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

Yulei Wang, Heguo Liu

Abstract

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

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

Abstract

Let be a prime and be a finite field of elements. Let denote the group algebra of the finite -group over the field and denote the group of normalized units in . Suppose that is a finite -group given by a central extension of the form and , and is odd. In this paper, the structure of is determined. And the relations of and , and are given. Furthermore, there is a direct proof for .
Paper Structure (7 sections, 32 theorems, 139 equations)

This paper contains 7 sections, 32 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The structure of the group
  4. The isomorphism type $A_1$ of $\zeta G$
  5. The isomorphism types $A_2$ and $A_3$ of $\zeta G$
  6. The isomorphism type $A_4$ of $\zeta G$
  7. The normalized unit groups

Key Result

theorem 1

Let $G$ be a finite $p$-group given by a central extension of the form and $G'=\langle c\rangle\cong \mathbb{Z}_p$, $n\geq m\geq 1$ and $p>2$. Suppose that the order of $G/\zeta G$ is $p^{2k}$ and $d(\zeta G)=r$. Then (1) $\zeta G=\langle z_1\rangle\times\langle z_2\rangle\times\cdots\times\langle z_r\rangle$ has four isomorphism types $A_i$, $i=1,2,3,4$, (i) $A_1\cong (3) For the isomorphism typ

Theorems & Definitions (52)

  • theorem 1
  • theorem 2
  • lemma 1: XuQ
  • lemma 2: XuQ
  • lemma 3: Gupta
  • lemma 4: Brauer
  • definition 1
  • lemma 5: Milies
  • lemma 6: BB1
  • lemma 7: BB1
  • ...and 42 more