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$$S$$-embedded subgroups of finite groups. (English. Russian original) Zbl 1255.20021
Algebra Logic 49, No. 4, 293-304 (2010); translation from Algebra Logika 49, No. 4, 433-450 (2010).
Summary: A subgroup $$H$$ of $$G$$ is said to be $$S$$-embedded in $$G$$ if $$G$$ has a normal subgroup $$N$$ such that $$HN$$ is $$s$$-permutable in $$G$$ and $$H\cap N\leqslant H_{sG}$$, where $$H_{sG}$$ is the largest $$s$$-permutable subgroup of $$G$$ contained in $$H$$. $$S$$-embedded subgroups are used to give novel characterizations for some classes of groups. New results are obtained and a number of previously known ones are generalized.

##### 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 20D25 Special subgroups (Frattini, Fitting, etc.)
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