π-quasinormally embedded and c-supplemented subgroup of finite group

On $W$-$S$-permutable subgroups of finite groups

“…A subgroup T of G is said to be S-permutable (S-quasinormal) in G if T is π(G)-quasinormal in G, where π(G) denote a set of primes dividing |G|. The relationship between the structure of a group G and its S-permutable subgroups has been extensively studied by many authors (for example, see [4], [5], [12], [17]). On the other hand, a subgroup H of a group G is C-supplemented in G if there exists a subgroup K of G such that G = HK and H ∩ K ≤ H G , where H G is the core of H in G [6].…”

“…(1) Let G = S 4 , the symmetric group of degree 4. Take H = (12) . Then it is easy to see H is W -S-permutable in G. But H is not nearly S-permutable in G since N S3 ( (12) ) does not contain any Sylow 3-subgroup of S 3 .…”