On subgroups and equivalent relations

• Main Authors: Baysic, Celestina, Chen Cen Sio, Dennis Tan
• Format: text
• Language: English
• Published: Animo Repository 1998
• Subjects: Group actions (Mathematics); Algebraic varieties; Topological transformation groups; Equivalence relations (Set theory)
• Online Access: https://animorepository.dlsu.edu.ph/etd_bachelors/16496
• Institution: De La Salle University

This paper is an exposition of "Subgroups and Equivalence Relations", by Pierre J. Malraison, Jr. (1977). It contains key results detailing the relationship between subgroups and equivalence relations. The results were obtained using the concepts of group action and commutator groups. 1) If G is a group and R is an equivalence relation on G, the following are equivalent: (i) R is closed under products in G x G; (ii) R is a subgroup of G x G; (iii) R is the relation of belonging to the same coset of some normal subgroup of G. 2) If the equivalence relation R is a subgroup of G x G, then the following are equivalent: i) R is normal in G x G. ii) G/[e] is abelian. iii) G' is a subgroup of [e].