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On weakly \(S\)-embedded and weakly \(\tau\)-embedded subgroups. (English. Russian original) Zbl 1311.20022
Sib. Math. J. 54, No. 5, 931-945 (2013); translation from Sib. Mat. Zh. 54, No. 5, 1162-1181 (2013).
In the paper under review, some subgroup embedding properties are studied in order to obtain information about groups in which some subgroups of prime power order satisfy these properties.
Let \(G\) be a finite group. A subgroup of \(G\) which permutes with all Sylow subgroups of \(G\) is called S-quasinormal (or S-permutable). A subgroup \(H\) of \(G\) is said to be S-quasinormally embedded (or S-permutably embedded) in \(G\) if every Sylow subgroup of \(H\) is a Sylow subgroup of some S-quasinormal subgroup of \(G\). A subgroup \(H\) of \(G\) is said to be weakly S-embedded in \(G\) if there exists a normal subgroup \(K\) such that \(HK\) is S-quasinormal in \(G\) and \(H\cap K\leq H_{seG}\), where \(H_{seG}\) is the subgroup generated by all subgroups of \(H\) which are S-quasinormally embedded in \(G\). A subgroup \(H\) of \(G\) is said to be \(\tau\)-quasinormal in \(G\) if \(H\) permutes with all Sylow \(q\)-subgroups \(Q\) of \(G\) such that \(\gcd(q,|H|)=1\) and \(\gcd(|H|,|Q^G|)\neq 1\). A subgroup \(H\) of a group \(G\) is said to be weakly \(\tau\)-embedded in \(G\) if there exists a normal subgroup \(K\) of \(G\) such that \(HK\) is S-quasinormal in \(G\) and \(H\cap K\leq H_{\tau G}\), where \(H_{\tau G}\) is the subgroup generated by all \(\tau\)-quasinormal subgroups of \(G\). The embedding properties analysed in this paper are weak S-embedding and weak \(\tau\)-embedding. We will denote by \(G^{\mathfrak N_p}\) the smallest normal subgroup \(N\) of \(G\) such that \(G/N\) is \(p\)-nilpotent.
The general hypothesis for Theorem 3.1 and Theorem 3.2 is that \(p\) is a prime divisor of \(|G|\) and that \(\gcd(|G|,(p-1)(p^2-1)\cdots (p^n-1))=1\) for some integer \(n\geq 1\). In Theorem 3.1, it is proved that \(G\) is \(p\)-nilpotent if and only if there exists a normal subgroup \(H\) of \(G\) such that \(G/H\) is \(p\)-nilpotent and for all Sylow \(p\)-subgroups \(P\) of \(H\), every \(n\)-maximal subgroup of \(P\) not containing \(P\cap G^{\mathfrak N_p}\) (if it exists) either has a \(p\)-nilpotent supplement in \(G\) or \(H\) is weakly S-embedded (and analogously with weakly \(\tau\)-embedded). A similar characterisation holds for subgroups \(L\) of order \(p^n\) or \(4\) (if \(p=2\) and \(n=1\), \(P\) is non-abelian and \(L\) is cyclic) not contained in \(Z_\infty(G)\) (Theorem 3.2). For \(A_4\)-free groups and primes \(p\) dividing \(|G|\) such that \(\gcd(|G|,p-1)=1\), a similar characterisation with \(2\)-maximal subgroups is obtained.
Recall that a saturated formation is a class of groups which is closed under epimorphic images, subdirect products, and Frattini extensions. For a saturated formation \(\mathfrak F\) containing all supersoluble groups, it is proved that \(G\in\mathfrak F\) if and only if there exists a normal subgroup \(H\) of \(G\) such that \(G/F\in\mathfrak F\) and, for any non-cyclic Sylow subgroup \(P\) of \(H\), each maximal subgroup of \(P\) either has a supersoluble supplement in \(G\) or is weakly S-embedded (or weakly \(\tau\)-embedded) in \(G\) (Theorem 3.4). A similar result is obtained with the condition that every cyclic subgroup \(L\) of \(P\) of prime order or order \(4\) (when \(p=2\) and \(P\) is non-abelian) not contained in \(Z_\infty(G)\) either has a supersoluble supplement in \(G\) or is weakly S-embedded (or weakly \(\tau\)-embedded), this is Theorem 3.5. Finally, similar results are obtained for Sylow subgroups of \(F^*(G)\), the generalised Fitting subgroup of \(G\) (Theorem 3.6).

MSC:
20D40 Products of subgroups of abstract finite groups
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20D25 Special subgroups (Frattini, Fitting, etc.)
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