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A NOTE ON NORMAL COMPLEMENTS FOR FINITE GROUPS

  • NING SU (a1), ADOLFO BALLESTER-BOLINCHES (a2) and HANGYANG MENG (a3)

Abstract

Assume that $G$ is a finite group and $H$ is a 2-nilpotent Sylow tower Hall subgroup of $G$ such that if $x$ and $y$ are $G$ -conjugate elements of $H\cap G^{\prime }$ of prime order or order 4, then $x$ and $y$ are $H$ -conjugate. We prove that there exists a normal subgroup $N$ of $G$ such that $G=HN$ and $H\cap N=1$ .

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The first author is supported by the National Natural Science Foundation of China (no. 11401597). The second and third authors have been supported by the grant MTM2014-54707-C3-1-P from the Ministerio de Economía y Competitividad, Spain, and FEDER, European Union. The second author is also supported by Prometeo/2017/057 of Generalitat (Valencian Community, Spain) and the third author is also supported by the China Scholarship Council, grant no. 201606890006.

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[1] Ballester-Bolinches, A., Ezquerro, L. M. and Skiba, A. N., ‘On subgroups of hypercentral type of finite groups’, Israel J. Math. 199 (2014), 259265.
[2] Doerk, K. and Hawkes, T., Finite Soluble Groups, De Gruyter Expositions in Mathematics, 4 (Walter de Gruyter, Berlin–New York, 1992).
[3] González-Sánchez, J., ‘A p-nilpotency criterion’, Arch. Math. (Basel) 94 (2010), 201205.
[4] Gorenstein, D., Finite Groups (Chelsea, New York, 1980).
[5] Isaacs, I. M., Character Theory of Finite Groups (Academic Press, New York, 1976).
[6] Rose, J. S., A Course on Group Theory (Cambridge University Press, London–New York–Melbourne, 1978).
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