Subgroups Inclusions in 3-Factors Direct Product

Abstract

The aim of this paper is to use a correspondent theorem to characterize containment of a degenerate $2$-factor injective subdirect products. Namely, let $\Omega,\Lambda$ be degenerate 2-factor injective subdirect products of $ M_{1}\times M_{2}\times M_{3}$, we provide necessary and sufficient conditions for $\Omega\leq \Lambda.$ Based on a decomposition of the inclusion order on the subgroup lattice of a subdirect product as a relation product of three smaller partial orders, we induce a matrix product of three incidence matrices.