bar{\mathfrak {p}}_{19}\}$ and three isolated points. So, although the group $A_6$ is simple, it is not minimally simple.

Notice that, by the proposition above, every connected component of $\mathrm{Spec}(\mathcal {B}(G))$ contains exactly one minimal prime $\bar{\mathfrak {p}}_i$ such that $H_i$ is a perfect group. Moreover, this group is the smallest subgroup $H_i$ for which $\bar{\mathfrak {p}}_i$ is in that connected component. Hence we can find the conjugacy classes of perfect subgroups of G using the following Algorithm 8.

The correctness of this algorithm follows from Dress (1969, Prop. 2), since a subgroup U of G which has no non-trivial solvable quotients is necessarily perfect, so that the perfect subgroups are the smallest elements of the equivalence classes corresponding to the connected components of $\mathrm{Spec}(\mathcal {B}(G))$.

Corollary 8.10

Using Algorithm 8, we can check effectively whether the group G is solvable: it suffices to check whether $H_1$, the trivial subgroup, is the only perfect subgroup of G.

One last job remains to be done: we would like to have an algorithm for computing the primitive idempotents in terms of the basis of $\mathcal {B}(G)$. Knowing the connected components of $\mathrm{Spec}(\mathcal {B}(G))$, they are easy to compute because their image in the ghost ring is the sum of the corresponding primitive idempotents $e_i$ of $\mathbb {Z}^s$. Thus we get Algorithm 9.

Another way to describe the primitive idempotents of $\mathcal {B}(G)$ is the formula

$$\begin{aligned} e_{G,H_i}^\pi \;=\; (\# N_G(H_i))^{-1} \cdot \sum _{D\le N_G(H_i)} \# D \cdot \chi (D,H_i) \cdot [G{/}D] \end{aligned}$$

proved in Yoshida (1983). However, for effectively evaluating this formula, in addition to identifying the perfect subgroups $H_i$ of G, we need to determine $N_G(H_i)$, and this is in general not possible using only the knowledge of $\mathcal {T}(G)$ (see Remark 3.10). Moreover, the formula uses the Möbius function of the subgroup lattice of G which is not easy to compute. Thus we believe that Algorithm 8 is much more efficient.

For an example, let us return to the alternating group $A_5$ once again.

Example 8.11

For the group $A_5$ discussed above, the connected components of $\mathrm{Spec}(\mathcal {B}(G))$ were shown to be indexed by $C_1=\{1,2,\dots ,8\}$ and $C_2=\{9\}$. Let us compute the corresponding primitive idempotents.

• (1) To find the idempotent for $C_1$, we have to write the tuple $(1,1,\dots ,1,0)$ as a linear combination of the rows of $\mathcal {T}(G)$. The result is the coefficient tuple $(1,-2,-1,0,0,1,1,1,0)$. Hence the first primitive idempotent is

  $$\begin{aligned} e_{C_1} = [G{/}H_1]- 2[G{/}H_2] -[G{/}H_3] +[G{/}H_6] + [G{/}H_7] + [G{/}H_8] \end{aligned}$$

• (2) For the component $C_2$, we have to write $(0,0,\dots ,0,1)$ as a linear combination of the rows of $\mathcal {T}(G)$. Alternatively, using $e_{C_2}=1-e_{C_1}$, we get

  $$\begin{aligned} e_{C_2} = -[G{/}H_1]+ 2[G/H_2] +[G/H_3] -[G/H_6] - [G/H_7] - [G/H_8] + [G/H_9] \end{aligned}$$

In this example the spectrum of $\mathcal {B}(G)$ consists of a connected component and a single irreducible component. As shown in Raggi-Cárdenas and Valero-Elizondo (2005), this is equivalent to G being perfect and all subgroups of G being solvable. Let us see an example of this type where G is not a simple group.

Example 8.12

The table of marks of the group $G=\mathrm{SL}_2(\mathbb {F}_5)$ is given by

$$\begin{aligned} \mathcal {T}(\mathrm{SL}_2(\mathbb {F}_5)) = \left( \begin{array}{llllllllllll} 120 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 60 &{}\quad 60 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 40 &{}\quad 0 &{}\quad 4 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 30 &{}\quad 30 &{}\quad 0 &{}\quad 2 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 24 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 4 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 20 &{}\quad 20 &{}\quad 2 &{}\quad 0 &{}\quad 0 &{}\quad 2 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 15 &{}\quad 15 &{}\quad 0 &{}\quad 3 &{}\quad 0 &{}\quad 0 &{}\quad 3 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 12 &{}\quad 12 &{}\quad 0 &{}\quad 0 &{}\quad 2 &{}\quad 0 &{}\quad 0 &{}\quad 2 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 10 &{}\quad 10 &{}\quad 1 &{}\quad 2 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 6 &{}\quad 6 &{}\quad 0 &{}\quad 2 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 \\ 5 &{}\quad 5 &{}\quad 2 &{}\quad 1 &{}\quad 0 &{}\quad 2 &{}\quad 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 1 &{}\quad 0 \\ 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 \\ \end{array} \right) \end{aligned}$$

The second row of this table shows that $H_2$ is a normal subgroup, whence G is not simple. The prime divisors of $\# G=120$ are 2, 3, and 5. Besides $H_1=\{e\}$, the only other perfect subgroup of G turns out to be G itself: there are no subgroups of index 2 or 3, and the subgroups of index 5 are equal to their normalizer.

Thus $\mathrm{Spec}(\mathcal {B}(G))$ has two connected components. When we reduce the table modulo 2, 3, and 5, we see that $\bar{\mathfrak {p}}_1,\dots , \bar{\mathfrak {p}}_{11}$ are all in one connected component, and the irreducible component of $\bar{\mathfrak {p}}_{12}$ is the second connected component of $\mathcal {B}(G)$.

Finally, we compute the primitive idempotents. For the first component we have to write $(1,1,\dots ,1,0)$ as a linear combination of the rows of $\mathcal {T}(G)$. We get

$$\begin{aligned} e_{C_1} = [G{/}H_2] - 2[G{/}H_4] - [G{/}H_6] + [G{/}H_9] + [G{/}H_{10} + [G{/}H_{11}] \end{aligned}$$

and the other primitive idempotent is then $e_{C_2}=[G{/}H_{12}] - e_{C_1}$.