Unit Groups of Group Algebras of Abelian Groups of order 32

TL;DR

This paper provides a complete characterization of the unit groups of group algebras over finite fields for all abelian groups of order 32, advancing understanding of algebraic structures in modular group algebras.

Contribution

It offers the first comprehensive description of unit groups for group algebras of all abelian groups of order 32 over finite fields of positive characteristic.

Findings

Complete classification of unit groups for abelian groups of order 32.

Explicit descriptions depend on the characteristic of the finite field.

Results extend previous partial classifications in modular group algebra theory.

Abstract

Let $F$ be a finite field of characteristic $p>0$ with $q = p^{n}$ elements. In this paper, a complete characterization of the unit groups $U(FG)$ of group algebras $FG$ for the abelian groups of order $32$, over finite field of characteristic $p>0$ has been obtained.