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.

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Acknowledgements

The authors would like to thank the referee for their valuable suggestions and useful comments contributed to the final version of this paper.

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Correspondence to Xiuyun Guo.

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The research of the work was partially supported by the National Natural Science Foundation of China (11771271)

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Zheng, T., Guo, X. The Normalizer Property for Finite Groups Whose Sylow 2-Subgroups are Abelian. Commun. Math. Stat. 9, 87–99 (2021). https://doi.org/10.1007/s40304-020-00211-w

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Keywords

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

Mathematics Subject Classification

  • 20C05
  • 16S34
  • 20C10