Finite soluble groups with permutable subnormal subgroups (Q5939573)

In this article all groups considered are finite. A \(PST\)-group is a group in which each subnormal subgroup permutes with every Sylow subgroup. (These are also the groups in which permutability with Sylow subgroups is transitive.) In the article under review the authors define various local properties which for soluble groups are equivalent to \(PST\). These are: (i) A group \(G\) has \(U_p^*\) if all its \(p\)-chief factors are of order \(p\) and are \(G\)-isomorphic: (ii) \(G\) is \(PST_p\) if each subnormal \(p'\)-perfect subgroup permutes with every Hall \(p'\)-subgroup: (iii) \(G\) has \({\mathbf H}_p\) if each normal subgroup of a Sylow \(p\)-subgroup \(P\) is pronormal in \(G\).NEWLINENEWLINENEWLINEThe authors prove that for soluble groups the properties (i), (ii), and (iii) for all subgroups are equivalent. At the global level this result translates intoNEWLINENEWLINENEWLINETheorem: The following are equivalent properties for a soluble group \(G\): (i) \(G\) is \(PST\); (ii) \(G\) is \(PST_p\) for all \(p\): (iii) \(G\) has \(U^*_p\) for all \(p\): (iv) all subgroups of \(G\) have \({\mathbf H}_p\) for all \(p\).NEWLINENEWLINENEWLINEThere are connections with the class of \(PT\)-groups, i.e. groups in which permutability is transitive. Characterizations of soluble \(PT\)-groups were given by \textit{J. C. Beidleman, B. Brewster} and \textit{D. J. S. Robinson} [in J. Algebra 222, No. 2, 400-412 (1999; Zbl 0948.20015)]. The authors point out that for a soluble group the only difference between \(PST\) and \(PT\) is that in the latter case Sylow subgroups are modular. In fact this statement is true for insoluble groups too, as is pointed out in a recent paper of the reviewer [J. Aust. Math. Soc. 70, No. 2, 143-159 (2001; Zbl 0997.20027)].