Joins of σ-subnormal subgroups

Abstract

Let $\mathit{\sigma }=\{{\mathit{\sigma }}_j:j\in J\}$ be a partition of the set $\mathbb{P}$ of all prime numbers. A subgroup X of a finite group G is σ-subnormal in G if there exists a chain of subgroups

$$X=X_0\le X_1\le \cdots \le X_n=G$$

such that, for each $1\le i\le n-1$, $X_{i-1}⊴X_i$ or $X_i∕{(X_{i-1})}_{X_i}$ is a ${\mathit{\sigma }}_{j_i}$-group for some $j_i\in J$. Skiba 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 a previous paper, we laid the foundation of a general theory of σ-subnormal subgroups (and σ-series) in locally finite groups. It turns out that the main difference between the finite and the locally finite case concerns the behavior of the join of σ-subnormal subgroups: in finite groups, σ-subnormal subgroups form a sublattice of the lattice of all subgroups, but this is no longer true for arbitrary locally finite groups. This situation is very similar to that concerned with subnormal subgroups; therefore, as in the case of subnormal subgroups, it makes sense to study the class ${\mathfrak{S}}_{\mathit{\sigma }}^{\mathrm{\infty }}$ (resp. ${\mathfrak{S}}_{\mathit{\sigma }}$) of locally finite groups in which the join of (resp. of finitely many) σ-subnormal subgroups is σ-subnormal. In particular, the aim of this paper is to study how much one can extend a group in one of these classes before going outside the same class (see, for example, Theorems 3.6, 3.8, 5.5, and 5.7). Furthermore, some σ-subnormality criteria for the join of two σ-subnormal subgroups are obtained: for example, similar to a celebrated theorem of Williams, we give necessary and sufficient conditions for a join of two σ-subnormal subgroups to always be σ-subnormal; as a consequence, we show that the join of two orthogonal σ-subnormal subgroups is σ-subnormal (this is the analog of a result of Roseblade).