Normalizer property of finite groups with almost simple subgroups

Jingjing Hai; Xian Ling; Jingjing Hai; Xian Ling

doi:10.3934/era.2022215

Electronic Research Archive

2022, Volume 30, Issue 11: 4232-4237. doi: 10.3934/era.2022215

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Research article

Normalizer property of finite groups with almost simple subgroups

Jingjing Hai ^{1
,
,},
Xian Ling ²

1.
College of Quality and Standardization, Qingdao University, Qingdao 266071, China
2.
College of Mathematics and Statistics, Qingdao University, Qingdao 266071, China

Received: 18 June 2022 Revised: 01 September 2022 Accepted: 03 September 2022 Published: 22 September 2022

In this paper, we prove that all Coleman automorphisms of extension of an almost simple group by an abelian group or a simple group are inner. Using our methods we also show that the Coleman automorphisms of $ 2 $-power order of an odd order group by an almost simple group are inner. In particular, these groups have the normalizer property.
- almost simple group,
- simple group,
- Coleman automorphism,
- $ q $-central automorphism,
- normalizer property
Citation: Jingjing Hai, Xian Ling. Normalizer property of finite groups with almost simple subgroups[J]. Electronic Research Archive, 2022, 30(11): 4232-4237. doi: 10.3934/era.2022215

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Abstract

In this paper, we prove that all Coleman automorphisms of extension of an almost simple group by an abelian group or a simple group are inner. Using our methods we also show that the Coleman automorphisms of $ 2 $-power order of an odd order group by an almost simple group are inner. In particular, these groups have the normalizer property.

References

[1]	S. K. Sehgal, Units in integral group rings, Longman Scientific and Technical Press, 1993. https://doi.org/0803.16022
[2]	S. Jackowski, Z. S. Marciniak, Group automorphisms inducing the identity map on cohomology, J. Pure Appl. Algebra, 44 (1987), 241–250. https://doi.org/10.1016/0022-4049(87)90028-4 doi: 10.1016/0022-4049(87)90028-4
[3]	M. Hertweck, W. Kimmerle, Coleman automorphisms of finite groups, Math. Z., 242 (2002), 203–215. https://doi.org/10.1007/s002090100318 doi: 10.1007/s002090100318
[4]	F. Gross, Automorphisms which centralize a Sylow $p$-subgroup, J. Algebra, 77 (1988), 202–233. https://doi.org/10.1016/0021-8693(82)90287-3 doi: 10.1016/0021-8693(82)90287-3
[5]	D. B. Coleman, On the modular group ring of a $p$-group, Proc. Amer. Math. Soc., 5 (1964), 511–514. https://doi.org/10.2307/2034735 doi: 10.2307/2034735
[6]	J. K. Hai, L. L. Zhao, Coleman automorphisms of extensions of finite characteristically simple groups by some finite groups, Algebra Colloq., 28 (2021), 561–568. https://doi.org/10.1142/S1005386721000444 doi: 10.1142/S1005386721000444
[7]	S. O. Juriaans, J. M. Miranda, J. R. Rob$\acute{e}$rio, Automorphisms of finite groups, Comm. Algebra, 32 (2004), 1705–1714. https://doi.org/10.1081/AGB-120029897 doi: 10.1081/AGB-120029897
[8]	Z. X. Li, Coleman automorphisms of permutational wreath products Ⅱ, Comm. Algebra, 46 (2020), 4473–4479. https://doi.org/10.1080/00927872.2018.1448836 doi: 10.1080/00927872.2018.1448836
[9]	T. Zheng, X. Y. Guo, The normalizer property for finite groups whose Sylow $2$-subgroups are abelian, Commun. Math. Stat., 9 (2021), 87–99. https://doi.org/10.1007/s40304-020-00211-w doi: 10.1007/s40304-020-00211-w
[10]	A. Van Antwerpen, Coleman automorphisms of finite groups and their minimal normal subgroups, J. Pure Appl. Algebra, 222 (2018), 3379–3394. https://doi.org/10.1016/j.jpaa.2017.12.013 doi: 10.1016/j.jpaa.2017.12.013
[11]	H. Kurzweil, B. Stellmacher, The Theory of Finite Groups: An Introduction, Springer-Verlag, New York, 2004. https://doi.org/10.1115/1.3259028
[12]	M. Hertweck, Class-preserving automorphisms of finite groups, J. Algebra, 241 (2001), 1–26. https://doi.org/10.1006/jabr.2001.8760 doi: 10.1006/jabr.2001.8760

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