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 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
and G′ ≅ ℤ2, n, m ⩾ 1. Furthermore, a conjecture is confirmed, i.e., the order of V*(FG) can be divisible by \(|F{|^{{1 \over 2}(|G| + |{\Omega _1}(G)|) - 1}}\), where Ω1(G) = {g ∈ G ∣ g2 = 1}.
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Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant No. 12171142). The authors cordially thank the referees for their time and helpful comments.
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Wang, Y., Liu, H. The unitary subgroups of group algebras for a class of finite 2-groups with the derived subgroup of order 2. Sci. China Math. (2024). https://doi.org/10.1007/s11425-023-2156-7
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DOI: https://doi.org/10.1007/s11425-023-2156-7