Finite simple groups the nilpotent residuals of all of whose subgroups are \(TI\)-subgroups (Q6635325)

A subgroup \(H\) of a finite group \(G\) is called \(\mathsf{TI}\)-group if \(H \cap H^{x}=1\) for every \(x \in G \setminus N_{G}(H)\). Let \(\Sigma=\{G \trianglelefteq G \mid G/N \; \mathrm{is \; nilpotent} \}\) and let \(G^{\mathcal{N}}=\bigcap_{N \in \Sigma} N\) be the nilpotent residual of \(G\).\N\NIn the paper under review, the authors show that if \(G\) is a non-abelian finite simple group and \(K^{\mathcal{N}}\) is a \(\mathsf{TI}\)-group for every proper subgroup \(K < G\), then \(G\) is isomorphic to either \(\mathrm{PSL}(2,7)\), \(\mathrm{PSL}(2,2^{p})\) (\(p\) a prime) or \(\mathrm{Sz}(2^{p})\) (\(p\) an odd prime). The proof makes use of the classification of finite simple groups.