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On the Nilpotent Residual Norm of a Group and the Structure of S-Groups

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Abstract

For a finite group G, we define the subgroup S(G) to be the intersection of the normalizers of the nilpotent residuals of all subgroups of G. Set \(S_0=1\) and define \(S_{i+1}(G)/S_i(G)=S(G/S_i(G))\) for \(i\ge 1\). The terminal term of this upper series is denoted by \(S_{\infty }(G)\). This upper series implies a lot of information on the structure of G. In this paper, we solve several basic problems on S(G). If \(G=S(G)\), we call the finite group G an S-group. The new class of S-groups are investigated and some open problems on S-groups are posed. Furthermore, we develop the research on \(S_{\infty }(G)\) by a new idea and unify some known results.

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Acknowledgements

The authors would like to thank the editors and referees for their suggestions and detail report.

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Authors and Affiliations

  1. College of Science, China Agricultural University, Beijing, 100083, China

    Zhencai Shen

  2. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, Fujian, China

    Ni Du

  3. Department of Mathematics, Southeastern Louisiana University, Hammond, LA, 70403, USA

    Gary L. Walls

Authors
  1. Zhencai Shen
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  2. Ni Du
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  3. Gary L. Walls
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Correspondence to Ni Du or Gary L. Walls.

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To celebrate the 80th birthday of Prof. Shirong Li.

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This work was completed with the support of the National Natural Science Foundation of China (no. 11771356) and the Natural Science Foundation of Fujian Province of China (no. 2019J01025)

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Shen, Z., Du, N. & Walls, G.L. On the Nilpotent Residual Norm of a Group and the Structure of S-Groups. Mediterr. J. Math. 19, 191 (2022). https://doi.org/10.1007/s00009-022-02101-7

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  • DOI: https://doi.org/10.1007/s00009-022-02101-7

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