The structure of non-nilpotent CTI-groups. (Q2842799)

Let \(G\) be a finite group. \(H\) is a TI (QTI-)subgroup of \(G\) if \(H\cap H^g\in\{1,H\}\) (\(C_G(x)\leq N_G(H)\) for any \(1\neq x\in H\)). If every (Abelian, cyclic) subgroup is a TI-subgroup then \(G\) is called a TI (ATI-, CTI-)group. If every (Abelian) subgroup is a QTI-subgroup then \(G\) is called a QTI (AQTI-)group. \textit{G. Walls} [Arch. Math. 32, 1-4 (1979; Zbl 0388.20011)] classified TI-groups, \textit{X. Guo, S. Li} and \textit{P. Flavell} [J. Algebra 307, No. 2, 565-569 (2007; Zbl 1116.20014)] ATI-groups and \textit{G. Qian} and \textit{F. Tang} [J. Algebra 320, No. 9, 3605-3611 (2008; Zbl 1178.20014)] AQTI-groups.NEWLINENEWLINE The authors in the present paper describe CTI-groups as follows. If \(G\) is a solvable CTI-group with trivial center and its Fitting \(F\) is a \(p\)-group then either \(G\) is isomorphic to \(S_4\) or \(F\) is a \(p\)-Sylow subgroup and \(G\) is Frobenius with kernel \(F\). If \(G\) is a non-solvable CTI-group then \(G\) is isomorphic to either \(\text{PSL}(2,q)\) or \(\text{PGL}(2,q)\) with \(q>3\) prime power.