A generalization of an Agrawal theorem on soluble <i>PST</i> -groups (Q6636338)

Let \(\sigma=\{\sigma_{i} \mid i \in I \}\) be a partition of all prime numbers. A finite group \(G\) is: \(\sigma\)-primary if \(G\) is a \(\sigma_{i}\)-group for some \(i \in I\); \(\sigma\)-nilpotent if \(G\) is the direct product of \(\sigma\)-primary groups; \(\sigma\)-soluble if every chief factor of \(G\) is \(\sigma\)-primary. A subgroup \(A \leq G\) is \(\sigma\)-permutable in \(G\) if \(G\) has a Hall \(\sigma_{i}\)-subgroup for all \(i \in I\) and \(A\) permutes with every such Hall subgroup of \(G\). The group \(G\) is said to be a \(P\sigma T\)-group if \(\sigma\)-permutability is a transitive relation in \(G\) and, in the case that \(|\sigma_{i}|=1\) for every \(i \in I\), \(G\) is said to be a \(PST\)-group.\N\NThe main result in this paper is (see Theorem B): Let \(G\) be a soluble group and let \(D=G^{\mathfrak{N}_{\sigma}}\) be the \(\sigma\)-nilpotent residual of \(G\) (that is, the smallest normal subgroup \(N\) of \(G\) with \(\sigma\)-nilpotent quotient \(G/N\)). Then any two of the following statements are equivalent: (i) every subnormal subgroup of \(G\) is \(\sigma\)-permutable in \(G\) and \(D\cap H_{i} \leq Z_{\infty}(H_{i})\) for every Hall \(\sigma_{i}\)-subgroup \(H_{i}\) of \(G\) and all \(i\in I\); (ii) \(D\) is an abelian Hall subgroup acted on by conjugation by \(G\) as a group of power automorphisms, \(O_{\sigma_{i}}(D)\) has a normal complement in a Hall \(\sigma_{i}\)-subgroup of \(G\) for all \(i \in I\), and \((p, |D|)=1\) for all primes \(p\) such that \((p-1,|G|)=1\); (iii) \(G\) is a \(P\sigma T\)-group.\N\NIn particular, the authors obtain a new characterization of soluble \(P\sigma T\)-groups which is a generalization of a result by \textit{R. K. Agrawal} [Proc. Am. Math. Soc. 47, 77--83 (1975; Zbl 0299.20014)] on soluble \(PST\)-groups.