Abstract
Let $G$ be a finite almost simple group. It is well known that $G$ can be generated by three elements, and in previous work we showed that 6 generators suffice for all maximal subgroups of $G$. In this paper, we consider subgroups at the next level of the subgroup lattice—the so-called second maximal subgroups. We prove that with the possible exception of some families of rank 1 groups of Lie type, the number of generators of every second maximal subgroup of $G$ is bounded by an absolute constant. We also show that such a bound holds without any exceptions if and only if there are only finitely many primes $r$ for which there is a prime power $q$ such that $(q^{r}-1)/(q-1)$ is prime. The latter statement is a formidable open problem in Number Theory. Applications to random generation and polynomial growth are also given.
Highlights
In recent years it has been shown that finite nonabelian simple groups share several fundamental generation properties with their maximal subgroups
We investigate analogous questions for subgroups lying deeper in the subgroup lattice of an almost simple group—namely, for second maximal subgroups
Combining Theorem 1 with results of Jaikin-Zapirain and Pyber [16], we extend this to second maximal subgroups, as follows
Summary
In recent years it has been shown that finite nonabelian simple groups share several fundamental generation properties with their maximal subgroups. (ii) There exists a constant c such that all second maximal subgroups of finite simple groups are generated by at most c elements. We show that ν(M) c for every second maximal subgroup M of an almost simple group if and only if the question (1) has a negative solution This follows by combining Theorem 3 with Corollary 8.2. A polynomial bound for second maximal subgroups was obtained in [8, Corollary 6] This was based on the random generation of maximal subgroups by a bounded number of elements, together with Lubotzky’s inequality mn(H ) nν(H)+3.5 for all finite groups H [26].
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