On solvability of finite groups with some ss-supplemented subgroups Czechoslovak Mathematical Journal, Vol. 65 (2015) Abstract. A subgroup H of a finite group G is said to be ss-supplemented in G if there exists a subgroup K of G such that G = HK and H ∩ K is s-permutable in K. In this paper, we first give an example to show that the conjecture in A. A. Heliel's paper (2014) has negative solutions. Next, we prove that a finite group G is solvable if every subgroup of odd prime order of G is ss-supplemented in G, and that G is solvable if and only if every Sylow subgroup of odd order of G is ss-supplemented in G. These results improve and extend recent and classical results in the literature.

For a finite group G, let v(G) denote the number of conjugacy classes of all non-normal subgroups of G, and let π(G) denote the set of primes dividing the order of G. In this note, we shall classify the finite groups G with v(G) ≤ |π(G)|.