## The influence of minimal subgroups on the structure of finite groups.(English)Zbl 0937.20008

A subgroup $$A$$ of a group $$G$$ is said to be seminormal in $$G$$ if there exists a subgroup $$B$$ of $$G$$ such that $$G=AB$$ and for each proper subgroup $$B_1$$ of $$B$$, we have that $$AB_1$$ is a proper subgroup of $$G$$.
The author presents an extension of earlier results of P. Wang on seminormality [J. Algebra 148, No. 8, 289-295 (1992; Zbl 0778.20011)] through the theory of formations. The main result is the following Theorem. Let $$F$$ be a subgroup-closed saturated formation of finite groups with the following property: a minimal non-$$F$$-group is soluble and its $$F$$-residual is a Sylow subgroup. If any cyclic subgroup of $$G$$ of order 4 is seminormal in $$G$$ and any minimal subgroup of $$G$$ is contained in the $$F$$-hypercentre of $$G$$, then $$G$$ is an $$F$$-group. The proof of the theorem is a rather easy application of the structure of the minimal non-$$F$$-groups.

### MSC:

 20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, $$\pi$$-length, ranks 20D40 Products of subgroups of abstract finite groups 20D25 Special subgroups (Frattini, Fitting, etc.) 20D35 Subnormal subgroups of abstract finite groups 20D15 Finite nilpotent groups, $$p$$-groups

Zbl 0778.20011