On $$\sigma $$-subnormality criteria in finite $$\sigma $$-soluble groups

Abstract

Let σ = {σ i : i ∈ I} be a partition of the set P of all prime numbers. A subgroup X of a finite groupwhere for every j = 1, . . . , n the subgroup X j−1 normal in X j or X j /Core X j (X j−1 ) is a σ i -group for some i ∈ I.In the special case that σ is the partition of P into sets containing exactly one prime each, the σ-subnormality reduces to the familiar case of subnormality.In this paper some σ-subnormality criteria for subgroups of σsoluble groups, or groups in which every chief factor is a σ i -group, for some σ i ∈ σ, are showed.

σ-Subnormality in locally finite groups

“…Second, σ-residuals have very nice permutability properties. Using Lemma 3 (5) of [2], the argument in Lemma 3.3 of [6] proves the following result. Lemma 2.3 Let σ = {σ i : i ∈ I} be a partition of P and let G be a σ-soluble finite group.…”

“…Now, in order to see that G σ = H σ , K σ , we only need to apply the above argument replacing σ-soluble residuals by σ-residuals, Corollary 6.5.48 of [4] by Lemma 3 (5) of [2], and to notice that locally σ-hypercentral groups are σ-hypercentral (so in this case R 1 = H σ , and so on). The above argument also yields that H σ K σ = K σ H σ : although the σ-residuals are not in general perfect, we may replace the theorem of Wielandt by Corollary 2.4.…”

“…. 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 σ j i ∈ σ. Skiba showed that the set of all σ-subnormal subgroups has a strong influence on the structure of a finite soluble group and this led many authors to investigate which of the most relevant theorems about subnormal subgroups have analogs in terms of σ-subnormal subgroups (see for instance [1], [2], [3]). It turns out that one of the main features of σ-subnormal subgroups (in finite groups) is that the join of two σ-subnormal subgroups is always σ-subnormal (see for instance [3]), so they form a sublattice of the lattice of all subgroups.…”