Generalized bases of finite groups

Abstract

Motivated by recent results on the minimal base of a permutation group, we introduce a new local invariant attached to arbitrary finite groups. More precisely, a subset $\Delta$ of a finite group G is called a p-base (where p is a prime) if $\langle \Delta \rangle$ is a p-group and $\mathrm {C}_G(\Delta )$ is p-nilpotent. Building on results of Halasi–Maróti, we prove that p-solvable groups possess p-bases of size 3 for every prime p. For other prominent groups, we exhibit p-bases of size 2. In fact, we conjecture the existence of p-bases of size 2 for every finite group. Finally, the notion of p-bases is generalized to blocks and fusion systems.

1 Introduction

Many algorithms in computational group theory depend on the existence of small bases. Here, a base of a permutation group G acting on a set $\Omega$ is a subset $\Delta \subseteq \Omega$ such that the pointwise stabilizer $G_\Delta$ is trivial (i.e. if $g\in G$ fixes every $\delta \in \Delta$, then $g=1$). The aim of this short note is to introduce a generalized base without the presence of a group action. To this end, let us first consider a finite group G acting faithfully by automorphisms on a p-group P. If p does not divide |G|, then G always admits a base of size 2 by a theorem of Halasi–Podoski [5]. Now suppose that G is p-solvable, P is elementary abelian, and G acts completely reducibly on P. Then G has a base of size 3 (2 if $p\ge 5$) by Halasi–Maróti [4]. In those situations, we may form the semidirect product $H:=P\rtimes G$. Now there exists $\Delta \subseteq P$ such that $|\Delta |\le 3$ and $\mathrm {C}_H(\Delta )=\mathrm {C}_H(\langle \Delta \rangle )\le P$. This motivates the following definition.

Definition 1

Let G be a finite group with Sylow p-subgroup P. A subset $\Delta \subseteq P$ is called a p-base of G if $\mathrm {C}_G(\Delta )$ is p-nilpotent, i.e. $\mathrm {C}_G(\Delta )$ has a normal p-complement.

Clearly, any generating set of P is a p-base of G since $\mathrm {C}_G(P)=\mathrm {Z}(P)\times \mathrm {O}_{p'}(\mathrm {C}_G(P))$ (this observation is generalized in Lemma 7 below).

Our main theorem extends the work of Halasi–Maróti as follows.

Theorem 2

Every p-solvable group has a p-base of size 3 (2 if $p\ge 5$).

Although Halasi–Maróti’s theorem does not extend to non-p-solvable groups, the situation for p-bases seems more fortunate. For instance, if V is a finite vector space in characteristic p, then every base of ${\text {GL}}(V)$ (under the natural action) contains a basis of V, so its size is at least $\dim V$. On the other hand, $G={\text {AGL}}(V)=V\rtimes {\text {GL}}(V)$ possesses a p-base of size 2. To see this, let P be the Sylow p-subgroup of ${\text {GL}}(V)$ consisting of the upper unitriangular matrices. Let $x\in P$ be a Jordan block of size $\dim V$. Then $\mathrm {C}_{{\text {GL}}(V)}(x)\le P\mathrm {Z}({\text {GL}}(V))$. For any $y\in \mathrm {C}_V(x)\setminus \{1\}$, we obtain a p-base $\Delta :=\{x,y\}$ such that $\mathrm {C}_G(\Delta )\le VP$. We have even found a p-base consisting of commuting elements. After checking many more cases, we believe that the following might hold.

Conjecture 3

Every finite group has a (commutative) p-base of size 2 for every prime p.

The role of the number 2 in Conjecture 3 appears somewhat arbitrary at first. There is, however, an elementary dual theorem: A finite group is p-nilpotent if and only if every 2-generated subgroup is p-nilpotent. This can be deduced from the structure of minimal non-p-nilpotent groups (see [6, Satz IV.5.4]). It is a much deeper theorem of Thompson [8] that the same result holds when “p-nilpotent” is replaced by “solvable”. Similarly, 2-generated subgroups play a role in the Baer–Suzuki theorem and its variations.

Apart from Theorem 2 we give some more evidence of Conjecture 3.

Theorem 4

Let G be a finite group with Sylow p-subgroup P. Then Conjecture 3 holds for G in the following cases:

• (i) P is abelian.

• (ii) G is a symmetric group or an alternating group.

• (iii) G is a general linear group, a special linear group, or a projective special linear group.

• (iv) G is a sporadic simple group or an automorphism group thereof.

Our results on (almost) simple groups carry over to the corresponding quasisimple groups by Lemma 8 below. The notion of p-bases generalizes to blocks of finite groups and even to fusion systems.

Definition 5

• Let B be a p-block of a finite group G with defect group D. A subset $\Delta \subseteq D$ is called base of B if B has a nilpotent Brauer correspondent in $\mathrm {C}_G(\Delta )$ (see [1, Definition IV.5.38]).

• Let ${\mathcal {F}}$ be a saturated fusion system on a finite p-group P. A subset $\Delta \subseteq P$ is called base of ${\mathcal {F}}$ if there exists a morphism $\varphi$ in ${\mathcal {F}}$ such that $\varphi (\langle \Delta \rangle )$ is fully ${\mathcal {F}}$-centralized and the centralizer fusion system ${\mathcal {C}}:=\mathrm {C}_{{\mathcal {F}}}(\varphi (\langle \Delta \rangle ))$ is trivial, i.e. ${\mathcal {C}}={\mathcal {F}}_{\mathrm {C}_P(\Delta )}(\mathrm {C}_P(\Delta ))$ (see [1, Definition I.5.3, Theorem I.5.5]).

By Brauer’s third main theorem, the bases of the principal p-block of G are the p-bases of G (see [1, Theorem IV.5.9]). Moreover, if ${\mathcal {F}}$ is the fusion system attached to an arbitrary block B, then the bases of B are the bases of ${\mathcal {F}}$ (see [1, Theorem IV.3.19]). By the existence of exotic fusion systems, the following conjecture strengthens Conjecture 3.

Conjecture 6

Every saturated fusion system has a base of size 2.

We show that Conjecture 6 holds for p-groups of order at most $p^4$.

2 Results

Proof of Theorem 2

Let G be a p-solvable group with Sylow p-subgroup P. Let $N:=\mathrm {O}_{p'}(G)$. For $Q\subseteq P$, $\mathrm {C}_G(Q)N/N$ is contained in $\mathrm {C}_{G/N}(QN/Q)$. Hence, $\mathrm {C}_G(Q)$ is p-nilpotent whenever $\mathrm {C}_{G/N}(QN/Q)$ is p-nilpotent. Thus, we may assume that $N=1$. Instead we consider $N:=\mathrm {O}_p(G)$. Since G is p-solvable, $N\ne 1$. We show by induction on |N| that there exists a p-base $\Delta \subseteq N$ such that $\mathrm {C}_G(\Delta )\le N$. By the Hall–Higman lemma (see [6, Hilfssatz VI.6.5]), $\mathrm {C}_{G/N}(N/\Phi (N))=N/\Phi (N)$ where $\Phi (N)$ denotes the Frattini subgroup of N. It follows that $\mathrm {O}_{p'}(G/\Phi (N))=1$. Hence, by induction, we may assume that N is elementary abelian. Then ${\overline{G}}:=G/N$ acts faithfully on N and it suffices to find a p-base $\Delta \subseteq N$ such that $\mathrm {C}_{{\overline{G}}}(\Delta )=1$. Thus, we may assume that $G=N\rtimes H$ where $\mathrm {C}_G(N)=N$ and $\mathrm {O}_p(H)=1$.

Note that $\Phi (G)\le \mathrm {F}(G)=N$ where $\mathrm {F}(G)$ is the Fitting subgroup of G. Since H is contained in a maximal subgroup of G, we even have $\Phi (G)<N$. Let $K\unlhd H$ be the kernel of the action of H on $N/\Phi (G)$. By way of contradiction, suppose that $K\ne 1$. Since K is p-solvable and $\mathrm {O}_p(K)\le \mathrm {O}_p(H)=1$, also $K_0:=\mathrm {O}_{p'}(K)\ne 1$. Now $K_0$ acts coprimely on N and we obtain

$$\begin{aligned}N=[K_0,N]\mathrm {C}_N(K_0)=\Phi (G)\mathrm {C}_N(K_0)\end{aligned}$$

as is well-known. Both $\Phi (G)$ and $\mathrm {C}_N(K_0)H$ lie in a maximal subgroup M of G. But then $G=NH=\Phi (G)\mathrm {C}_N(K_0)H\le M$, a contradiction. Therefore, H acts faithfully on $N/\Phi (G)$ and we may assume that $\Phi (G)=1$. Then there exist maximal subgroups $M_1,\ldots ,M_n$ of G such that $N_i:=M_i\cap N<N$ for $i=1,\ldots ,n$ and $\bigcap _{i=1}^nN_i=1$. Since $G=M_iN$, the quotients $N/N_i$ are simple $\mathbb {F}_pH$-modules and N embeds into $N/N_1\times \cdots \times N/N_n$. Hence, the action of H on N is faithful and completely reducible. Now, by the main result of Halasi–Maróti [4], there exists a p-base with the desired properties. $\square$

Next we work towards Theorem 4.

Lemma 7

Let P be a Sylow p-subgroup of G. Let $Q\unlhd P$ such that $\mathrm {C}_P(Q)\le Q$. Then every generating set of Q is a p-base of G.

Proof

Since $P\in {\text {Syl}}_p(\mathrm {N}_G(Q))$, we have $\mathrm {Z}(Q)=\mathrm {C}_P(Q)\in {\text {Syl}}_p(\mathrm {C}_G(Q))$ and therefore $\mathrm {C}_G(Q)=\mathrm {Z}(Q)\times \mathrm {O}_{p'}(\mathrm {C}_G(Q))$ by the Schur–Zassenhaus theorem.$\square$

Lemma 8

Let $\Delta$ be a p-base of G and let $N\le \mathrm {Z}(G)$. Then ${\overline{\Delta }}:=\{xN:x\in \Delta \}$ is a p-base of G/N.

Proof

Let $gN\in \mathrm {C}_{G/N}({\overline{\Delta }})$. Then g normalizes the nilpotent group $\langle \Delta \rangle N$. Hence, g acts on the unique Sylow p-subgroup P of $\langle \Delta \rangle N$. Since g centralizes

$$\begin{aligned}\langle {\overline{\Delta }}\rangle =\langle \Delta \rangle N/N=PN/N\cong P/P\cap N\end{aligned}$$

and $P\cap N\le N\le \mathrm {Z}(G)$, g induces a p-element in ${\text {Aut}}(P)$ and also in ${\text {Aut}}(\langle \Delta \rangle N)$. Consequently, there exists a p-subgroup $Q\le \mathrm {N}_G(\langle \Delta \rangle N)$ such that $\mathrm {C}_{G/N}({\overline{\Delta }})=Q\mathrm {C}_G(\Delta N)/N=Q\mathrm {C}_G(\Delta )/N$. Since $\mathrm {C}_G(\Delta )$ is p-nilpotent, so is $Q\mathrm {C}_G(\Delta )$ and the claim follows. $\square$

The following implies the first part of Theorem 4.

Proposition 9

Let P be a Sylow p-subgroup of G with nilpotency class c. Then G has a p-base of size 2c.

Proof

The $p'$-group $\mathrm {N}_G(\mathrm {Z}(P))/\mathrm {C}_G(\mathrm {Z}(P))$ acts faithfully on $\mathrm {Z}(P)$. By Halasi–Podoski [5], there exists $\Delta _0=\{x,y\}\subseteq \mathrm {Z}(P)$ such that $\mathrm {N}_H(\mathrm {Z}(P))\le \mathrm {C}_H(\mathrm {Z}(P))$ where $H:=\mathrm {C}_G(\Delta _0)$. If $c=1$, then $P=\mathrm {Z}(P)$ is abelian and Burnside’s transfer theorem implies that H is p-nilpotent. Hence, let $c>1$. By a well-known fusion argument of Burnside, elements of $\mathrm {Z}(P)$ are conjugate in H if and only if they are conjugate in $\mathrm {N}_H(\mathrm {Z}(P))$. Consequently, all elements of $\mathrm {Z}(P)$ are isolated in our situation. By the $\mathrm {Z}^*$-theorem (assuming the classification of finite simple groups), we obtain

$$\begin{aligned}\mathrm {Z}(H/\mathrm {O}_{p'}(H))=\mathrm {Z}(P)\mathrm {O}_{p'}(H)/\mathrm {O}_{p'}(H).\end{aligned}$$

The group ${\overline{H}}:=H/\mathrm {Z}(P)\mathrm {O}_{p'}(H)$ has Sylow p-subgroup ${\overline{P}}\cong P/\mathrm {Z}(P)$ of nilpotency class $c-1$. By induction on c, there exists a p-base $\overline{\Delta _1}\subseteq {\overline{P}}$ of ${\overline{H}}$ of size $2(c-1)$. We may choose $\Delta _1\subseteq P$ such that $\overline{\Delta _1}=\{{\overline{x}}:x\in \Delta _1\}$. Since $\overline{\mathrm {C}_H(\Delta _1)}\le \mathrm {C}_{{\overline{H}}}(\overline{\Delta _1})$ is p-nilpotent, so is

$$\begin{aligned}\bigl (\mathrm {C}_H(\Delta _1)\mathrm {Z}(P)\mathrm {O}_{p'}(H)/\mathrm {O}_{p'}(H)\bigr )/\mathrm {Z}(H/\mathrm {O}_{p'}(H)).\end{aligned}$$

It follows that $\mathrm {C}_H(\Delta _1)\mathrm {Z}(P)\mathrm {O}_{p'}(H)/\mathrm {O}_{p'}(H)$ and $\mathrm {C}_H(\Delta _1)=\mathrm {C}_G(\Delta _0\cup \Delta _1)$ are p-nilpotent as well. Hence, $\Delta :=\Delta _0\cup \Delta _1$ is a p-base of G of size (at most) 2c.

$\square$

Proposition 10

The symmetric and alternating groups $S_n$ and $A_n$ have commutative p-bases of size 2 for every prime p.

Proof

Let $n=\sum _{i=0}^k a_ip^i$ be the p-adic expansion of n. Suppose first that $G=S_n$. Let

$$\begin{aligned}x=\prod _{i=0}^k\prod _{j=1}^{a_i}x_{ij}\in G\end{aligned}$$

be a product of disjoint cycles $x_{ij}$ where $x_{ij}$ has length $p^i$ for $j=1,\ldots ,a_i$. Then x is a p-element and

$$\begin{aligned}\mathrm {C}_G(x)\cong \prod _{i=0}^kC_{p^i}\wr S_{a_i}.\end{aligned}$$

Since $a_i<p$, $P:=\langle x_{ij}:i=0,\ldots ,k,j=1,\ldots ,a_i\rangle$ is an abelian Sylow p-subgroup of $\mathrm {C}_G(x)$. Let $y:=\prod _{i=0}^k\prod _{j=1}^{a_i}x_{ij}^{j}\in P$. It is easy to see that $\Delta :=\{x,y\}$ is a commutative p-base of G with $\mathrm {C}_G(\Delta )=P$.

Now let $G=A_n$. If $p>2$, then x, y lie in $A_n$ as constructed above and the claim follows from $\mathrm {C}_{A_n}(\Delta )\le \mathrm {C}_{S_n}(\Delta )$. Hence, let $p=2$. If $\sum _{i=1}^ka_i\equiv 0\pmod {2}$, then we still have $x\in A_n$ and $\mathrm {C}_G(x)=\langle x_{ij}:i,j\rangle$ is already a 2-group. Thus, we have a 2-base of size 1 in this case. In the remaining case, let $m\ge 1$ be minimal such that $a_m=1$. We adjust our definition of x by replacing $x_{m1}$ with a disjoint product of two cycles of length $2^{m-1}$. Then $x\in A_n$ and $\mathrm {C}_G(x)$ is a 2-group or a direct product of a 2-group and $S_3$ (the latter case happens if and only if $m=1=a_0$). We clearly find a 2-element $y\in \mathrm {C}_G(x)$ such that $\mathrm {C}_G(x,y)$ is a 2-group. $\square$

The following elementary facts are well-known, but we provide proofs for the convenience of the reader.

Lemma 11

Let p be a prime and let q be a prime power such that $p\not \mid q$. Let $e\mid p-1$ be the multiplicative order of q modulo p. Let $p^s$ be the p-part of $q^e-1$. Then for every $n\ge 1$, the polynomial $X^{p^n}-1$ decomposes as

$$\begin{aligned}X^{p^n}-1=(X-1)\prod _{k=1}^{(p^s-1)/e}\gamma _{0,k}\prod _{i=1}^{n-s}\prod _{k=1}^{\varphi (p^s)/e}\gamma _{i,k}\end{aligned}$$

where the $\gamma _{i,k}$ are pairwise coprime polynomials in $\mathbb {F}_q[X]$ of degree $ep^i$ for $i=0,\ldots ,n-s$.

Proof

Let $\zeta$ be a primitive root of $X^{p^n}-1$ in some finite field extension of $\mathbb {F}_q$. Then

$$\begin{aligned}X^{p^n}-1=\prod _{k=0}^{p^n-1}(X-\zeta ^k).\end{aligned}$$

Recall that $\mathbb {F}_q$ is the fixed field under the Frobenius automorphism $c\mapsto c^q$. Hence, the irreducible divisors of $X^{p^n}-1$ in $\mathbb {F}_q[X]$ correspond to the orbits of $\langle q+p^n\mathbb {Z}\rangle$ on $\mathbb {Z}/p^n\mathbb {Z}$ via multiplication. The trivial orbit corresponds to $X-1$. For $i=1,\ldots ,s$, the order of q modulo $p^i$ is e by the definition of s. This yields $(p^s-1)/e$ non-trivial orbits of length e in $p^{n-s}\mathbb {Z}/p^n\mathbb {Z}$. The corresponding irreducible factors are denoted by $\gamma _{0,k}$ for $k=1,\ldots ,(p^s-1)/e$.

For $i\ge 1$, the order of q modulo $p^{s+i}$ divides $ep^i$ (it can be smaller if $p=2$ and $s=1$). We partition $(p^{n-s-i}\mathbb {Z}/p^n\mathbb {Z})^\times$ into $\varphi (p^{s+i})/(ep^i)=\varphi (p^s)/e$ unions of orbits under $\langle q+p^n\mathbb {Z}\rangle$ such that each union has size $ep^i$. The corresponding polynomials $\gamma _{i,1},\ldots ,\gamma _{i,\varphi (p^s)/e}$ are pairwise coprime (but not necessarily irreducible). $\square$

Lemma 12

Let A be an $n\times n$-matrix over an arbitrary field F such that the minimal polynomial of A has degree n. Then every matrix commuting with A is a polynomial in A.

Proof

By hypothesis, A is similar to a companion matrix. Hence, there exists a vector $v\in F^n$ such that $\{v,Av,\ldots ,A^{n-1}v\}$ is a basis of $F^n$. Let $B\in F^{n\times n}$ such that $AB=BA$. There exist $a_0,\ldots ,a_{n-1}\in F$ such that $Bv=a_0v+\cdots +a_{n-1}A^{n-1}v$. Set $\gamma :=a_0+a_1X+\cdots +a_{n-1}X^{n-1}$. Then

$$\begin{aligned}BA^iv=A^iBv=a_0A^iv+\cdots +a_{n-1}A^{n-1}A^iv=\gamma (A)A^iv\end{aligned}$$

for $i=0,\ldots ,n-1$. Since $\{v,Av,\ldots ,A^{n-1}v\}$ is a basis, we obtain $B=\gamma (A)$ as desired. $\square$

Proposition 13

The groups ${\text {GL}}(n,q)$, ${\text {SL}}(n,q)$, and ${\text {PSL}}(n,q)$ possess commutative p-bases of size 2 for every prime p.

Proof

Let q be a prime power. By Lemma 8, it suffices to consider ${\text {GL}}(n,q)$ and ${\text {SL}}(n,q)$. Suppose first that $p\mid q$. Let $x\in G:={\text {GL}}(n,q)$ be a Jordan block of size $n\times n$ with eigenvalue 1. Then x is a p-element since $x^{p^n}-1=(x-1)^{p^n}=0$. Moreover, $\mathrm {C}_G(x)$ consists of polynomials in x by Lemma 12. In particular, $\mathrm {C}_G(x)$ is abelian and therefore p-nilpotent. Hence, we found a p-base of size 1. Since $(q-1,p)=1$, this is also a p-base of ${\text {SL}}(n,q)$.

Now let $p\not \mid q$. We “linearize” the argument from Proposition 10. Let e and s be as in Lemma 11. Let $0\le a_0\le e-1$ be such that $n\equiv a_0\pmod {e}$. Let

$$\begin{aligned}\frac{n-a_0}{e}=\sum _{i=0}^{r}a_{i+1}p^i\end{aligned}$$

be the p-adic expansion. Let $M_i\in {\text {GL}}(ep^i,q)$ be the companion matrix of the polynomial $\gamma _{i,1}$ from Lemma 11 for $i=0,\ldots ,r$. Let $G_i:={\text {GL}}(ea_{i+1}p^i,q)$ and $x_i:={\text {diag}}(M_i,\ldots ,M_i)\in G_i$. Then the minimal polynomial of

$$\begin{aligned}x:={\text {diag}}(1_{a_0},x_0,\ldots ,x_r)\in G\end{aligned}$$

divides $X^{p^{r+s}}-1$ by Lemma 11. In particular, x is a p-element. Since the $\gamma _{i,1}$ are pairwise coprime, it follows that

$$\begin{aligned}\mathrm {C}_G(x)={\text {GL}}(a_0,q)\times \prod _{i=0}^r\mathrm {C}_{G_i}(x_i).\end{aligned}$$

Since $a_0<e$, ${\text {GL}}(a_0,q)$ is a $p'$-group. By Lemma 12, every matrix commuting with $M_i$ is a polynomial in $M_i$. Hence, the elements of $\mathrm {C}_{G_i}(x_i)$ have the form $A=(A_{kl})_{1\le k,l\le a_{i+1}}$ where each block $A_{kl}$ is a polynomial in $M_i$. We define

$$\begin{aligned}y_i:={\text {diag}}(M_i,M_i^2,\ldots , M_i^{a_{i+1}})\in \mathrm {C}_{G_i}(x_i)\end{aligned}$$

and $y:={\text {diag}}(1_{a_0},y_0,\ldots ,y_r)\in \mathrm {C}_G(x)$. Let $A=(A_{kl})\in \mathrm {C}_{G_i}(x_i,y_i)$. We want to show that $A_{kl}=0$ for $k\ne l$. To this end, we may assume that $k<l$ and $A_{kl}=\rho (M_i)$ where $\rho \in \mathbb {F}_q[X]$ with $\deg (\rho )<\deg (\gamma _{i,1})=ep^i$. Since $A\in \mathrm {C}_{G_i}(x_i,y_i)$, we have $M_i^kA_{kl}=M_i^lA_{kl}$ and $(M^{l-k}-1)A_{kl}=0$. It follows that the minimal polynomial $\gamma _{i,1}$ of $M_i$ divides $(X^{l-k}-1)\rho$. By way of contradiction, we assume that $\rho \ne 0$. Then $\gamma _{i,1}$ divides $X^{l-k}-1$ and $X^{p^{r+s}}-1$. However, $l-k\le a_{i+1}<p$ and $\gamma _{i1}$ must divide $X-1$. This contradicts the definition of $\gamma _{i,1}$ in Lemma 11. Hence, $A_{kl}=0$ for $k\ne l$. We have shown that the elements of $\mathrm {C}_G(x,y)$ have the form

$$\begin{aligned}L\oplus \bigoplus _{i=0}^r\bigoplus _{j=1}^{a_{i+1}}L_{ij}\end{aligned}$$

where $L\in {\text {GL}}(a_0,q)$ and each $L_{ij}$ is a polynomial in $M_i$. In particular, $\mathrm {C}_G(x,y)$ is a direct product of a $p'$-group and an abelian group. Consequently, $\mathrm {C}_G(x,y)$ is p-nilpotent.

Now let $G:={\text {SL}}(n,q)$. If $p\not \mid q-1$, then the p-base of ${\text {GL}}(n,q)$ constructed above already lies in G. Thus, we may assume that $p\mid q-1$. Then $e=1$ and $a_0=0$ with the notation above. We now have the polynomials $\gamma _{i,k}$ with $i=0,\ldots ,r$ and $k=1,\ldots ,p-1\le \varphi (p^s)$ at our disposal. Let $M_{i,k}$ be the companion matrix of $\gamma _{i,k}$. Define

$$\begin{aligned}x_i:={\text {diag}}(M_{i,1},\ldots ,M_{i,a_{i+1}})\end{aligned}$$

for $i=0,\ldots ,r$. Then the minimal polynomial of $x:={\text {diag}}(x_0,\ldots ,x_r)\in {\text {GL}}(n,q)$ has degree n and therefore $\mathrm {C}_{{\text {GL}}(n,q)}(x)$ is abelian by Lemma 12. Let $i\ge 0$ be minimal such that $a_{i+1}>0$. We replace the block $M_{i,1}$ of x by the companion matrix of $X^{p^i}-1$. Then, by Lemma 11, the minimal polynomial of x still has degree n. Moreover, x has at least one block B of size $1\times 1$. We may modify B such that $\det (x)=1$. After doing so, it may happen that B occurs twice in x. In this case, $\mathrm {C}_G(x)\le {\text {GL}}(2,q)\times H$ where H is abelian. Then the matrix

$$\begin{aligned}y:={\left\{ \begin{array}{ll} \begin{pmatrix}0&{}-1\\ 1&{}0\end{pmatrix}\oplus 1_{n-2}&{}\text {if }p=2,\\ {\text {diag}}(M_{0,1},M_{0,1}^{-1},1_{n-2})&{}\text {if }p>2 \end{array}\right. } \end{aligned}$$

lies in $\mathrm {C}_G(x)$ and $\mathrm {C}_G(x,y)$ is abelian. Hence, $\{x,y\}$ is a p-base of G. $\square$

Proposition 13 can probably be generalized to classical groups. The next result completes the proof of Theorem 4.

Proposition 14

Let S be a sporadic simple group and $G\in \{S,S.2\}$. Then G has a commutative p-base of size 2 for every prime p.

Proof

If $p^4$ does not divide |G|, then the claim follows from Lemma 7. So we may assume that $p^4$ divides |G|. From the character tables in the Atlas [2], we often find p-elements $x\in G$ such that $\mathrm {C}_G(x)$ is already a p-group. In this case, we found a p-base of size 1 and we are done. If G admits a permutation representation of “moderate” degree (including $Co_1$), then the claim can be shown directly in GAP [3]. In the remaining cases, we use the Atlas to find p-elements with small centralizers:

• $G=Ly$, $p=2$: There exists an involution $x\in G$ such that $\mathrm {C}_G(x)=2.A_{11}$. By the proof of Proposition 10, there exists $y\in A_{11}$ such that $\mathrm {C}_{A_{11}}(y)$ is a 2-group. We identify y with a preimage in $\mathrm {C}_G(x)$. Then $\mathrm {C}_G(x,y)$ is a 2-group.

• $G=Ly$, $p=3$: Here we find $x\in G$ of order 3 such that $\mathrm {C}_G(x)=3.McL$. Since McL contains a 3-element y such that $\mathrm {C}_{McL}(y)$ is a 3-group, the claim follows.

• $G=Th$, $p=2$: There exists an involution $x\in G$ such that $\mathrm {C}_G(x)=2^{1+8}_+.A_9$. As before, we find $y\in \mathrm {C}_G(x)$ such that $\mathrm {C}_G(x,y)$ is a 2-group.

• $G=M$, $p=5$: There exists a 5-element $x\in G$ such that $\mathrm {C}_G(x)=C_5\times HN$. Since there is also a 5-element $y\in HN$ such that $\mathrm {C}_{HN}(y)$ is a 5-group, the claim follows.

• $G=M$, $p=7$: In this case there exists a radical subgroup $Q\le G$ such that $\mathrm {C}_G(Q)=Q\cong C_7\times C_7$ by Wilson [9, Theorem 7] (this group was missing in the list of local subgroups in the Atlas). Any generating set of Q of size 2 is a desired p-base of G.

• $G=HN.2$, $p=3$: There exists an element $x\in G$ of order 9 such that $|\mathrm {C}_G(x)|=54$. Clearly, we find $y\in \mathrm {C}_G(x)$ such that $\mathrm {C}_G(x,y)$ is 3-nilpotent. $\square$

Finally, we consider a special case of Conjecture 6.

Proposition 15

Let ${\mathcal {F}}$ be a saturated fusion system on a p-group P of order at most $p^4$. Then ${\mathcal {F}}$ has a base of size 2.

Proof

Recall that $A:={\text {Out}}_{{\mathcal {F}}}(P)$ is a $p'$-group and there is a well-defined action of A on P by the Schur–Zassenhaus theorem. If ${\mathcal {F}}$ is the fusion system of the group $P\rtimes A$, then the claim follows from Halasi–Podoski [5] as before. We may therefore assume that P contains an ${\mathcal {F}}$-essential subgroup. In particular, P is non-abelian. Let $Q<P$ be a maximal subgroup of P containing $\mathrm {Z}(P)$. The fusion system $\mathrm {C}_{{\mathcal {F}}}(Q)$ on $\mathrm {C}_P(Q)=\mathrm {Z}(Q)$ is trivial by definition. Hence, we are done whenever Q is generated by two elements.

It remains to deal with the case where $|P|=p^4$ and all maximal subgroups containing $\mathrm {Z}(P)$ are elementary abelian of rank 3. Since two such maximal subgroups intersect in $\mathrm {Z}(P)$, we obtain that $|\mathrm {Z}(P)|=p^2$ and $|P'|=p$ by [7, Lemma 1.9] for instance. By the first part of the proof, we may choose an ${\mathcal {F}}$-essential subgroup Q such that $\mathrm {Z}(P)<Q<P$. Let $A:={\text {Aut}}_{{\mathcal {F}}}(Q)$. Since Q is essential, P/Q is a non-normal Sylow p-subgroup of A (see [1, Proposition I.2.5]). Moreover, $[P,Q]=P'$ has order p. By [7, Lemma 1.11], there exists an A-invariant decomposition

$$\begin{aligned}Q=\langle x,y\rangle \times \langle z\rangle .\end{aligned}$$

We may choose those elements such that $\Delta :=\{xz,y\}\nsubseteq \mathrm {Z}(P)$. Then $\mathrm {C}_P(\Delta )=Q$ and $\mathrm {C}_A(\Delta )=1$. Let $\varphi :S\rightarrow T$ be a morphism in ${\mathcal {C}}:=\mathrm {C}_{{\mathcal {F}}}(\Delta )$ where $S,T\le Q$. Then $\varphi$ extends to a morphism ${\hat{\varphi }}:S\langle \Delta \rangle \rightarrow T\langle \Delta \rangle$ in ${\mathcal {F}}$ such that ${\hat{\varphi }}(x)=x$ for all $x\in \langle \Delta \rangle$. Hence, if $S\le \langle \Delta \rangle$, then $\varphi ={\text {id}}$. Otherwise, $S\langle \Delta \rangle =Q$ and $\hat{\varphi } \in \mathrm {C}_A(\Delta )=1$ since morphisms are always injective. In any case, ${\mathcal {C}}$ is the trivial fusion system and $\Delta$ is a base of ${\mathcal {F}}$. $\square$