# zbMATH — the first resource for mathematics

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
Full Text: