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Weakly \(s\)-supplementally embedded minimal subgroups of finite groups. (English) Zbl 1234.20022

Many papers have dealt with the influence of subgroup embedding properties on the structure of a finite group. Recall that a subgroup \(H\) of a finite group \(G\) is said to be \(s\)-quasinormally embedded (or \(s\)-permutably embedded) in \(G\) if every Sylow subgroup of \(H\) is a Sylow subgroup of an \(s\)-quasinormal (or \(s\)-permutable) subgroup of \(G\) [see A. Ballester-Bolinches and M. C. Pedraza-Aguilera, J. Pure Appl. Algebra 127, No. 2, 113-118 (Zbl 0928.20020)].
In this paper, a subgroup \(H\) of a finite group \(G\) is said to be ‘weakly \(s\)-supplementally embedded’ in \(G\) if there exists a subgroup \(T\) of \(G\) and an \(s\)-permutably embedded subgroup \(H_{se}\) of \(G\) contained in \(H\) such that \(G=HT\) and \(H\cap T\leq H_{se}\). Weakly \(s\)-supplemented subgroups (subgroups \(H\) such that there exists a subgroup \(T\) of \(G\) such that \(HT=G\) and \(H\cap T\) is contained in the largest \(s\)-permutable subgroup of \(G\) contained in \(H\), [see A. N. Skiba, J. Algebra 315, No. 1, 192-209 (Zbl 1130.20019)]) are examples of weakly \(s\)-supplementally embedded subgroups.
The authors study groups in which some subgroups of prime order and of order \(4\) are weakly \(s\)-supplementally embedded. They obtain in Theorem 3.1 that if \(H\) is a normal subgroup of \(G\), \(G/H\) is supersoluble, and every cyclic subgroup \(\langle x\rangle\) of every non-cyclic Sylow subgroup of \(H\) with prime order or of order \(4\) (if the Sylow \(2\)-subgroup of \(H\) is non-Abelian) which does not have a supersoluble supplement in \(G\) is weakly \(s\)-supplementally embedded in \(G\), then \(G\) is supersoluble.
This result is generalised to saturated formations containing the class of all supersoluble groups in Theorem 3.2. Theorem 3.3 is a nilpotency criterion: Let \(N\) be a normal subgroup of a group \(G\) such that \(G/N\) is nilpotent. If every cyclic subgroup of \(N\) with prime order is contained in the hypercentre \(Z_\infty(G)\) of \(G\) and every cyclic subgroup of \(N\) with order \(4\) not having a supersoluble supplement in \(G\) is weakly \(s\)-supplementally embedded in \(G\), then \(G\) is nilpotent.
Finally in Theorem 3.4, for a saturated formation \(\mathfrak F\) containing the class \(\mathfrak N\) of all nilpotent groups, it is proved that if every cyclic subgroup of the \(\mathfrak F\)-residual \(G^{\mathfrak F}\) with order \(4\) is weakly \(s\)-supplementally embedded in \(G\), then \(G\in\mathfrak F\) if and only if every cyclic subgroup of \(G^{\mathfrak F}\) of prime order lies in the \(\mathfrak F\)-hypercentre \(Z_{\mathfrak F}(G)\) of \(G\).

MSC:

20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20D40 Products of subgroups of abstract finite groups
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
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