NEW CHARACTERIZATIONS OF SOLUBILITY OF FINITE GROUPS - Li, Jinbao; Shi, Wujie; Chen, Guiyun; Yu, Dapeng (Italian Journal of Pure and Applied Mathematics, 2014)
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 ∩ T ≤ HsG, where HsG 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 primes power orders, where primes belong to 3,5. In particular, a counterexample is given to show that the conjecture, proposed by Heliel at the end of [A.A. Heliel, A note on c-supplemented subgroups of finite groups, Comm. Algebra, 42 (2014), 1650-1656] and related to c-supplemented subgroups of primes power orders, is negative.