The unitary subgroups of group algebras of a class of finite 2-groups with derived subgroup of order 2

2023/05/09 by Yulei Wang, Wang, Yulei, Heguo Liu +1
Mathematics · #20C05 #20D15 #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.2305.05158

openalex publication_date 2023/05/09 · openalex created_date 2023/05/12 · openalex updated_date 2026/07/28

Abstract

Let p be a prime and F be a finite field of characteristic p. Suppose that FG is the group algebra of the finite p-group G over the field F. Let V(FG) denote the group of normalized units in FG and let V_*(FG) denote the unitary subgroup of V(FG). If p is odd, then the order of V_*(FG) is |F|^(|G|-1)/2. However, the case when p=2 still is open. In this paper, the order of V_*(FG) is computed when G is a nonabelian 2-group given by a central extension of the form 1\longrightarrow ℤ_2ⁿ× ℤ_2^m \longrightarrow G \longrightarrow ℤ₂× ⋯× ℤ₂ \longrightarrow 1 and G'≅ ℤ₂, n, m≥ 1. Further, a conjecture is confirmed, namely, the order of V_*(FG) can be divisible by |F|^{(1)/(2)(|G|+|Ω₁(G)|)-1}, where Ω₁(G)=\g∈ G | g²=1\.

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