Twisted centrally large subgroups of finite groups

发布时间：2023-04-25 作者：浏览次数：

Speaker:	安立坚教授	DateTime:	2023年4月27日（周四）17：00-18：00
Brief Introduction to Speaker: 安立坚山西师范大学教授		Place:	6号楼二楼学术报告厅
Abstract:Let G be a finite group. A subgroup Q of G is defined, by Glauberman, to be a centrally large subgroup, or CL-subgroup, of G if \|Q\| · \|Z(Q)\| > \|Q∗\| · \|Z(Q∗)\|for every subgroup Q∗ of G. A CL-subgroup of G that is minimal under inclusion is called a minimal CL-subgroup of G. Glauberman [Centrally large subgroups of finite p-groups, J. Algebra, 300(2006), 480-508] studied some properties of CLsubgroups, especially minimal CL-subgroups. He showed that if Q1 and Q2 are a pair of minimal CL-subgroups, then Qi = (Q1 ∩ Q2)Z(Qi) for i = 1, 2. We say that two CL-subgroups Q1 and Q2 (not necessarily minimal) are twisted CLsubgroups if they satisfy the above property. In this paper, we extend Glauberman's work to prove some further properties of CL-subgroups, especially twisted CLsubgroups. In addition, we give a negative answer to a conjecture proposed by Scoppola et al.