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New characterizations of solubility of finite groups. (English) Zbl 1333.20025
Summary: A subgroup \(H\) of a group \(G\) is said to be \(S\)-supplemented in \(G\) if there exists a subgroup \(T\) of \(G\) such that \(G=HT\) and \(H\cap T\leq H_{sG}\), where \(H_{sG}\) denotes the subgroup of \(H\) generated by all those subgroups of \(H\) which are \(S\)-permutable in \(G\). In this paper, two new characterizations of solubility of finite groups are presented in terms of \(S\)-supplemented subgroups of prime power orders, where primes belong to \(\{3,5\}\). In particular, a counterexample is given to show that the conjecture, proposed by A. A. Heliel at the end of [Commun. Algebra 42, No. 4, 1650-1656 (2014; Zbl 1291.20020)] and related to \(c\)-supplemented subgroups of prime power orders, is negative.

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
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