The unit group of the group ring over ℤ_n

Abstract

Let ${\mathbb{Z}}_n$ be the ring of integers modulo n. Let C_t , E_m , and $F_{r,s}$ respectively denote the cyclic group of order t, the elementary abelian 2-group of order $2^m$, and the abelian group of exponent 4 with order $2^r{4}^s$. In this article, we find the structure and generators of the unit group $\mathbb{V}({\mathbb{Z}}_n{C}_2).$ We also solve the normal complement problem in $\mathbb{V}({\mathbb{Z}}_n{C}_2)$. Additionally, we provide a normal complement of E_m in $\mathbb{V}({\mathbb{Z}}_{2^n}{E}_m).$ At the end, we determine the structure of $\mathbb{V}({\mathbb{Z}}_{p^n}{F}_{r,s})$ for an odd prime p and establish that $F_{r,s}$ does not have a normal complement in $\mathbb{V}({\mathbb{Z}}_{p^n}{F}_{r,s})$.

Keywords:

• Cyclic group
• elementary abelian 2-group
• group of exponent 4
• normal complement
• unit group

2020 Mathematics Subject Classification:

• 16U60
• 20C05