On large orbits of subgroups of linear groups

Abstract

The main aim of this paper is to prove an orbit theorem and to apply it to obtain a result that can be regarded as a significant step towards the solution of Gluck’s conjecture on large character degrees of finite solvable groups.

On large orbits of supersoluble subgroups of linear groups

“…The following notation and arguments appear in [11,Section 3]. We summarise them here for the benefit of the reader.…”

“…As in [11,Section 3], we are interested here in regular orbits of a group G on completely reducible G-modules V over finite fields and so, in looking for regular orbits of G on V , we can assume without loss of generality that the field is a prime field.…”

“…Theorem 1 ([11,Theorem A]). If G is a finite soluble group, V is a faithful completely reducible G-module (possibly of mixed characteristic) and H is a subgroup of G such that the semidirect product V H is S 4 -free, then H has at least two regular orbits on V ⊕ V .…”

“…By [11,Corollary 3], the answer is affirmative if V is of odd order. Therefore it will be enough to prove Theorem A for a module V over a field of characteristic 2.…”

On two questions from the Kourovka Notebook

“…Lemma 21 (see [11,Theorem A] 3 Proof of Theorem A Assume we are trying to prove a result of the following type: Let G be a soluble group and let H be an F-prefrattini subgroup of G.…”