The normal complement problem and the structure of the unitary subgroup

Abstract

Let p be an odd prime and G be a finite split metabelian p-group of exponent p. In this article, we obtain a normal complement of G in $V\left(FG\right),$ where F is the field with p elements. Further, assume that $G=A⋊C_3,$ where A is a finite abelian p-group and $3\hspace{0.5em}|\hspace{0.5em}p-1.$ If F is any finite field of characteristic p, then we prove that G does not have a normal complement in $V\left(FG\right)$ and obtain the structure of the unitary subgroup $V_{*}\left(FG\right).$

Communicated by Sudarshan Kumar Sehgal

Keywords:

• Conjugacy class
• group ring
• normal complement
• unit group

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 16U60
• 20C05
• 20E45

Acknowledgment

The authors would like to thank the referee for valuable suggestions and comments.