A class of finite p-groups and the normalized unit groups of group algebras

Abstract

Let p be a prime and ${\mathbb{F}}_p$ be a finite field of p elements. Let ${\mathbb{F}}_{p}G$ denote the group algebra of the finite p-group G over the field ${\mathbb{F}}_p$ and $V({\mathbb{F}}_{p}G)$ denote the group of normalized units in ${\mathbb{F}}_{p}G$. Suppose that G is a finite p-group given by a central extension of the form $1\to {\mathbb{Z}}_{p^n}\times {\mathbb{Z}}_{p^m}\to G\to {\mathbb{Z}}_p\times \cdots \times {\mathbb{Z}}_p\to 1$ and $G\prime \cong {\mathbb{Z}}_p,n,m\ge 1$ and p is odd. In this paper, the structure of G is determined. And the relations of $V{({\mathbb{F}}_{p}G)}^{p^l}$ and $G^{p^l},{\Omega }_l(V({\mathbb{F}}_{p}G))$ and ${\Omega }_l(G)$ are given. Furthermore, there is a direct proof for $V{({\mathbb{F}}_{p}G)}^p\cap G=G^p$.

KEYWORDS:

• Central product
• derived subgroup
• finite p-group
• normalized unit

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 20C05
• 20D15

Disclosure statement

The authors declare no conflict of interest.

Additional information

Funding

Supported by National Natural Science Foundation of China (Grant No. 12171142).