On factorizations of finite groups with \(\mathcal F\)-hypercentral intersections of the factors. (English) Zbl 1241.20029
Summary: A chief factor \(H/K\) of a group \(G\) is called \(\mathcal F\)-central if \([H/K](G/C_G(H/K))\in\mathcal F\). The product of all such normal subgroups of \(G\), whose \(G\)-chief factors are \(\mathcal F\)-central in \(G\), is called the \(\mathcal F\)-hypercentre of \(G\) and denoted by \(Z_{\mathcal F}(G)\). The finite groups \(G\) with factorizations \(G=AB\), where \(A\cap B\leq Z_{\mathcal F}(G)\) for some class of groups \(\mathcal F\), are studied. Some known results about factorizations of groups are generalized.
MSC:
20D40 | Products of subgroups of abstract finite groups |
20D30 | Series and lattices of subgroups |
20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |
20D25 | Special subgroups (Frattini, Fitting, etc.) |
Keywords:
chief factors; finite groups; products of subgroups; hypercentre; factorizations of groups; solvable groups; supersolvable groups; nilpotent groupsReferences:
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