On π-Solvable and Locally π-Solvable Groups with Factorization

Abstract

We prove that, in a locally π-solvable group G = AB with locally normal subgroups A and B, there exist pairwise-permutable Sylow π′- and p-subgroups A _π′, A _p and B _π′, B _p , p ∈ π, of the subgroups A and B, respectively, such that A _π′ B _π′ is a Sylow π′-subgroup of the group G and, for an arbitrary nonempty set σ $\subseteq$ π,

$$\left( {\prod\nolimits_{p \in {\sigma }} {A_p } } \right)\left( {\prod\nolimits_{p \in {\sigma }} {B_p } } \right)\quad {and}\quad \left( {A_{{\pi }\prime } \prod\nolimits_{p \in {\sigma }} {A_p } } \right)\left( {B_{{\pi }\prime } \prod\nolimits_{p \in {\sigma }} {B_p } } \right)$$

are Sylow σ- and π′ ∪ σ-subgroups, respectively, of the group G.