Finite groups in which \(\tau\)-quasinormality is a transitive relation.
In the paper and in this review we deal only with finite groups. The well-known fact that normality is not a transitive relation in general has motivated the study of groups in which normality is a transitive relation, the T-groups. Later the condition of normality has been replaced by other weaker properties, like permutability or Sylow permutability (S-permutability in short). Recall that a subgroup \(H\) of a group \(G\) is permutable (respectively, S-permutable or S-quasinormal) in \(G\) when \(H\) permutes with all subgroups (respectively, all Sylow subgroups) of \(G\). Groups in which permutability (respectively, S-permutability) is transitive have been called PT-groups (respectively, PST-groups). The classes of T-groups, PT-groups, and PST-groups have been widely studied. The present paper deals with the effect of imposing on a subgroup to permute with the members of a smaller family of subgroups. For instance, a subgroup \(H\) of a group \(G\) is S-semipermutable (or S-semiquasinormal or S-seminormal) in \(G\) when \(H\) permutes with all Sylow \(p\)-subgroups \(P\) of \(G\) for the primes \(p\) not dividing \(|H|\). The authors introduced [in Acta Math. Hung. 125, No. 3, 237-248 (2009; Zbl 1207.20009)] another interesting embedding property: a subgroup \(H\) of a group \(G\) is \(\tau\)-quasinormal in \(G\) when \(H\) permutes with all Sylow \(p\)-subgroups \(P\) of \(G\) such that \(p\) does not divide \(|H|\) but \((|H|,|P^G|)\neq 1\). S-permutability implies S-semipermutability and S-semipermutability implies \(\tau\)-quasinormality, but none of the converses is true. As a generalisation of T-groups, in this paper groups in which \(\tau\)-quasinormality is a transitive relation are considered. They are called TQT-groups and characterised as the supersoluble groups \(G\) which have an Abelian normal Hall subgroup of odd order \(D\) such that \(G/D\) is nilpotent, every subgroup of \(D\) is normal in \(G\), and every subgroup of \(G\) is \(\tau\)-quasinormal in \(G\). Moreover, in this case, if \(p\notin\pi(D)\) and \(q\notin\pi(D)\), then the order of \([P,Q]\) divides \(r\), where \(r\in\pi(D)\) and \(P\) and \(Q\) are respectively Sylow \(p\)- and \(q\)-subgroups of \(G\), and if \([P,Q]\) has order \(r\), then \(r>p\), \(r>q\), and the Sylow \(r\)-subgroup of \(G\) is cyclic. The authors show that the class of soluble SBT-groups (an SBT-group is a group in which S-semipermutability is transitive) is a proper subgroup of the class of TQT-groups. The proof of this characterisation relies on some new nice characterisations of the class of soluble PST-groups in terms of this embedding property: these groups can be characterised as the groups \(G\) in which every subgroup of the generalised Fitting subgroup of \(G\) is \(\tau\)-quasinormal in \(G\), as the supersoluble groups \(G\) in which every subgroup of the Fitting subgroup of \(G\) is \(\tau\)-quasinormal in \(G\), apart from the well-known characterisation of \textit{R. K. Agrawal} [Proc.\ Am.\ Math.\ Soc. 47, 77-83 (1975; Zbl 0299.20014)] as the supersoluble groups \(G\) with an Abelian normal Hall subgroup \(D\) of odd order such that every subgroup of \(D\) is normal in \(G\). The other main result of the paper, which is also needed in the proof of the previous results, is Theorem~1.3: Let \(E\) be a normal subgroup of a group \(G\). Suppose that for every non-cyclic Sylow \(p\)-subgroup \(P\) of \(F^*(E)\) (the generalised Fitting subgroup of \(E\)) there is a number \(p^k\) such that \(1<p^k<|P|\) and every subgroup of \(P\) with order \(p^k\) and every cyclic subgroup of \(P\) with order \(4\) (if \(p^k=2\) and \(P\) is not Abelian) is \(\tau\)-quasinormal in \(G\). Then each \(G\)-chief factor below \(E\) is cyclic. Theorem~1.3 generalises many results obtained by several authors about S-permutable or S-semipermutable subgroups.
scientific article; zbMATH DE number 5849507
scientific article