In this paper, we give some new conditions of the existence of Hall subgroups in non-soluble finite groups, and so the famous Hall theorem and Schur-Zassenhaus theorem are generalized.
Letσ={σi|i∈I}be some partition of the set P of all primes and G afinite group.A set H of subgroups of G is said to be a complete Hallσ-set of G ifevery member≠1 of H is a Hallσi-subgroup of G for some i c l and H containsexactly one Hallσi-subgroup of G for every i such thatσi∩π(G)≠Ø.A subgroupA of G is said to be H-permutable if A permutes with all members of the completeHallσ-set H of G.In this paper,we study the structure of G under the assuming thatsome subgroups of G areσ-permutable.
A subgroup E of a finite group G is called hypercyclically embedded in G if every chief factor of G below E is cyclic.Let A be a subgroup of a group G.Then we call any chief factor H/AG of G a G-boundary factor of A.For any G-boundary factor H/AG of A,we call the subgroup(A∩H)/AG of G/AG a G-trace of A.On the basis of these notions,we give some new characterizations of hypercyclically embedded subgroups.
