The Normalizer Property for Finite Groups Whose Sylow 2-Subgroups are Abelian
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The Normalizer Property for Finite Groups Whose Sylow 2-Subgroups are Abelian Tao Zheng1 · Xiuyun Guo1 Received: 13 November 2019 / Revised: 8 January 2020 / Accepted: 4 March 2020 © School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature 2020
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 Outc (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 Mathematics Subject Classification 20C05 · 16S34 · 20C10
1 Introduction Let G be a finite group and let ZG be its integral group ring over Z. Denote by U (ZG) the group of units of ZG. A once long-standing problem (called the normalizer problem) [23, Problem 43] is whether NU (ZG) (G) = G · CU (ZG) (G) for any finite group G, where NU (ZG) (G) and CU (ZG) (G) are the normalizer and the centralizer of G in U (ZG), respectively. If this equality holds for G, then we say that G has the
The research of the work was partially supported by the National Natural Science Foundation of China (11771271).
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Xiuyun Guo [email protected] Department of Mathematics, Shanghai University, Shanghai 200444, People’s Republic of China
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T. Zheng, X. Guo
normalizer property. For any u ∈ NU (ZG) (G), we use ϕu to denote the automorphism of G induced by u through conjugation, i.e., ϕu (g) = u −1 gu for all g ∈ G. All such automorphisms of G form a subgroup of Aut(G), denoted by AutZ (G). It is easy to see that the normalizer property for a group G is equivalent to AutZ (G) = I nn(G) [14]. It is Coleman who first confirmed that finite nilpotent groups have the normalizer property [2], and later this result was extended to any finite 2-closed group by Jackowski and Marciniak [14]. In spite of a counterexample for the normalizer problem constructed by Hertweck [8], there has been a lot of research to find the classes of finite groups so that they have the normalizer property (for example, see [1,6,7,16–18,24]). It has been proved that Coleman automorphisms and class-preserving automorphisms of finite groups have intimate connection with the normalizer problem. Recall that an automorphism ρ of a finite group G is called a Coleman automorphism if the restriction of ρ to each Sylow subgroup of G equals the restriction of some inner automorphism of G. All such automorphisms of G form a subgroup of Aut(G), denoted by AutCol (G). An
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