σ-Subnormality in locally finite groups.

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]