Variations on a theme of Steinberg

AbstractThe Steinberg tensor product theorem is a fundamental tool for studying irreducible representations of simple algebraic groups over fields of positive characteristic. This paper is concerned with extending the result, replacing the target group SL(V) by an arbitrary simple algebraic group

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 algebraic groups

Let G be a connected algebraic group. An unrefinable chain of G is a chain of subgroups G=G0>G1>⋯>Gt=1 , where each Gi is a maximal connected subgroup of Gi−1 . We introduce the notion of the length (respectively, depth) of G, defined as the maximal (respectively, minimal) length of such a chain. Working over an algebraically closed field, we calculate the length of a connected group G in terms of the dimension of its unipotent radical Ru(G) and the dimension of a Borel subgroup B of the reductive quotient G/Ru(G) . In particular, a simple algebraic group of rank r has length dimB+r , which gives a natural extension of a theorem of Solomon and Turull on finite quasisimple groups of Lie type. We then deduce that the length of any connected algebraic group G exceeds 12dimG . We also study the depth of simple algebraic groups. In characteristic zero, we show that the depth of such a group is at most 6 (this bound is sharp). In the positive characteristic setting, we calculate the exact depth of each exceptional algebraic group and we prove that the depth of a classical group (over a fixed algebraically closed field of positive characteristic) tends to infinity with the rank of the group. Finally we study the chain difference of an algebraic group, which is the difference between its length and its depth. In particular we prove that, for any connected algebraic group G with soluble radical R(G), the dimension of G / R(G) is bounded above in terms of the chain difference of G

Unipotent class representatives for finite classical groups

We describe explicitly representatives of the conjugacy classes of unipotent elements of the finite classical groups

The classification of 3/2-transitive permutation groups and 1/2-transitive linear groups

A linear group G ≤ GL(V ), where V is a finite vector space, is called 12 -transitive if all the G-orbits on the set of nonzero vectors have the same size. We complete the classification of all the 12-transitive linear groups. As a consequence we complete the determination of the finite 32-transitive permutation groups – the transitive groups for which a point-stabilizer has all its nontrivial orbits of the same size. We also determine the (k +1 2)-transitive groups for integers k ≥ 2

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