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.

Keywords

Class-preserving automorphism / Coleman automorphism / Normalizer property / AZ-group

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Funding

National Natural Science Foundation of China(11771271)