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

2024/01/01 by Yulei Wang, Wang, Yulei, Heguo Liu +1
Mathematics · #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.2401.00638

openalex publication_date 2024/01/01 · openalex created_date 2024/01/03 · openalex updated_date 2026/07/28

Abstract

Let p be a prime and \mathbbF_p be a finite field of p elements. Let \mathbbF_pG denote the group algebra of the finite p-group G over the field \mathbbF_p and V(\mathbbF_pG) denote the group of normalized units in \mathbbF_pG. Suppose that G is a finite p-group given by a central extension of the form 1\longrightarrow ℤ_pⁿ× ℤ_p^m \longrightarrow G \longrightarrow ℤ_p× ⋯× ℤ_p \longrightarrow 1 and G'≅ ℤ_p, n, m≥ 1 and p is odd. In this paper, the structure of G is determined. And the relations of V(\mathbbF_pG)^pl and G^pl, Ω_l(V(\mathbbF_pG)) and Ω_l(G) are given. Furthermore, there is a direct proof for V(\mathbbF_pG)^p\bigcap G=G^p.

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