Let G be a finite group, H a subgroup of G and H sG the subgroup of H generated by all those subgroups of H which are s-permutable in G. Then we say that H is weakly s-permutable in G if G has a subnormal subgroup T such that H T = G and T ∩ H H sG . We fix in every non-cyclic Sylow subgroup P of G a subgroup D satisfying 1 < |D| < |P | and study the structure of G under the assumption that all subgroups H with |H | = |D| are weakly s-permutable in G.

Let σ = {σ i |i ∈ I} be some partition of the set of all primes P, G a finite group and σHall σ-set and A permutes with each Hall σ i -subgroup H of G, that is, AH = HA for all i ∈ I.We characterize finite groups with distributive lattice of the σ-permutable subgroups. 0

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Let X be a non-empty subset of a group G. Then we call a subgroup A of G a X-semipermutable subgroup of G if A has a supplement T in G such that for every subgroup T 1 of T there exists an element x ∈ X such that AT x 1 = T x 1 A. In this paper, we study the properties of X-semipermutable subgroups. In particular, a new version of the famous Schur-Zassenhaus Theorem in terms of X-semipermutable subgroups is given.

Let [Formula: see text] be some partition of the set [Formula: see text] of all primes, that is, [Formula: see text] and [Formula: see text] for all [Formula: see text]. We say that a finite group [Formula: see text] is [Formula: see text]-soluble if every chief factor [Formula: see text] of [Formula: see text] is a [Formula: see text]-group for some [Formula: see text]. We give some characterizations of finite [Formula: see text]-soluble groups.