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

作　　者：Yulei Wang Heguo Liu 

机构地区：[1]Department of Mathematics,Henan University of Technology,Zhengzhou 450001,China [2]Department of Mathematics,Hainan University,Haikou 570228,China

出　　处：《Science China Mathematics》2024年第9期1989-2018,共30页中国科学（数学）（英文版）

基　　金：supported by National Natural Science Foundation of China(Grant No.12171142)。

摘　　要：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 1→Z_(2^(n))×Z_(2^(m))→G→Z_(2)×…×Z_(2)→1 and G'≌Z_(2),n,m≥1.Furthermore,a conjecture is confirmed,i.e.,the order of V_(*)(FG)can be divisible by|F|^(1/2(|G|+|Ω1(G)|)-1),where Ω_(1)(G)={g∈G|g^(2)=1}.

关 键 词：normalized unit unitary subgroup inner abelian p-group central extension 

分 类 号：O152[理学—数学]