On the normal complement problem in modular and semisimple group algebras

Abstract

Let p and q be odd primes such that $p=2q+1.$ Let F be the field with p elements and $G=A⋊C_q$ be a group, where A is an abelian group of order $p^m.$ In this article, we prove that if $m\ge (q-1),$ then G does not have a normal complement in $\mathcal{U}(FG).$ Further, for any integer $n\ge 4,$ we prove that if F is a finite field such that $\mathit{char}(F)>n,$ then $S_n$ and $A_n$ do not have a normal complement in $\mathcal{U}(F{S}_n)$ and $\mathcal{U}(F{A}_n),$ respectively.

Keywords:

• Conjugacy class
• group ring
• normal complement
• standard representation
• unit group
• Wedderburn-Artin theorem

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 16U60
• 20C05
• 20E45

Acknowledgement

The author would like to thank the referee for the valuable comments and suggestions which improved the presentation of the article.

Additional information

Funding

This project is funded by CSIR, India (File no. 09/092(1066)/2020-EMR-I).