Criteria for permutability to be transitive in finite groups (Q1969361)

A subgroup \(H\) of a group \(G\) is said to be permutable in \(G\) if \(HK=KH\) for all subgroups \(K\) of \(G\). The authors study \(PT\)-groups, that is finite groups \(G\) such that \(H\) permutable in \(K\) and \(K\) permutable in \(G\) imply that \(H\) is permutable in \(G\). The structure of soluble \(PT\)-groups was determined by Zacher. The authors introduce the condition \(X_p\): a finite group \(G\) satisfies \(X_p\) if and only if each subgroup of a Sylow \(p\)-subgroup \(P\) of \(G\) is permutable in the normalizer \(N_G(P)\). Using Zacher's theorem they show that \(G\) is a soluble \(PT\)-group if and only if it satisfies \(X_p\) for all primes \(p\). For this they have to study the property \(X_p\). They show that a finite group \(G\) satisfies \(X_p\) if and only if either \(G\) is \(p\)-nilpotent and has modular Sylow \(p\)-subgroups (this latter condition is missing in the statement of Theorem B of the paper) or \(G\) has an Abelian Sylow \(p\)-subgroup \(P\) and every subgroup of \(P\) is normal in \(N_G(P)\). And if \(p\) is the smallest prime divisor of \(|G|\), then \(G\) has \(X_p\) if and only if \(G\) is \(p\)-nilpotent and has modular Sylow \(p\)-subgroups. Reviewer's remark: The proof of this last result could be shortened considerably by using Lemma 2.3.5 in the reviewer's book ``Subgroup lattices of groups'' (1994; Zbl 0843.20003).