Propermutable characterizations of finite soluble \(PST\)-groups and \(PT\)-groups.
For \(H,X\) subgroups of \(G\) we say that \(H\) is \(X\)-propermutable in \(G\) if there is a subgroup \(B\) of \(G\) such that \(G=N_G(H)B\) and \(H^xB=BH^x\) for some \(x\in X\); for \(X=1\) the groups are just called propermutable. \(H\) is completely propermutable if \(H\) is propermutable in every \(K\) with \(H\subseteq K\subseteq G\). The author shows: (i) The soluble group \(G\) is a PST-group iff all Hall subgroups and all of their maximal subgroups are \(Z_\infty(G)\)-propermutable in \(G\). (ii) A soluble group of odd order is a PT-group iff all Hall subgroups and all subnormal subgroups of \(G\) are completely propermutable in \(G\) (Theorems A, B).
scientific article; zbMATH DE number 6526921
scientific article