Twisted centrally large subgroups of finite groups. (English) Zbl 1514.20080
Summary: 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 \(| Q | \cdot | Z(Q) | \geqslant | Q^\ast | \cdot | Z( Q^\ast) |\) for every subgroup \(Q^\ast\) of \(G\). A CL-subgroup of \(G\) that is minimal under inclusion is called a minimal CL-subgroup of \(G\). G. Glauberman [J. Algebra 300, No. 2, 480–508 (2006; Zbl 1103.20013)] studied some properties of CL-subgroups, especially minimal CL-subgroups. He showed that if \(Q_1\) and \(Q_2\) are a pair of minimal CL-subgroups, then
\[
Q_i = (Q_1 \cap Q_2) Z( Q_i) \text{ for } i = 1, 2 . \tag{\( \ast \)}
\]
We say that two CL-subgroups \(Q_1\) and \(Q_2\) (not necessarily minimal) are twisted CL-subgroups if they satisfy the above property \(( \ast )\). In this paper, we extend Glauberman’s work to prove some further properties of CL-subgroups, especially twisted CL-subgroups. In addition, we give a negative answer to a conjecture proposed by A. Morresi Zuccari et al. [J. Algebra 502, 262–276 (2018; Zbl 1403.20032)].
MSC:
| 20D30 | Series and lattices of subgroups |
| 20D15 | Finite nilpotent groups, \(p\)-groups |
| 20D60 | Arithmetic and combinatorial problems involving abstract finite groups |
Keywords:
centrally large subgroup; Chermak-Delgado lattice; isoclinic group; twisted CL-pairs; generalized extremal CL-subgroupsReferences:
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