Finite groups in which \(\sigma\)-quasinormality is a transitive relation (Q6597523)

Let \(G\) be a finite group. A subgroup \(A \leq G\) is called quasinormal or subnormal if \(A\) permutes with every subgroup of \(G\) (that is, \(AH=HA\) for every \(H \leq G\)). Quasinormal subgroups have many interesting and useful for applications properties. For instance, if \(A\) is quasinormal in \(G\), then: \(A\) is subnormal in \(G\) [\textit{O. Ore}, Duke math. J. 5, 431--460 (1939; JFM 65.0065.06)], \(A/A_{G}\) is nilpotent [\textit{N. Ito} and \textit{J. Szep}, Acta Sci. Math. 23, 168--170 (1962; Zbl 0112.02106)], every chief factor \(H/K\) of \(G\) between \(A_{G}\) and \(A_{G}\) is central [\textit{R. Maier} and \textit{P. Schmid}, Math. Z. 131, 269--272 (1973; Zbl 0259.20017)].\N\NLet \(\sigma=\{ \sigma_{i} \mid i \in I \}\) be a partition of the set of all primes. A subgroup \(A\) of \(G\) is said to be: (i) \(\sigma\)-subnormal in \(G\) if there is a subgroup chain \(A=A_{0} \leq A_{1}\leq \ldots \leq A_{n}=G\) such that either \(A_{r-1} \trianglelefteq A_{r}\) or \(A_{r}/(A_{r-1})_{A_{r}}\) is a \(\sigma_{i}\)-group, \(i=i(r)\) for all \(r=1,2,\ldots,n\). (ii) modular in \(G\) if the following conditions are held: (a) \(\langle X, A \cap Z\rangle=\langle X, A \rangle \cap Z\) for all \(X \leq Z \leq G\); (b) \(\langle A, Y \cap Z\rangle = \langle A, Y \rangle \cap Z\) for all \(Y,Z \leq G\), such that \(A \leq Z\); (iii) \(\sigma\)-quasinormal in \(G\) if \(A\) is \(\sigma\)-subnormal and modular in \(G\).\N\NThe article under review is devoted to the study of the structure of finite groups in which \(\sigma\)-quasinormality (respectively, modularity) is a transitive relation. The authors also re-obtain some known results.