Affiliation:
1. Laboratory of Algebra, Geometry and Application, University of Yaoundé I, P.O. Box : 812, Yaoundé, Cameroon
Abstract
Over the past years various authors have investigated the famous elementary result in group theory called Goursat's lemma for characterizing the subgroups of the direct product $A\times B$ of two groups $A,B$. Given a homomorphic relation $\rho = (R,A,B)$ where $A$ and $B$ are groups and $R$ is a subgroup of $A\times B.$ What can one say about the structure of $\rho$. In 1950 Riguet proved a theorem that allows us to obtain a characterization of $\rho$ induces by examining the sections of the direct factors. The purpose of this paper is two-fold. A first and more concrete aim is to provide a containment relation property between homomorphic relation. Indeed if $\rho,\sigma$ are homomorphic relations, we provide necessary and sufficient conditions for $\sigma\leq\rho$. A second and more abstract aim is to introduce a generalization of some notions in homological algebra. We define the concepts of $\theta$-exact. We also obtain some interesting results. We use these results to find a generalization of Lambek Lemma.
Cited by
1 articles.
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1. On the extensions of Goursat’s theorem to direct products of n R-algebras;THE 3RD INTERNATIONAL CONFERENCE ON MATHEMATICS AND ITS APPLICATIONS (ICOMATHAPP) 2022: The Latest Trends and Opportunities of Research on Mathematics and Mathematics Education;2024