Let σ={σi:i∈I} be a partition of the set all prime numbers. A subgroup H finite group G is said to σ-subnormal in if can joined by chain subgroups H=H0⊆H1⊆⋯⊆Hn=G where, for every j=1,⋯,n, Hj−1 normal Hj or Hj/CoreHj(Hj−1) σi-group some i∈I. B soluble normalising Nσ-residual non-σ-subnormal G, where Nσ saturated formation σ-nilpotent groups. We show that normalises does not have section σ-residu...

Let K[G] be the group algebra of a locally finite group G over a field K of characteristic p > 0. In this paper, we show that K[G] is semiprimitive if and only if G has no locally subnormal subgroup of order divisible by p. Thus we settle the semiprimitivity problem for such group algebras by verifying a conjecture which dates back to the middle 1970’s. Of course, if G has a locally subnormal s...

Let σ = {σi|i ∈ I } be a partition of the set all primes ℙ and G finite group. A ℋ subgroups is said to complete Hall σ-set if every member ≠1 σi-subgroup for some i contains exactly one such that σi ⌒ π(G) ≠ ∅. group σ-primary it σi-group i. subgroup be: σ-permutable in possesses AH x H xA G; σ-subnormal there chain A0 ≤ A1 … At either Ai − 1 ⊴ or /(Ai 1)Ai 1, …, t;

A transitively normal subgroup of a group G is one that is normal in every subgroup in which it is subnormal. This concept is obviously related to the transitivity of normality because the latter holds in every subgroup of a group G if and only if every subgroup of G is transitively normal. In this paper we describe the structure of a group whose cyclic subgroups (or a part of them) are transit...