A base of a permutation group G is a sequence B of points from the permutation domain such that only the identity of G fixes B pointwise. Answering a question of Pyber, we prove that all primitive solvable permutation groups have a base of size at most four.

The Minimal Base Size of Primitive Solvable Permutation Groups

Orbit sizes, character degrees and Sylow subgroups

A pair (G. K) in whichG is a finite group andK◃G, 1<K<G, is said to satisfy (F2) if |C
 G
 (x)|=|C
 G/K
 (xK)| for allx∈G/K. First we survey all the examples known to us of such pairs in whichG is neither ap-group nor a Frobenius group with Frobenius kernelK. Then we show that under certain restrictions there are, essentially, all the possible examples.

Generalized Frobenius groups

Let G be a solvable group of automorphisms of a finite group K. If |G| and |K| are coprime, then there exists an orbit of G on K of size at least √|G|. It is also proved that in a π-solvable group, the largest normal π-subgroup is the intersection of at most three Hall π-subgroups.

/pdf/large-orbits-in-coprime-actions-of-solvable-groups-tkll2tpfru.pdf

Large orbits in coprime actions of solvable groups

We discuss some length questions in the theory of finite solvable groups for which character theory has proven useful We discuss some open problems, some techniques of proof for various character and module theoretic results arising in this context, as well as the applications of these results to the length problems themselves As this is an expository paper, most of our results appeared elsewhere An exception is a linear bound on the Fitting height of groups accepting a fixed point free abelian group of automorphisms of square-free odd exponent, which, although it follows easily from known results, does not seem to have been noticed before

Character Theory and Length Problems

(See p. 447 of [4].) He shows in Theorem B of [4] that IG:F(G)I < b(G)13/2 holds for a solvable group G. Here we prove his conjecture with the additional assumption that GI is odd. As a corollary, we show that if G is a (solvable) group of odd order, then G has an abelian subgroup A such that IG:AI _< b(G)6. This improves (for groups of odd order) the bound [G:AI < b(G)21/2 obtained in Theorem C of [4]. Our method consists of obtaining a regular orbit theorem for groups of odd order acting on vector spaces of odd characteristic (see Lemma 2.1 below). As Gluck points out, this method cannot be used for even order solvable groups (see 19. 447 of [4]). Our regular orbit theorem may be also used to prove a conjecture due to Huppert in some cases. Let G be a finite solvable group and, for a natural number n, let r(n) denote the set of prime divisors of n. For a group H, we write 7r(H)for r(lH[). Define

Alberto Espuelas

Papers

Large character degrees of groups of odd order

Regular orbits on symplectic modules

The Complement of a Frobenius-Wielandt Group

The fitting length of the Hughes subgroup

The existence of regular orbits