A note on the Wedderburn decomposition of the group algebra of wreath product
Gaurav MITTAL, Rajendra Kumar SHARMA
Received December 22, 2025. Published online May 11, 2026.
Abstract: Let $\mathbb{F}_q$ denote a finite field with $q=p^k$ for a prime $p$ and $k\in\mathbb{Z}^+$. Additionally, let $n$ and $t$ be positive integers such that $\gcd(p, n)=\gcd(p,t)=1$. In this paper, we study the Wedderburn decompositions of the semisimple group algebras $\mathbb{F}_qG_1$ and $\mathbb{F}_qG_2$, where $G_1=A {\rm Wr} C_n$ denotes the wreath product of an abelian group of order $t$ and a cyclic group of order $n$, and $G_2=H {\rm Wr} C_n$. Here $H$ denotes a non-abelian group of order $t$ with the assumption that it has a unique irreducible character of degree $>1$. As a by product, we give the description of the unit groups of $\mathbb{F}_qG_1$ and $\mathbb{F}_qG_2$.
Keywords: group algebra; unit; Wedderburn decomposition; representation theory
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Affiliations: Gaurav Mittal (corresponding author), Defence Research & Development Organisation (DRDO), Near Metcalfe House, New Delhi, 110054, India, e-mail: gaurav.mittaltwins@yahoo.com; Rajendra Kumar Sharma, Department of Mathematics, Delhi Technological University, Shahbad Daulatpur, Main Bawana Road, New Delhi, 110042, India, e-mail: rksharmaiitd@gmail.com
