Search

Let G be a finite group, H a subgroup of G and p a prime number. We say that H is weakly sp-permutable in G if G has a subnormal subgroup K such that G = HK, H s G ≤ K and | H ∩ K : H s G | is a p ′ -number, where HsG is the subgroup of H generated by all those subgroups of H which are s-permutable in G. In this paper, we investigate the structure of a group G under the assumption that certain subgroups of G are weakly sp-permutable in G. Some recent results are extended and generalized. Communicated by Alexander Olshanskii [ABSTRACT FROM AUTHOR]

Let σ = { σ j : j ∈ J } be a partition of the set P of all prime numbers. A subgroup X of a finite group G is σ-subnormal in G if there exists a chain of subgroups X = X 0 ≤ X 1 ≤ ... ≤ X n = G such that, for each 1 ≤ i ≤ n − 1 , X i − 1 ⊴ X i or X i / (X i − 1) X i is a σ j i -group for some j i ∈ J. Skiba [18] studied the main properties of σ -subnormal subgroups in finite groups and showed that the set of all σ -subnormal subgroups plays a very relevant role in the structure of a finite soluble group. In this paper we lay the foundation of a general theory of σ -subnormal subgroups (and σ -series) in locally finite groups. Although in finite groups, σ -subnormal subgroups form a sublattice of the lattice of all subgroups (see for instance [3]), this is no longer true for locally finite groups; in fact, the join of σ -subnormal subgroups is not always σ -subnormal, but this is the case (for example) whenever the join of subnormal subgroups is subnormal (see Theorem 3.16). We provide many criteria to determining when a subgroup is σ -subnormal starting from the much weaker concept of σ-seriality (see Section 2). These criteria are particularly useful when employed to investigate the join of σ -subnormal subgroups — we show for example that if two σ -subnormal subgroups H and K of a locally finite group G are such that H K = K H , then HK is σ -subnormal in G (see Theorem 3.15) — but they are also fit to show that on some occasions σ -seriality coincides with σ -subnormality — this is the case of linear groups (see Theorem 3.35). [ABSTRACT FROM AUTHOR]

Let p be a prime number. A positive integer m is said to be a p ′ -number provided p ∤ m . Let G be a finite group, and let H be a subgroup of G. We say that H is weakly s p -permutable in G if G has a subnormal subgroup K such that G = HK, H sG ≤ K and | H ∩ K : H sG | is a p ′ -number, where H sG is the subgroup of H generated by all those subgroups of H which are s-permutable in G. We study the structure of a finite group G of even order all of whose fourth maximal subgroups are weakly s 2 -permutable in G. [ABSTRACT FROM AUTHOR]