Finite groups in which SS-permutability is a transitive relation.

*(English)*Zbl 1316.20021A subgroup \(H\) of a finite group \(G\) is called semipermutable in \(G\) if \(H\) permutes with every subgroup \(X\) for which \((|H|,|X|)=1\). A subgroup \(H\) of \(G\) is called SS-permutable [NSS-permutable] in \(G\) if \(H\) has a [normal] complement \(K\) in \(G\) such that \(H\) permutes with every Sylow subgroup of \(K\). A group is called a BT-group [SST-group, NSST-group] if semipermutability [SS-permutability, NSS-permutability] is a transitive relation on the subgroups of \(G\).

The authors prove several theorems on the structure of finite SST-groups. For example, the authors show (Theorem 1.4) that if a finite group is an SST-group [NSST-group] then all its chief factors are simple.

The main result of the paper is the following. Theorem 1.7. Let \(G\) be a finite solvable group. Then the following are equivalent. (1) \(G\) is an SST-group. (2) \(G\) is an NSST-group. (3) Every subgroup of \(G\) is SS-permutable in \(G\). (4) Every subgroup of \(G\) is NSS-permutable in \(G\). (5) Every subgroup of \(G\) of prime power order is SS-permutable in \(G\). (6) Every subgroup of \(G\) of prime power order is NSS-permutable in \(G\). (7) Every cyclic subgroup of \(G\) of prime power order is SS-permutable in \(G\). (8) Every cyclic subgroup of \(G\) of prime power order is NSS-permutable in \(G\).

As a corollary, every finite solvable SST-group is a BT-group, but the converse does not necessarily hold. The authors give a characterization for a finite solvable BT-group to be an SST-group (Theorem 1.13). They provide some further structural theorems for finite solvable SST-groups and for finite solvable BT-groups.

The authors prove several theorems on the structure of finite SST-groups. For example, the authors show (Theorem 1.4) that if a finite group is an SST-group [NSST-group] then all its chief factors are simple.

The main result of the paper is the following. Theorem 1.7. Let \(G\) be a finite solvable group. Then the following are equivalent. (1) \(G\) is an SST-group. (2) \(G\) is an NSST-group. (3) Every subgroup of \(G\) is SS-permutable in \(G\). (4) Every subgroup of \(G\) is NSS-permutable in \(G\). (5) Every subgroup of \(G\) of prime power order is SS-permutable in \(G\). (6) Every subgroup of \(G\) of prime power order is NSS-permutable in \(G\). (7) Every cyclic subgroup of \(G\) of prime power order is SS-permutable in \(G\). (8) Every cyclic subgroup of \(G\) of prime power order is NSS-permutable in \(G\).

As a corollary, every finite solvable SST-group is a BT-group, but the converse does not necessarily hold. The authors give a characterization for a finite solvable BT-group to be an SST-group (Theorem 1.13). They provide some further structural theorems for finite solvable SST-groups and for finite solvable BT-groups.

Reviewer: Gábor Horváth (Debrecen)

##### MSC:

20D40 | Products of subgroups of abstract finite groups |

20D20 | Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure |

20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |

20D35 | Subnormal subgroups of abstract finite groups |

##### Keywords:

finite groups; SS-permutability; S-semipermutability; PST-groups; BT-groups; SST-groups; permutable subgroups; supplemented subgroups; Sylow subgroups; transitive permutability; semipermutable subgroups; finite solvable groups##### References:

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