Generation of second maximal subgroups and the existence of special primes

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

The length and depth of compact Lie groups

Let G be a connected Lie group. An unrefinable chain of G is defined to be a chain of subgroups G=G0>G1>⋯>Gt=1 , where each Gi is a maximal connected subgroup of Gi−1 . In this paper, we introduce the notion of the length (respectively, depth) of G, defined as the maximal (respectively, minimal) length of such a chain, and we establish several new results for compact groups. In particular, we compute the exact length and depth of every compact simple Lie group, and draw conclusions for arbitrary connected compact Lie groups G. We obtain best possible bounds on the length of G in terms of its dimension, and characterize the connected compact Lie groups that have equal length and depth. The latter result generalizes a well known theorem of Iwasawa for finite groups. More generally, we establish a best possible upper bound on dimG′ in terms of the chain difference of G, which is its length minus its depth

Orbit Closures and Invariants

The first author would like to thank Sebastian Herpel for the conversations we had which led to the first iteration of some of the ideas in this paper, and also Stephen Donkin for some very helpful nudges towards the right literature. All three authors acknowledge the funding of EPSRC grant EP/L005328/1. We would like to thank the anonymous referee for their very insightful comments and for pointing out a subtle gap in the proof of Theorem 1.1.Peer reviewedPublisher PD

Finite groups, minimal bases and the intersection number

Let G be a finite group and recall that the Frattini subgroup Frat(G) is the intersection of all the maximal subgroups of G. In this paper, we investigate the intersection number of G, denoted alpha(G), which is the minimal number of maximal subgroups whose intersection coincides with Frat(G). In earlier work, we studied alpha(G) in the special case where G is simple and here we extend the analysis to almost simple groups. In particular, we prove that alpha(G) <= 4 for every almost simple group G, which is best possible. We also establish new results on the intersection number of arbitrary finite groups, obtaining upper bounds that are defined in terms of the chief factors of the group. Finally, for almost simple groups G we present best possible bounds on a related invariant beta(G), which we call the base number of G. In this setting, beta(G) is the minimal base size of G as we range over all faithful primitive actions of the group and we prove that the bound beta(G) <= 4 is optimal. Along the way, we study bases for the primitive action of the symmetric group S-ab on the set of partitions of [1, ab] into a parts of size a >= b, determining the exact base size for a b. This extends earlier work of Benbenishty, Cohen and Niemeyer

The depth of a finite simple group

We introduce the notion of the depth of a finite group G, defined as the minimal length of an unrefinable chain of subgroups from G to the trivial subgroup. In this paper we investigate the depth of (non-abelian) finite simple groups. We determine the simple groups of minimal depth, and show, somewhat surprisingly, that alternating groups have bounded depth. We also establish general upper bounds on the depth of simple groups of Lie type, and study the relation between the depth and the much studied notion of the length of simple groups. The proofs of our main theorems depend (among other tools) on a deep number-theoretic result, namely, Helfgott’s recent solution of the ternary Goldbach conjecture