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Finite groups in which SS-permutability is a transitive relation. (English) Zbl 1316.20021
A 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.

##### 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
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