A note on Itô’s theorem of p-nilpotence

Abstract

The norm N(G) of a group G is the intersection of the normalizer of every subgroup of G. In this paper, we generalize Itô’s theorem and prove that the p-length of a p-soluble group is not greater than 1 if its Sylow p-subgroups P satisfy $\Omega (P)\le N(P)$ or $\Omega (O_p(G))\le N(P)$ and $O_{p\prime }(G)=1.$ Furthermore, some characteristics of nilpotent group are obtained.

Keywords:

• Finite group
• minimal subgroup
• norm
• p-length

2020 Mathematics Subject Classification:

• 20D15

Acknowledgments

The authors were grateful to the referees for their careful reading and helpful suggestions.

Additional information

Funding

This project was supported by NSFC10.13039/501100001809 (11601245 and 11471055).