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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

Citations:

Zbl 0778.20011
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