Homomorphic Relations and Goursat Lemma
Open Access
- 20 June 2022
- journal article
- Published by Earthline Publishers in Earthline Journal of Mathematical Sciences
- Vol. 10 (1), 169-181
- https://doi.org/10.34198/ejms.10122.169181
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.
Keywords
This publication has 11 references indexed in Scilit:
- Decomposition of Goursat Matrices and Subgroups of Zm x ZnEarthline Journal of Mathematical Sciences, 2021
- Anticommutativity and n-schemesEarthline Journal of Mathematical Sciences, 2021
- Triangular Scheme Revisited in the Light of n-permutable CategoriesEarthline Journal of Mathematical Sciences, 2021
- Subgroups of Finite Abelian Groups Having Rank Two via Goursat’s LemmaTatra Mountains Mathematical Publications, 2014
- Subgroups of direct products of groups, ideals and subrings of direct products of rings, and Goursat’s lemmaContemporary Mathematics, 2009
- A GENERALIZATION OF HOMOLOGICAL ALGEBRAJournal of the Korean Mathematical Society, 2002
- On the ubiquity of Mal′cev operationsContemporary Mathematics, 1992
- A Course in Universal AlgebraPublished by Springer Science and Business Media LLC ,1981
- Goursats Theorem and the Zassenhaus LemmaCanadian Journal of Mathematics, 1958
- Sur les substitutions orthogonales et les divisions régulières de l'espaceAnnales Scientifiques de lʼÉcole Normale Supérieure, 1889