On the generalized norms of a group

Abstract

Let G be a finite group. Inspired by the properties of D(G), we define the $D^{*}$-norm, denoted by $D^{*}(G),$ to be the intersection of the normalizers of the derived subgroups of all subgroups H of G such that H is generated by two elements of G and $H\prime$ is nilpotent. Set $D_0^{*}(G)$ = 1. We define $D_i^{*}(G)$/$D_{i-1}^{*}(G)$ = $D^{*}(G/D_{i-1}^{*}(G))$ for $i\ge 1$ and we denote by $D_{\infty }^{*}(G)$ the terminal term of the ascending series. In this paper, we mainly show that $D^{*}(G)$ = D(G).

Keywords:

• Derived subgroup
• norm
• solvable group

2020 Mathematics Subject Classification:

• 20D10
• 20D15

Acknowledgments

The authors are grateful to Prof. Shirong Li who provided profound suggestions. The authors are grateful to the referee who provided profound suggestions and detailed report.

Additional information

Funding

The project is supported by the Natural Science Foundation of China (No. 11401116).