Abstract

We survey the recent completion of the determination of the maximal 2-signalizers and the normalizers of Sylow 2-subgroups in all finite simple groups.

Introduction

In this survey we discuss the recent completion of the determination of the maximal 2-signalizers and the normalizers of Sylow 2-subgroups in all finite simple groups (see).

The notion of signalizer introduced by Thompson in plays an important rôle in finite group theory, in particular, in the method of signalizer functor. If G is a finite group, p is a prime and P is a Sylow p-subgroup of G, then any P-invariant p′-subgroup of G is called a P-signalizer or simply a p-signalizer. The case of 2-signalizers is of the greatest interest. In, the second author disproved Thompson's conjecture on the commutativity of 2-signalizers in finite simple groups. He constructed certain examples of finite simple groups with nilpotent 2-signalizers of any class of nilpotency and even with non-nilpotent 2-signalizers. In connection with announcing the classification of finite simple groups, Gorenstein posed the problem of the study of the properties of signalizers in known finite simple groups. It was known that the 2-signalizers in finite simple groups of Lie type over a field of even characteristic are trivial (see [9, Thm. 4.254]) and a non-trivial 2-signalizer in the alternating group An exists only in the case n ≡ 3 (mod 4) and has order 3 (see [9, Thm. 4.255(i)]).