NEW CHARACTERIZATIONS OF SOLUBILITY OF FINITE GROUPS

Abstract

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 ≤ 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 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.