The normalizer problem for finite groups having centerless normal subgroups

Liang Zhang; Jinke Hai; Liang Zhang; Jinke Hai

doi:10.3934/era.2026242

Electronic Research Archive

2026, Volume 34, Issue 8: 5421-5427. doi: 10.3934/era.2026242

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Research article

The normalizer problem for finite groups having centerless normal subgroups

Liang Zhang ¹,
Jinke Hai ^{1,2
,
,}

1.
College of Mathematics and Statistics, Yili Normal University, Yining 835000, China
2.
College of Mathematics and Statistics, Qingdao University, Qingdao 266071, China

Received: 30 March 2026 Revised: 04 June 2026 Accepted: 16 June 2026 Published: 29 June 2026

Let $ E $ be an extension of a centerless finite group by another finite group. Under some conditions, it is proved that $ E $ has the normalizer property. As an application, we prove that the holomorphs of all the alternating groups and the sporadic simple groups have the normalizer property.
- holomorph,
- normalizer property,
- sporadic simple groups
Citation: Liang Zhang, Jinke Hai. The normalizer problem for finite groups having centerless normal subgroups[J]. Electronic Research Archive, 2026, 34(8): 5421-5427. doi: 10.3934/era.2026242

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Abstract

Let $ E $ be an extension of a centerless finite group by another finite group. Under some conditions, it is proved that $ E $ has the normalizer property. As an application, we prove that the holomorphs of all the alternating groups and the sporadic simple groups have the normalizer property.

References

[1]	S. K. Sehgal, Units in Integral Group Rings, Longman Scientific and Technical Press, 1993.
[2]	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
[3]	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
[4]	J. D. Guo, L. Zhang, J. K. Hai, The normalizer problem for finite groups having normal 2-complements, Electron. Res. Arch., 32 (2024), 4905–4912. https://doi.org/10.3934/era.2024225 doi: 10.3934/era.2024225
[5]	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
[6]	S. O. Juriaans, J. M. Miranda, J. R. Rob$\acute{e}$rio, Automorphisms of finite groups, Commun. Algebra, 32 (2004), 1705–1714. https://doi.org/10.1081/AGB-120029897 doi: 10.1081/AGB-120029897
[7]	T. P. Lobão, S. K. Sehgal, The normalizer property for integral group rings of complete monomial groups, Commun. Algebra, 31 (2003), 2971–2983. https://doi.org/10.1081/agb-120021903 doi: 10.1081/agb-120021903
[8]	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
[9]	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
[10]	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
[11]	Z. X. Li, J. K. Hai, Coleman automorphisms of holomorphs of finite abelian groups, Acta Math. Sin., 30 (2014), 1549–1554. https://doi.org/10.1007/s10114-014-2143-6 doi: 10.1007/s10114-014-2143-6
[12]	M. Hertweck, A counterexample to the isomorphism problem for integral group rings, Ann. Math., 154 (2001), 115–138. https://doi.org/10.2307/3062112 doi: 10.2307/3062112
[13]	Z. Y. Chen, H. X. Zhang, H. Chen, ADI compact difference scheme for the two-dimensional integro-differential equation with two fractional Riemann-Liouville integral kernels, Fractal Fractional, 8 (2024), 707–726. https://doi.org/10.3390/fractalfract8120707 doi: 10.3390/fractalfract8120707
[14]	X. H. Yang, Y. Shi, H. X. Zhang, Z. M. Zhang, Analysis of a new time-space two-grid method for the two-dimensional nonlinear nonlocal mobile/immobile transport model, J. Sci. Comput., 107 (2026), 96–132. https://doi.org/10.1007/s10915-026-03314-8 doi: 10.1007/s10915-026-03314-8
[15]	J. S. Rose, A Course on Group Theory, Cambridge, Cambridge University Press, 1978.
[16]	J. H. Conway, R. T. Curttis, S. P. Norton, R. A. Parker, R. A. Wilson, Atlas of Finite Groups, Clarendon Press, Oxford, 1986.
[17]	J. K. Hai, J. D. Guo, The normalizer property for the integral group ring of the wreath product of two symmetric groups $S_{k}$ and $S_{n}$, Commun. Algebra, 45 (2017), 1278–1283. https://doi.org/10.1080/00927872.2016.1175613 doi: 10.1080/00927872.2016.1175613
[18]	H. Kurzweil, B. Stellmacher, The Theory of Finite Groups: An Introduction, Springer-Verlag, New York, 2004. https://doi.org/10.1115/1.3259028

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