The normalizer problem for finite groups having normal $ 2 $-complements

Jidong Guo; Liang Zhang; Jinke Hai; Jidong Guo; Liang Zhang; Jinke Hai

doi:10.3934/era.2024225

Electronic Research Archive

2024, Volume 32, Issue 8: 4905-4912. doi: 10.3934/era.2024225

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Research article

The normalizer problem for finite groups having normal $ 2 $-complements

College of Mathematics and Statistics, Yili Normal University, Yining 835000, China

Received: 07 May 2024 Revised: 02 August 2024 Accepted: 08 August 2024 Published: 15 August 2024

Assume that $ H $ is a finite group that has a normal $ 2 $-complement. Under some conditions, it is proven that the normalizer property holds for $ H $. In particular, if there is a nilpotent subgroup of index $ 2 $ in $ H $, then $ H $ has the normalizer property. The result of Li, Sehgal and Parmenter, stating that the normalizer property holds for finite groups that have an abelian subgroup of index $ 2 $ is generalized.
- the normalizer property,
- Coleman automorphisms,
- class-preserving automorphisms,
- normal $ 2 $-complements
Citation: Jidong Guo, Liang Zhang, Jinke Hai. The normalizer problem for finite groups having normal $ 2 $-complements[J]. Electronic Research Archive, 2024, 32(8): 4905-4912. doi: 10.3934/era.2024225

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Abstract

Assume that $ H $ is a finite group that has a normal $ 2 $-complement. Under some conditions, it is proven that the normalizer property holds for $ H $. In particular, if there is a nilpotent subgroup of index $ 2 $ in $ H $, then $ H $ has the normalizer property. The result of Li, Sehgal and Parmenter, stating that the normalizer property holds for finite groups that have an abelian subgroup of index $ 2 $ is generalized.

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