Bernd Stellmacher

A characteristic subgroup of Σ4-free groups

LetS be a finite non-trivial 2-group. It is shown that there exists a nontrivial characteristic subgroupW(S) inS satisfying:W(S) is normal inH for every finite Σ4-free groupsH withSεSyl2(H) andCH(O2(H))≤O2(H).

F-stability in finite groups

Let G be a finite group, S ∈ Syl P (G), and S be the set subgroups containing S. For M ∈ S and V = Ω 1 Z(Op(M)), the paper discusses the action of M on V. Apart from other results, it is shown that for groups of parabolic characteristic p either S is contained in a unique maximal p-local subgroup, or there exists a maximal p-local subgroup in M ∈ S such that V is a nearly quadratic 2F-module for M.

Transfer and p-Factor Groups

To search for nontrivial proper normal subgroups is often the first step in the investigation of a finite group. For example, if the group G has such a normal subgroup N, then in proofs by induction one frequently gets information about N and G/N, allowing one to derive the desired result for G (e.g., 6.1.2 on page 122).

Action and Conjugation

The notion of an action plays an important role in the theory of finite groups. The first section of this chapter introduces the basic ideas and results concerning group actions. In the other two sections the action on cosets is used to prove important theorems of Sylow, Schur-Zassenhaus and Gaschutz.

Groups Acting on Groups

The action of a group A on a set G is described by a homomorphism

p-Groups and Nilpotent Groups

\pi :A \to {S_{G}}

Operieren und Konjugieren

; see Section 3.1. Suppose that G is not only a set but also a group. Then Aut G ≤ S G , and we say that π describes the action of A on the group G if Im π is a subgroup of Aut G. In other words, in this case the action of A on G not only satisfies O 1 and O 2 but also