1 Introduction↩︎

One of the major areas of research in finite group theory is the investigation of the maximal subgroups of the finite simple groups [1]. When possible, it is desirable to have results that are not in the form of a classification, but crisp statements capturing uniform structural constraints. It is interesting that this can be done with respect to some natural parameter: in 2013, Burness, Liebeck and Shalev [2] proved that every maximal subgroup of every finite simple group can be generated by at most \(4\) elements. The same result for almost simple groups (with \(5\) in place of \(4\)) was proven by Lucchini, Marion and Tracey in 2020 [3]. A different type of constraint was noted by the second author in [4]. He observed that, if \(G\) is an almost simple group with a solvable maximal subgroup \(H\), then the derived length of \(H\) is at most \(10\) [4]. This is sharp if \(H\) is a solvable maximal subgroup of the sporadic Fischer group \(\mathrm{Fi}_{23}\).

In this paper, we show that the latter result is just a glimpse of a much more general phenomenon, which, similarly to the theorems in [2], [3], covers all maximal subgroups of all almost simple groups. In fact, we prove that all maximal subgroups are nearly perfect, in the sense that their derived series stabilize in a bounded number of steps. We write \(H^{(n)}\) for the \(n\)-th term of the derived series of \(H=H^{(0)}\).

We recover [4] when \(H\) is solvable (and \(H^{(10)}\) is trivial). The proof of Theorem 1 relies on the Classification of the Finite Simple Groups, which allows us to split into cases where the structure of \(H\) is well understood. When \(H\) is solvable, the work of Li and Zhang [5] yields immediate strong restrictions, but in the general case the analysis is much more delicate. One of our tools is the observation that, for any normal subgroup \(N\) of \(H\), if both \(N^{(a)}\) and \((H/N)^{(b)}\) are perfect then so is \(H^{(a+b)}\) (Proposition 4).

It is natural to ask whether Theorem 1 can be extended to maximal subgroups of perfect groups. In general, the answer is no:

However, the maximal subgroup \(H\) in Example 1 contains a large normal subgroup of \(G\), which is the source of its large derived length. In Section 4, we show that this is the only possible obstruction:

The proof of Theorem 2 is a combination of Theorem 1 and the O’Nan-Scott theorem. Generalizing [6], it was shown in [4] that in a primitive group with solvable stabilizer, there exist two points whose pointwise stabilizer has derived length bounded by an absolute constant. The results in this paper suggest the following further generalization to all primitive groups:

We point out in Example 3 that the constant in Conjecture 3 is at least \(7\).

2 Preliminaries↩︎

2.1 A generalized derived length↩︎

Every group in this paper is finite. For a finite group \(G\), let \(G=G^{(0)}\) and \[G^{(n+1)} \> = \> [G^{(n)},G^{(n)}]\] for any \(n \geq 0\) (we also write \(G'=G^{(1)}\)). Let \(G^{(\infty)} = \cap_{n \geq 1} G^{(n)}\) be the perfect core (or solvable residual) of \(G\). Since \(G\) is finite, \(G^{(\infty)}\) is the last term of the derived series, or equivalently it is the unique largest perfect subgroup (note that a group generated by perfect subgroups is perfect).

In order to deal with our main results, it is convenient to write \(\mathrm{dl^*}(G)\) for the minimum \(n\) such that \(G^{(n)}=G^{(\infty)}\) (this is also the minimum \(n\) such that \(G^{(n)} = G^{(n+1)}\)). We may also write \(\mathrm{dl}(G)\) if \(G\) is solvable, i.e. if \(G^{(\infty)} =1\). Observe that \(\mathrm{dl^*}(G) = \mathrm{dl}(G/G^{(\infty)})\) and that \(\mathrm{dl^*}(G) =0\) if and only if \(G\) is a (possibly trivial) perfect group.

A key difference between \(\mathrm{dl^*}(G)\) and the genuine derived length, which is only defined for solvable groups, is that the first can increase on subgroups in general (just take a perfect group and a non-trivial solvable subgroup). However, \(\mathrm{dl^*}(G)\) behaves well with respect to group extensions.

Corollary 1 is particularly useful when some large abelian section appears in the subnormal series. Large \(p\)-sections are also nice, as explained in the following remark:

We report another useful lemma.

A few times in the paper we deal with the derived subgroup of a particular wreath product. We isolate the following statement.

2.2 Finite simple groups↩︎

Let \(T\) be a non-abelian finite simple group. The outer automorphism group \(\mathop{\mathrm{Out}}(T)\) is the quotient of \(\mathop{\mathrm{Aut}}(T)\) by the normal subgroup of the inner automorphisms (which is isomorphic to \(T\)). The following result is a consequence of the Classification of the Finite Simple Groups.

A finite group \(G\) is almost simple if there exists a non-abelian simple group \(T\) such that \(T \unlhd G \leqslant \mathop{\mathrm{Aut}}(T)\). By Lemma 5, the almost simple groups are nearly perfect in the sense that \(\mathrm{dl^*}(G) \leq 3\), and Theorem 1 says that this property (with \(10\) in place of \(3\)) is inherited by the maximal subgroups.

We now give some preliminaries for the classical groups of Lie type. We follow the notation used in [1], which we summarize below. Let \(V\) be a vector space over a finite field \(\mathbb{F}\), and let \(\kappa\) be either the zero form, or a non-degenerate unitary, symplectic, or quadratic form on \(V\). We say that \(\kappa\) is of type \(\mathbf{L}, \mathbf{U}, \mathbf{S}\) or \(\mathbf{O}\), respectively. We may refer to type \(\mathbf{U}\) as \(\mathbf{L}^-\). Additionally, type \(\mathbf{O}\) splits into three further types: \(\mathbf{O}^+\) and \(\mathbf{O}^-\), when \(\dim(V)\) is even; and \(\mathbf{O}^\circ\), when \(\dim(V)\) is odd. We will refer to \(\mathbf{O}^\circ\) simply as \(\mathbf{O}\). As in [1], we define:

• \(S = S(V, \mathbb{F}, \kappa)\) the group of special \(\kappa\)-isometries of \(V\);

• \(I = I(V, \mathbb{F}, \kappa)\) the group of \(\kappa\)-isometries of \(V\);

• \(\Delta = \Delta(V, \mathbb{F}, \kappa)\) the group of \(\kappa\)-similarities of \(V\);

• \(\Gamma = \Gamma(V, \mathbb{F}, \kappa)\) the group of \(\kappa\)-semisimilarities of \(V\);

• If \(\kappa\) is of type \(\mathbf{O}\), let \(\Omega = \Omega(V, \mathbb{F}, \kappa)\) be an index \(2\) subgroup of \(S\), as defined in [1]. Otherwise, let \(\Omega = S\);

• If \(\kappa\) is of type \(\mathbf{L}\) with \(\dim(V) \geq 3\), let \(\Sigma = \Sigma(V, \mathbb{F}, \kappa) = \langle \Gamma, \iota \rangle\), where \(\iota\) is the inverse transpose map on \(\Gamma\). Otherwise, let \(\Sigma = \Gamma\).

We have the chain of groups \[\Omega \> \leqslant \> S \> \leqslant \> I \> \leqslant \> \Delta \> \leqslant \> \Gamma \> \leqslant \> \Sigma,\] with \(\Gamma\) acting on the vector space \(V\). For example, if \(V=\mathbb{F}_q^n\) and \(\kappa\) is the zero form, then \(\Omega = S = \mathop{\mathrm{SL}}_n(q)\), \(I = \Delta = \mathop{\mathrm{GL}}_n(q)\), \(\Gamma = \mathop{\mathrm{\Gamma L}}_n(q)\) and \(\Sigma = \langle \Gamma, \iota \rangle\).

For any subgroup \(K\) of \(\Sigma\), we write \(\overline{K}\) to denote the image of \(K\) under the quotient by the scalars of \(\Sigma\). Similarly, for any subgroup \(K\) of \(\overline{\Sigma}\), we use \(K^\wedge\) to denote the preimage of \(K\) in \(\Sigma\).

3 Proof of Theorem 1↩︎

Let \(G\) be an almost simple group with socle \(T\), and let \(H\) be a maximal subgroup of \(G\). We show that \(\mathrm{dl^*}(H) \leq 10\). By Lemma 5 we have that \(H/T \cap H\) is solvable (as a subgroup of \(G/T\)), and so \(H^{(\infty)} \subseteq T \cap H\). It follows by Lemma 1 and Proposition 4 that \[\mathrm{dl^*}(T \cap H) \> \leq \> \mathrm{dl^*}(H) \> \leq \> \mathrm{dl^*}(T \cap H) + \mathrm{dl}(G/T) .\] In most cases, it will be sufficient to use the weaker bound \[\mathrm{dl^*}(H) \> \leq \> \mathrm{dl^*}(T \cap H) + \mathrm{dl}( \mathop{\mathrm{Out}}(T) ) .\] Note that, in general, \(T \cap H\) is not a maximal subgroup of \(T\).

Using the Classification of the Finite Simple Groups, we distinguish various cases.

3.1 \(T\) is alternating↩︎

We address this case with the following proposition. To simplify the argument, we use the main theorem of [9].

In the exceptional case where \(T=\mathop{\mathrm{Alt}}(6)\), it can be checked easily that \(\mathrm{dl^*}(H) \leq 3\).

3.2 \(T\) is a classical group↩︎

Let \(T\) be a classical simple group over \(\mathbb{F} = \mathbb{F}_{q^u}\) where \(q=p^r\) and \(u\) is either \(2\) if \(T\) is of type \(\mathbf{U}\), or \(1\) otherwise. Then there exists a classical geometry \((V, \mathbb{F}, \kappa)\) such that \(T = \overline{\Omega}(V, \mathbb{F}, \kappa)\), and \(\overline{\Omega} \leqslant G \leqslant \mathop{\mathrm{Aut}}(\overline{\Omega})\). We assume first that \(G \subseteq \overline{\Sigma}\), and treat the exceptional cases at the end of this section. We split into different types of maximal subgroups \(H<G\) by using Aschbacher’s Theorem (see [1]). We assume that \(H\) is not solvable, as this case is already treated in [4]. Finally, we delay the case where \(H\) is parabolic to Section 3.4, where we are able to consider all groups of Lie type simultaneously.

1. \(\mathscr C_1\): We only treat the non-parabolic cases. Suppose \(H\) is of type \(\mathscr C_1\). We assume first that \(G\) is of type \(\mathbf{L}\), and \(H\) is of type \(\mathop{\mathrm{GL}}_m(q) \oplus \mathop{\mathrm{GL}}_{n-m}(q)\), so \(H\) stabilizes a vector space decomposition \(V=V_1 \oplus V_2\) with \(\dim(V_1) = m\) and \(\dim(V_2) = n-m\). Recall that \(H^\wedge\) denotes the preimage of the subgroup \(H\) of \(\overline{\Sigma}\) in \(\Sigma\). Then, we have that \[\mathop{\mathrm{SL}}_m(q) \times \mathop{\mathrm{SL}}_{n-m}(q) \leqslant H^\wedge \cap \Gamma \leqslant \mathop{\mathrm{\Gamma L}}_m(q) \times \mathop{\mathrm{\Gamma L}}_{n-m}(q).\] Since \(\mathop{\mathrm{\Gamma L}}_i(q)/\mathop{\mathrm{SL}}_i(q)\) is solvable for all \(i\), this implies that \((\mathop{\mathrm{\Gamma L}}_m(q) \times \mathop{\mathrm{\Gamma L}}_{n-m}(q))^{(\infty)}\) is contained in \(H^\wedge \cap \Gamma\). As such, by Lemma 1, \[\begin{align} \mathrm{dl^*}(H \cap \overline{\Gamma}) & \leq \mathrm{dl^*}(H^\wedge \cap \Gamma)\\ & \leq \mathrm{dl^*}(\mathop{\mathrm{\Gamma L}}_m(q) \times \mathop{\mathrm{\Gamma L}}_{n-m}(q))\\ & \leq \max(\mathrm{dl^*}(\mathop{\mathrm{\Gamma L}}_m(q)), \mathrm{dl^*}(\mathop{\mathrm{\Gamma L}}_{n-m}(q))) \\ & \leq 4, \end{align}\] where the final inequality is by Lemma 6. As such, by Proposition 4, \[\mathrm{dl^*}(H) \leq \mathrm{dl^*}(H \cap \overline{\Gamma}) + \mathrm{dl}(\overline{\Sigma}/\overline{\Gamma}) \leq 5.\]

   Assume that \(H\) is of type \(\mathop{\mathrm{GU}}_m(q) \perp \mathop{\mathrm{GU}}_{n-m}(q)\), \(\mathop{\mathrm{Sp}}_m(q) \perp \mathop{\mathrm{Sp}}_{n-m}(q)\) or \(\relax_m^{\epsilon_1}(q) \perp \relax_{n-m}^{\epsilon_2}(q)\). In these cases \(G\) is not of type \(\mathbf{L}\), and as such \(\Sigma = \Gamma\). There exist non-degenerate vector subspaces \(V_1, V_2\) of \(V\) with \(V = V_1 \perp V_2\), and \(H\) equal to \(N_G(V_1)\). Define forms \(\kappa_i\) on each \(V_i\) by restricting \(\kappa\) to \(V_i\). Then, \[\Omega(V_1, \mathbb{F}, \kappa_1) \times \Omega(V_2, \mathbb{F}, \kappa_2) \leqslant H^{\wedge} \leqslant \Gamma(V_1, \mathbb{F}, \kappa_1) \times \Gamma(V_2, \mathbb{F}, \kappa_2).\] The groups \(\Gamma(V_i, \mathbb{F}, \kappa_i)/\Omega(V_i, \mathbb{F}, \kappa_i)\) are solvable, and hence \[\mathrm{dl^*}(H) \leq \mathrm{dl^*}(H^\wedge) \leq \max_i \left( \mathrm{dl^*}(\Gamma(V_i, \mathbb{F}, \kappa_i))\right) \leq 6.\]

   Finally, if \(H\) is of type \(\mathop{\mathrm{Sp}}_{n-2}(q)\) then \(H\) is a classical group of type \(\mathbf{S}\), and so \(\mathrm{dl^*}(H) \leq 3\).

2. \(\mathscr C_2\): Suppose \(H\) is of type \(\mathop{\mathrm{GL}}_m(q) \wr_t \mathop{\mathrm{Sym}}(t)\). Then \(H\) stabilizes a vector space decomposition \(V=V_1 \oplus \cdots \oplus V_t\) with \(\dim(V_i) = m\) for all \(i\). Thus, \[\mathop{\mathrm{SL}}_m(q) \wr_t \mathop{\mathrm{Alt}}(t) \leqslant H^{\wedge} \cap \Gamma \leqslant \mathop{\mathrm{\Gamma L}}_m(q) \wr_t \mathop{\mathrm{Sym}}(t).\] Let \(J\) be the projection of \(H\) to \(\mathop{\mathrm{Sym}}(t)\), which must be either \(\mathop{\mathrm{Alt}}(t)\) or \(\mathop{\mathrm{Sym}}(t)\). Hence, by Proposition 4, \[\mathrm{dl^*}(H^\wedge \cap \Gamma) \leq \mathrm{dl^*}(\mathop{\mathrm{\Gamma L}}_m(q)^t \cap H) + \mathrm{dl^*}(J).\] However, \(\mathop{\mathrm{\Gamma L}}_m(q)^t \cap H\) contains \(\mathop{\mathrm{SL}}_m(q)^t\), and hence contains \((\mathop{\mathrm{\Gamma L}}_m(q)^t)^{(\infty)}\). Thus, by Lemma 1, \[\begin{align} \mathrm{dl^*}(H^\wedge \cap \Gamma) & \leq \mathrm{dl^*}(\mathop{\mathrm{\Gamma L}}_m(q)^t) + \mathrm{dl^*}(J) \\ & \leq \mathrm{dl^*}(\mathop{\mathrm{\Gamma L}}_m(q)) + \mathrm{dl^*}(\mathop{\mathrm{Sym}}(t)). \end{align}\] If \(\mathop{\mathrm{PSL}}_m(q)\) is a simple group, then \(\mathrm{dl^*}(\mathop{\mathrm{\Gamma L}}_m(q)) \leq 3\) by Lemma 6. Since \(\mathrm{dl^*}(\mathop{\mathrm{Sym}}(t)) \leq 3\) for all \(t\), this implies that \(\mathrm{dl^*}(H^{\wedge} \cap \Gamma) \leq 6\). On the other hand, if \(\mathop{\mathrm{PSL}}_m(q)\) is not simple we have that \(\mathrm{dl^*}(\mathop{\mathrm{\Gamma L}}_m(q)) \leq 4\). However, in this case we may assume that \(\mathop{\mathrm{Alt}}(t)\) is simple, so that \(H\) is non-solvable, and hence \(\mathrm{dl^*}(\mathop{\mathrm{Sym}}(t)) = 1\). Hence we have that \(\mathrm{dl^*}(H^\wedge \cap \Gamma) \leq 5\). Thus, by Proposition 4, \[\mathrm{dl^*}(H) \leq \mathrm{dl^*}(H^{\wedge}) \leq \mathrm{dl^*}(H^{\wedge} \cap \Gamma) + \mathrm{dl}(\Sigma/\Gamma) \leq 7.\]

   Suppose next that \(H\) is of type \(\mathop{\mathrm{GU}}_m(q) \wr_t \mathop{\mathrm{Sym}}(t)\), \(\mathop{\mathrm{Sp}}_m(q)\wr_t \mathop{\mathrm{Sym}}(t)\), or \(\relax_m^\epsilon(q) \wr_t \mathop{\mathrm{Sym}}(t)\). Note that \(G\) is not of type \(\mathbf{L}\) in this case. Then there exist some pairwise isometric, non-degenerate vector spaces \(V_1, \dots, V_t\) of \(V\) such that \(V = V_1 \perp \cdots \perp V_t\), with \(H\) equal to \(N_G\{V_1, \dots, V_t\}\). Define forms \(\kappa_i\) on each \(V_i\) by restricting \(\kappa\) to \(V_i\), and let \(\Omega_i = \Omega(V_i, \mathbb{F}, \kappa_i)\), and \(\Gamma_i = \Gamma(V_i, \mathbb{F}, \kappa_i)\) for each \(i\). Note that all of the groups \(\Omega_i\) are isomorphic, as are all of the groups \(\Gamma_i\). Then, \[\Omega_1 \wr_t \mathop{\mathrm{Alt}}(t) \leqslant H^{\wedge} \leqslant \Gamma_1 \wr_t \mathop{\mathrm{Sym}}(t).\] Hence, similarly to the \(\mathbf{L}\) case, \[\begin{align} \mathrm{dl^*}(H^{\wedge}) & \leq \mathrm{dl^*}(\Gamma_1) + \mathrm{dl^*}(\mathop{\mathrm{Sym}}(t)) \\ & \leq \begin{cases*} 3 + 3 & if \overline{\Omega}_1 is simple\\ 6 + 1 & if t \geq 5. \end{cases*} \end{align}\] So \(\mathrm{dl^*}(H) \leq 7\), assuming \(H\) is non-solvable.

   If \(H\) is of type \(\mathop{\mathrm{GL}}_{n/2}(q^u).2\), then \(\mathop{\mathrm{SL}}_{n/2}(q^u) \leqslant H^\wedge \cap \Omega \leqslant \mathop{\mathrm{GL}}_{n/2}(q^u).2\), and so by Proposition 4 and Lemma 6 we have that \[\begin{align} \mathrm{dl^*}(H) & \leq \mathrm{dl^*}(H^\wedge \cap \Omega) + \mathrm{dl}(\overline{\Sigma}/\overline{\Omega}) \\ & \leq \mathrm{dl^*}(\mathop{\mathrm{GL}}_{n/2}(q^u)) + \mathrm{dl}(C_2) + \mathrm{dl}(\overline{\Sigma}/\overline{\Omega})\\ & \leq 4 + 1 + 2. \end{align}\]

   If \(H\) is of type \(\relax_{n/2}(q)^2\), then \(H^\wedge\) is contained in \(\mathop{\mathrm{\Gamma O}}_{n/2}(q)^2 : C_2\) and contains \(\Omega_{n/2}(q)^2\), so as previously we have that \(\mathrm{dl^*}(H^\wedge) \leq 6\).

3. \(\mathscr C_3\): For \(H\) in this class there exists a field extension \(\mathbb{F}_\sharp\) of \(\mathbb{F}\) and a classical geometry \((V_\sharp, \mathbb{F}_\sharp, \kappa_\sharp)\) such that \[\Omega(V_\sharp, \mathbb{F}_\sharp, \kappa_\sharp) \leqslant H^\wedge \cap \Gamma \leqslant \Gamma(V_\sharp, \mathbb{F}_\sharp, \kappa_\sharp).\] When \(G\) is not of type \(\mathbf{L}\), we have that \(\Sigma = \Gamma\), and so \(H^{\wedge} = H^\wedge \cap \Gamma\). Thus, by Lemmas 1 and 6, \[\mathrm{dl^*}(H) \leq \mathrm{dl^*}(\Gamma(V_\sharp, \mathbb{F}_\sharp, \kappa_\sharp)) \leq 6.\] On the other hand, if \(G\) is of type \(\mathbf{L}\), then \(\Sigma = \Gamma :2\). Note that, by [1], the geometry \((V_\sharp, \mathbb{F}_\sharp, \kappa_\sharp)\) is also of type \(\mathbf{L}\) in this case. Hence, by Lemma 6 and Proposition 4, \[\mathrm{dl^*}(H) \leq \mathrm{dl^*}(\Gamma(V_\sharp, \mathbb{F}_\sharp, \kappa_\sharp)) + 1 \leq 5.\]

4. \(\mathscr C_4\): Assume \(H\) is of type \(\mathop{\mathrm{GL}}_{n_1}(q) \otimes \mathop{\mathrm{GL}}_{n_2}(q)\), where \(n_1 n_2 = n\). Then, \(H\) stabilizes a tensor product decomposition \(V=V_1 \otimes V_2\), with \(\dim(V_i) = n_i\). We have \[\mathop{\mathrm{PSL}}_{n_1}(q) \times \mathop{\mathrm{PSL}}_{n_2}(q) \leqslant H \cap \overline{\Gamma} \leqslant \mathop{\mathrm{P \Gamma L}}_{n_1}(q) \times \mathop{\mathrm{P \Gamma L}}_{n_2}(q).\] Thus, as in the \(\mathscr{C}_1\) case, this implies that \(\mathrm{dl^*}(H \cap \overline{\Gamma}) \leq 3\) and so \(\mathrm{dl^*}(H) \leq 4\).

   If \(H\) is instead of type \(\mathop{\mathrm{GU}}_{n_1}(q) \otimes \mathop{\mathrm{GU}}_{n_2}(q)\), \(\mathop{\mathrm{Sp}}_{n_1}(q) \otimes \relax_{n_2}^\epsilon (q)\), \(\mathop{\mathrm{Sp}}_{n_1}(q) \otimes \mathop{\mathrm{Sp}}_{n_2} (q)\) or \(\relax_{n_1}^{\epsilon_1}(q) \otimes \relax_{n_2}^{\epsilon_2} (q)\), then \(H\) stabilizes a vector space decomposition \(V=V_1 \otimes V_2\) for some vector spaces \(V_1, V_2\) with forms \(\kappa_i\) on each \(V_i\). Thus, \[\overline{\Omega}(V_1, \mathbb{F}, \kappa_1) \times \overline{\Omega}(V_2, \mathbb{F}, \kappa_2) \leqslant H \leqslant \overline{\Gamma}(V_1, \mathbb{F}, \kappa_1) \times \overline{\Gamma}(V_2, \mathbb{F}, \kappa_2).\] We hence obtain that \(\mathrm{dl^*}(H) \leq 6\), as before.

5. \(\mathscr C_5\): For \(H\) in this class, there exists a subfield \(\mathbb{F}_\sharp\) of \(\mathbb{F}\) and a classical geometry \((V_\sharp, \mathbb{F}_\sharp, \kappa_\sharp)\) as described in [1], such that \[\overline{\Omega}(V_\sharp, \mathbb{F}_\sharp, \kappa_\sharp) \leqslant H \cap \overline{\Delta} \leqslant \overline{\Delta}(V_\sharp, \mathbb{F}_\sharp, \kappa_\sharp)\] So \(\mathrm{dl^*}(H \cap \overline{\Delta}) \leq 5\) and hence \(\mathrm{dl^*}(H) \leq 6\).

6. \(\mathscr C_6\): Recall we exclude any types for which \(H\) is always solvable. If \(H\) is of type \((4 \circ 2 ^{1+2m}). \mathop{\mathrm{Sp}}_{2m}(2)\) with \(m > 1\), then \(H \leqslant (2^{2m} . \mathop{\mathrm{Sp}}_{2m}(2)) \times 2\). If \(m=2\), then \(\mathop{\mathrm{Sp}}_4(2)' = \mathop{\mathrm{Alt}}(6)\), and \(\mathop{\mathrm{Alt}}(6)\) acts irreducibly on the natural module of \(\mathop{\mathrm{Sp}}_4(2)\). Hence, \(H^{(\infty)}\) contains \(2^4.\mathop{\mathrm{Alt}}(6)\), so \(\mathrm{dl^*}(H) \leq 1\). Otherwise, if \(m \geq 3\), then \(H^{(\infty)}\) contains \(2^{2m}.\mathop{\mathrm{Sp}}_{2m}(2)\), so \(\mathrm{dl^*}(H) \leq 1\).

   Similarly, if \(H\) is of type \(r^{1+2m}.\mathop{\mathrm{Sp}}_{2m}(r)\), with \(r\) an odd prime and \(m > 1\), or if it is of type \(2_\pm^{1+2m}.\relax_{2m}^\pm(2)\) with \(m > 2\), then \(\mathrm{dl^*}(H) \leq 1\).

7. \(\mathscr C_7\): Assume first that \(H\) is of type \(\mathop{\mathrm{GL}}_m(q) \wr_t \mathop{\mathrm{Sym}}(t)\). This implies that \(H\) stabilizes a tensor product decomposition \(V = V_1 \otimes \cdots \otimes V_t\) for some vector spaces \(V_i\) with \(\dim(V_i) = m\). Note that \(\mathop{\mathrm{PSL}}_m(q)\) must be simple in this case. Then, \[\mathop{\mathrm{PSL}}_m(q) \wr_t \mathop{\mathrm{Alt}}(t) \leqslant H \cap \overline{\Gamma} \leqslant \mathop{\mathrm{P \Gamma L}}_m(q) \wr_t \mathop{\mathrm{Sym}}(t).\] Thus, as in the \(\mathscr C_2\) case, we can obtain that \(\mathrm{dl^*}(H) \leq 6\). Similarly, for the other types of \(H\) in this class, \(H\) stabilizes a decomposition \(V=V_1 \otimes \cdots \otimes V_t\). For each \(i\) we can define a form \(\kappa_i\) on \(V_i\) as in [1], with \(V_1, \dots, V_t\) pairwise similar. Then, \[\overline{\Omega}(V_1, \mathbb{F}, \kappa_1) \wr_t \mathop{\mathrm{Alt}}(t) \leqslant H \leqslant \overline{\Gamma}(V_1, \mathbb{F}, \kappa_1) \wr_t \mathop{\mathrm{Sym}}(t).\] Note that \(\overline{\Omega}(V_1, \mathbb{F}, \kappa_1)\) must be non-solvable, by definition. Hence \(\mathrm{dl^*}(H) \leq 5\).

8. \(\mathscr C_8\): Let \((V_\sharp, \mathbb{F}_\sharp, \kappa_\sharp)\) be a classical geometry as described in [1], and let \(\Gamma_\sharp = \Gamma(V_\sharp, \mathbb{F}_\sharp, \kappa_\sharp)\). Then, \(H \leqslant \overline{\Gamma}_\sharp . 2\), and hence \(\mathrm{dl^*}(H) \leq 6\).

9. \(\mathscr S\): In this case \(\mathrm{dl^*}(H) \leq 3\) by Lemma 5.

Finally, we consider the case where \(G\) is not a subgroup of \(\overline{\Sigma}\). As mentioned in Remark 7, we must have that \(T = \overline{\Omega}\) is equal to \(\mathop{\mathrm{PSp}}_4(q)\) or \(\mathop{\mathrm{P \Omega}}_8^+(q)\). Additionally, the group \(\mathop{\mathrm{Aut}}(\overline{\Omega})\) is generated by \(\overline{\Sigma}\) and a graph automorphism of order \(2\) or \(3\) respectively.

If \(H \cap \overline{\Sigma}\) is solvable, then so is \(H\) and we are done. If \(H \cap \overline{\Sigma}\) is one of the non-solvable groups considered above, then we have proven that \(\mathrm{dl^*}(H \cap \overline{\Sigma}) \leq 7\), and hence \(\mathrm{dl^*}(H) \leq 8\). Thus, we can assume that \(H \cap \overline{\Sigma}\) is not in the classes \(\mathscr{C}_1\) to \(\mathscr{C}_8\), nor \(\mathscr{S}\).

We use [11] to study the remaining cases. If \(T = \mathop{\mathrm{PSp}}_4(q)\), then \(H\) is solvable. Hence we assume \(T = \mathop{\mathrm{P \Omega}}_8^+(q)\). Let \(d=(q-1,2)\). We have the following possibilities for \(H^{\wedge} \cap \Omega\), assuming it is non-solvable and not contained in \(\mathscr C_1\)-\(\mathscr C_8\) nor \(\mathscr S\): \(d \times G_2(q)\); \((\Omega_2^\pm(q) \times \frac{1}{d} \mathop{\mathrm{GL}}_3(q)).[2d]\); and \(2^4.2^6.\relax_3(2)\). We treat each case individually.

If \(H^\wedge \cap \Omega = d \times G_2(q)\), we can assume that \(G_2(q)\) is not solvable, and hence \(\mathrm{dl^*}(H^{\wedge} \cap \Omega) =1\). Thus, by Proposition 4 and Lemma 5, \(\mathrm{dl^*}(H) \leq 4\). If \(H^\wedge \cap \Omega = (\Omega_2^\pm(q) \times \frac{1}{d} \mathop{\mathrm{GL}}_3(q)).[2d]\), then \(\mathrm{dl^*}(H \cap \overline{\Omega}) \leq 3\) and so \(\mathrm{dl^*}(H) \leq 6\). If \(H^\wedge \cap \Omega = 2^4.2^6.\relax_3(2)\), then \(\mathrm{dl^*}(H \cap \overline{\Omega}) \leq 2 + \mathrm{dl^*}(\relax_3(2)) = 2\), and so \(\mathrm{dl^*}(H) \leq 5\).
To conclude this part, we observe that there are infinitely many almost simple classical groups \(G\) and maximal subgroups \(H\) with \(\mathrm{dl^*}(H)=6\):

3.3 \(T\) is exceptional↩︎

By Remark 6, we can assume that \(T\) is one among \[F_4(q) , \> ^2F_4(q) , \> E_6(q) , \> ^2E_6(q) , \> E_7(q) , \>orE_8(q) .\]

Let \(\widetilde{G}\) be the algebraic group corresponding to \(T\), and let \(\sigma\) be a Steinberg endomorphism such that \(\widetilde{G}_\sigma' = T\). Note that \(\widetilde{G}_\sigma\) is equal to \(\operatorname{Inndiag}(T)\), the subgroup of \(\mathop{\mathrm{Aut}}(T)\) generated by \(T\) and the diagonal automorphisms of \(T\).

We will use the following result from [12].

We start analyzing the various possibilities. If \(H\) is of type (i), and \(D\) is not a torus, the possibilities for \(N_{\widetilde{G}_\sigma}(D_\sigma)\) are given in [13]. By Proposition 4 we have \(\mathrm{dl^*}(H) \leq \mathrm{dl^*}(H \cap \widetilde{G} _\sigma) + \mathrm{dl^*}(H/ H \cap \widetilde{G}_\sigma)\). Now \(H \cap \widetilde{G} _\sigma = N_{\widetilde{G}_\sigma}(D_\sigma) \cap G\), so \(N_{\widetilde{G}_\sigma}(D_\sigma)^{(\infty)} \subseteq H \cap \widetilde{G}_\sigma\) and \(\mathrm{dl^*}(H \cap \widetilde{G}_\sigma) \leq \mathrm{dl^*}(N_{\widetilde{G}_\sigma}(D_\sigma))\) by Lemma 1. Moreover, \(H/ H \cap \widetilde{G}_\sigma\) is isomorphic to a subgroup of the solvable group \(G \widetilde{G}_\sigma/ \widetilde{G}_\sigma\). Therefore \[\mathrm{dl^*}(H) \> \leq \> \mathrm{dl^*}(N_{\widetilde{G}_\sigma}(D_\sigma)) + \mathrm{dl}(G/ G \cap \widetilde{G}_\sigma).\] Using this inequality, we can prove that \(\mathrm{dl^*}(H) \leq 9\) for all groups in [13]. We remark on one of the more complex cases: suppose \(T=E_8(q)\) and \(N_{\widetilde{G}_\sigma}(D_\sigma)= d^4.(\relax_2(q))^8.d^4.\mathop{\mathrm{AGL}}_3(2)\). If \(q\) is \(2\) or \(3\), we can directly construct this group in Magma [15], and hence show that \(\mathrm{dl^*}(N_{\widetilde{G}_\sigma}(D_\sigma)) \leq 1\). When \(q \geq 4\) instead, we can show that \(d^4:\mathop{\mathrm{AGL}}_3(2)\) is perfect and hence \(N_{\widetilde{G}_\sigma}(D_\sigma)\) is also perfect. The other cases are easy, with the largest bounds on \(\mathrm{dl^*}(H)\) usually occurring when \(q\) is \(2\) or \(3\).

If \(H\) is of type (i) and \(D\) is a maximal torus, we similarly consider every case in [13]. In each case we can show that \(\mathrm{dl^*}(N_{\widetilde{G}_\sigma}(D_\sigma)/D_\sigma) \leq 4\), so by Corollary 1 we have that \(\mathrm{dl^*}(H) \leq \mathrm{dl^*}(D_\sigma) + 4 + \mathrm{dl^*}(G/(\widetilde{G}_\sigma \cap G)) \leq 7\).

If \(H\) is of type (ii), then there exists \(q_0 = q^{1/r}\), with \(S\) equal to one of \(F_4(q_0)\), \({}^2F_4(q_0)\), \(E_6(q_0)\), \({}^2E_6(q_0)\), \(E_7(q_0)\) or \(E_8(q_0)\), such that \(S \leqslant T \cap H \leqslant \mathop{\mathrm{Aut}}(S)\). Since \(S\) is always simple, we have \(\mathrm{dl^*}(T \cap H) \leq 2\), and thus \(\mathrm{dl^*}(H) \leq 4\).

Next, if \(H\) is of type (iii), then \(F^*(H)\) is the direct product of two or three non-abelian simple groups which are given in [12]. Moreover, \(H\) is a subgroup of \(\mathop{\mathrm{Aut}}(F^*(H))\) containing \(F^*(H)\), and it is easy to see that \(\mathrm{dl^*}(H) \leq 5\). For instance, suppose that \(T = E_8(q)\), and \(F^*(H) = \relax_2(q) \times G_2(q)^2\) where \(q > 3\). Then \(H\) is contained in \(\mathop{\mathrm{Aut}}(F^*(H)) = \mathop{\mathrm{Aut}}(\relax_2(q)) \times (\mathop{\mathrm{Aut}}(G_2(q)) \wr C_2)\), and contains \(\mathop{\mathrm{Aut}}(F^*(H))^{(\infty)} = F^*(H)\). As such, by Lemma 1, \[\mathrm{dl^*}(H) \leq \mathrm{dl^*}(\mathop{\mathrm{Aut}}(F^*(H))) \leq \max \left( \mathrm{dl}(\mathop{\mathrm{Out}}(\relax_2(q))), \mathrm{dl}(\mathop{\mathrm{Out}}(G_2(q)) \wr C_2) \right) .\] Now \(\mathrm{dl}(\mathop{\mathrm{Out}}(\relax_2(q))) \leq 2\) by Lemma 5, and \[\mathrm{dl}(\mathop{\mathrm{Out}}(G_2(q)) \wr C_2) \leq \mathrm{dl}(\mathop{\mathrm{Out}}(G_2(q)) + 1 \leq 3.\] Hence, \(\mathrm{dl^*}(H) \leq 5\).

If \(H\) is of type (iv) then \(H \cap \widetilde{G}_\sigma = N_{\widetilde{G}_\sigma}(E)\) for some elementary abelian group \(E\), and \(N_{\widetilde{G}_\sigma}(E)\) is described in [14]. As such, we can find that \[\begin{align} \mathrm{dl^*}(H) & \leq \mathrm{dl^*}(C_{\widetilde{G} _\sigma}(E)) + \mathrm{dl^*}(N_{\widetilde{G}_\sigma}(E)/C_{\widetilde{G}_\sigma}(E)) + \mathrm{dl}(G/(G \cap \widetilde{G}_\sigma)) \\ & \leq 2 + 2 + 2 = 6. \end{align}\]

If \(H\) is of type (v), then \(\mathrm{dl^*}(T \cap H) \leq 4\) and so \(\mathrm{dl^*}(H) \leq 6\). Finally, if \(H\) is of type (vi) then \(\mathrm{dl^*}(H) \leq 3\) by Lemma 5.

3.4 \(H\) is a parabolic subgroup↩︎

Let \(G\) be a group of Lie type with socle \(T\) and let \(H<G\) be a parabolic maximal subgroup. As in the previous section, let \(\widetilde{G}\) be the algebraic simple group corresponding to \(T\). Let \(\Phi\) be a root system of \(\widetilde{G}\), and \(\Pi\) its simple system. For any subset \(J \subset \Pi\), let \(\Phi_J\) be the root subsystem of \(\Phi\) spanned by \(J\), and let \(\Phi_J^+, \Phi^+\) denote their respective positive systems. For a root \(\alpha \in \Phi^+ \setminus \Phi_J^+\), we can write \(\alpha = \sum_{\beta \in J} b_\beta \beta + \sum_{\beta \in \Pi \setminus J} c_\beta \beta\), and we say \[\begin{align} \operatorname{height}(\alpha) & = \sum b_\beta + \sum c_\beta , \\ \operatorname{level}(\alpha) & = \sum c_\beta . \end{align}\] For instance, if \(G\) is of type \(E_6\), then the root of \(\Phi\) of largest height is \[\dynkin[edge length=.5cm, labels={1, 2, 2, 3, 2, 1}, label directions={below,right,below,below,below,below}] E{6},\] which has height \(11\). If \(J\) is, for example, \(\Pi\) with the middle node removed, then the level of this root is \(3\), and this is the largest possible level of a root in this case.

For any root \(\alpha \in \Phi\), let \(X_\alpha\) be the root subgroup of \(T\) corresponding to \(\alpha\). Since \(H\) is a parabolic subgroup of \(G\), there exists a subset \(J\) of \(\Pi\) such that \(T \cap H\) is conjugate to \(P_J\), where \(P_J = U \rtimes L\) with \[U \> = \> \prod_{\alpha \in \Phi^+ \setminus \Phi_J^+} X_\alpha\] a \(p\)-group, and \[L \> = \> \langle \prod_{\alpha \in J} X_\alpha, K \rangle\] where \(K\) is a Cartan subgroup of \(T\).

By Proposition 4 we have \[\mathrm{dl^*}(T \cap H) \> \leq \> \mathrm{dl}(U) + \mathrm{dl^*}(L) ,\] and we will bound \(\mathrm{dl}(U)\) and \(\mathrm{dl^*}(L)\) separately.

By [18], the subgroup \(L_0 = \prod_{\alpha \in J} X_\alpha\) is a product of Chevalley groups and is normal in \(L\). That is, \(L_0 = G_1 \circ \cdots \circ G_m\) for some groups \(G_i\) which are either quasisimple, or one of the following: \[\mathop{\mathrm{SL}}_2(2), \> \mathop{\mathrm{SL}}_2(3), \> \mathop{\mathrm{SU}}_3(2), \> B_2(2), \text{ or } {}^2B_2(2).\] In fact, each \(G_i\) corresponds to a connected component of the Dynkin diagram of \(\Phi_J\).

By combining Lemmas 7 and 8, we always obtain \(\mathrm{dl^*}(T \cap H) \leq 7\). By Lemma 5, this gives \(\mathrm{dl^*}(H) \leq 9\) if \(T\) is not \(\mathop{\mathrm{P \Omega}}_8^+(q)\) for \(q\) odd. If \(T=\mathop{\mathrm{P \Omega}}_8^+(q)\), then we actually obtain \(\mathrm{dl^*}(T \cap H) \leq 6\) and therefore \(\mathrm{dl^*}(H) \leq 9\). This concludes the proof of Theorem 1 for groups of Lie type.

3.5 \(T\) is sporadic↩︎

Let \(G\) be an almost simple group with sporadic socle \(T\), and let \(H\) be a maximal subgroup of \(G\). To read Table 2, let \[\mathrm{dl^*Max}(G) \> := \> \max \{ \mathrm{dl^*}(H) : His a maximal subgroup ofG \} ,\] so that Theorem 1 states that \(\mathrm{dl^*Max}(G) \leq 10\) if \(G\) is almost simple.

Let us describe how we obtain these values.

We use the \(\mathbb{A}\mathbb{T}\mathbb{L}\mathbb{A}\mathbb{S}\) (see also the library CTblLib (v1.3.9) in [19]), and we combine this data with Corollary 1 and Remark 5. For instance, if \(G = \mathbb{B}\) and \(H\) is \([2^{30}].\relax_5(2)\), then \(\mathrm{dl}([2^{30}]) \leq 5\) and hence \(\mathrm{dl^*}(H) \leq 5\). More precisely, if \(T\) is not one of \(\mathop{\mathrm{Fi}}_{24}'\), \(\mathbb{B}\) or \(\mathbb{M}\), then the maximal subgroups of \(G\) can be obtained by [19], and thus \(\mathrm{dl^*Max}(G)\) can be computed. The equality in Theorem 1 is attained when \(H = \texttt{AtlasSubgroup(``Fi23'',7)}\), which is a solvable subgroup with shape \(3^{1+8}.2^{1+6}.3^{1+2}.2S_4\).

If \(T=\mathop{\mathrm{Fi}}_{24}'\), the maximal subgroups of \(G=\mathop{\mathrm{Aut}}(T)\) can be obtained from the \(\mathbb{A}\mathbb{T}\mathbb{L}\mathbb{A}\mathbb{S}\). Thus, we can compute \(\mathrm{dl^*}(H) \leq 4\) in this case (and \(4\) is attained). If a maximal subgroup of \(T\) is equal to the intersection of a maximal subgroup of \(G\) with \(T\), then \(\mathrm{dl^*}(H) \leq 4\) by Lemma 1. Using the classification of the maximal subgroups of \(\mathop{\mathrm{Fi}}_{24}'\) given in the \(\mathbb{A}\mathbb{T}\mathbb{L}\mathbb{A}\mathbb{S}\), it is easy to bound \(\mathrm{dl^*}(H) \leq 4\) in the remaining cases.

If \(G=\mathbb{M}\), arguing as above we see that \(\mathrm{dl^*}(H) \leq 6\) unless \(H\) is \(H_1 = 3^2.3^5.3^{10}.(\mathrm{M_{11}} \times \mathop{\mathrm{GL}}_2(3))\). In fact \(H_1\) is available in Magma and we can compute \(\mathrm{dl^*}(H_1) =4\). Finding the exact value of \(\mathrm{dl^*Max}(\mathbb{M})\) seems to be cumbersome with our current knowledge.

Finally, if \(G=\mathbb{B}\), then \(\mathrm{dl^*}(H) \leq 6\) unless \(H\) is \(H_2 = 3^2.3^3.3^6.(\mathop{\mathrm{Sym}}(4) \times \mathop{\mathrm{GL}}_2(3))\). From \(H_1\) in \(\mathbb{M}\), we obtain \(H_2\) by centralizing and then factoring out an involution in the \(\mathrm{M_{11}}\) (see [20]). This allows us to compute \(\mathrm{dl}(H_2)=6\).

4 Perfect groups↩︎

We now deal with primitive perfect groups.

We would like to point out the following result for quasisimple groups, i.e. perfect central extensions of non-abelian simple groups. For completeness, we also discuss the minimum number of generators, which we denote by \(d(G)\).

We do not know whether the bound for \(d(H)\) in Proposition 12 can be improved to \(4\).

To conclude the paper, we make a comment on the constant in Conjecture 3, i.e. what can be achieved by stabilizing two points in an arbitrary primitive group. For \(G=\mathop{\mathrm{Fi}}_{23}\) and the solvable maximal subgroup \(H\) with \(\mathrm{dl}(H) =10\), we can find \(g \in G\) such that \(\mathrm{dl}(H \cap H^g) =2\). However, if \(G=\mathop{\mathrm{Aut}}(\mathop{\mathrm{P \Omega}}_8^+(3))\) and \(H\) is the solvable maximal subgroup of index \(36\,400\), then the corresponding primitive group has rank \(5\) and the derived lengths of stabilizers of two points are \([8,6,6,5,5]\). In particular, \(\mathrm{dl}(H \cap H^g) \geq 5\) for all \(g \in G\). We can do better with groups of product type:

Acknowledgments: We thank the referee for their helpful comments, which significantly improved the presentation of the paper.

References↩︎