LOCALIZED NEARLY m-EMBEDDED PROPERTY OF SOME SUBGROUPS OF FINITE GROUPS

Abstract

Let A be a subgroup of a finite group G and Σ : G₀ ≤ G₁ ≤ · · · ≤ G_n some subgroup series of G.  Suppose that for each pair (K,H) such that K is a maximal subgroup of H and G_i−1 ≤ K < H ≤ G_i, for some i, either A ∩ H = A ∩ K or AH = AK.  Then A is said to be Σ-embedded in G; A is said to be nearly m-embedded in G if G has a subgroup T and a {1 ≤ G}-embedded subgroup C in G such that G = AT and T ∩ A ≤ C ≤ A.  In this paper, we localize the above conditions in the G-normalizer of Sylow subgroups of the group G. Some new characterizations of some classes of finite groups are given.