The Normalizer Property for Finite Groups Whose Sylow 2-Subgroups are Abelian

Abstract

In this paper we mainly investigate the Coleman automorphisms and class-preserving automorphisms of finite AZ-groups and finite groups related to AZ-groups. For example, we first prove that $$Out_c(G)$$ of an AZ-group G must be a $$2'$$ -group and therefore the normalizer property holds for G. Then we find some classes of finite groups such that the intersection of their outer class-preserving automorphism groups and outer Coleman automorphism groups is $$2'$$ -groups, and therefore, the normalizer property holds for these kinds of finite groups. Finally, we show that the normalizer property holds for the wreath products of AZ-groups by rational permutation groups under some conditions.