σ-Subnormality in locally finite groups

Abstract

Let σ={σj: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 subgroupsX=X0≤X1≤…≤Xn=G such that, for each 1≤i≤n−1, Xi−1⊴Xi or Xi/(Xi−1)Xi is a σji-group for some ji∈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 HK=KH, 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).