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The unitary subgroups of group algebras of a class of finite p-groups

Yulei Wang

https://orcid.org/0000-0003-4539-9134

Department of Mathematics, Henan University of Technology, Zhengzhou 450001, P. R. China

E-mail Address: yulwang@163.com

Corresponding author.

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and

Heguo Liu

Department of Mathematics, Hubei University, Wuhan 430062, P. R. China

E-mail Address: ghliu@hubu.edu.cn

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https://doi.org/10.1142/S0219498823500433Cited by:4 (Source: Crossref)

Abstract

Let p be a prime and let F be a finite field of characteristic p. Let FG denote the group algebra of the finite p-group G over the field F and let $V (F G)$ denote the group of normalized units in FG. The anti-automorphism $g \mapsto g^{- 1}$ of G extends linearly to an anti-automorphism $a \mapsto a^{*}$ of FG. An element $u \in V (F G)$ is called unitary if $u^{*} = u^{- 1}$ . All unitary elements of $V (F G)$ form a subgroup which is denoted by $V_{*} (F G)$ . If p is odd, the order of $V_{*} (F G)$ is $| F |^{(| G | - 1) / 2}$ . However, to compute the order of $V_{*} (F G)$ still is open when $p = 2$ . In this paper, the order of $V_{*} (F G)$ is computed when G is a nonabelian $2$ -group given by a central extension of the following form:

1 \to ℤ 2 m \to G \to ℤ 2 \times \dots \times ℤ 2 \to 1 <math display="block" altimg="eq-00014.gif"><mrow><mn>1</mn><mo class="MathClass-rel">\to</mo><msub><mrow><mi>ℤ</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup></mrow></msub><mo class="MathClass-rel">\to</mo><mi>G</mi><mo class="MathClass-rel">\to</mo><msub><mrow><mi>ℤ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy="false" class="MathClass-bin">\times</mo><mo class="MathClass-rel">\dots</mo><mo stretchy="false" class="MathClass-bin">\times</mo><msub><mrow><mi>ℤ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo class="MathClass-rel">\to</mo><mn>1</mn></mrow></math>

and

$G^{'} ≅ ℤ_{2}$ ,

$m \geq 1$ . Further, a conjecture is confirmed, namely, the order of

$V_{*} (F G)$ can be divided by

$| F |^{\frac{1}{2} (| G | + | Ω_{1} |) - 1}$ , where

$Ω_{1} = {g \in G | g^{2} = 1}$ .

Communicated by M. L. Lewis

Keywords:

AMSC: 20C05, 20D15

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