Semigroup presentations for congruences on groups

Abstract

We consider a congruence ? on a group G as a subsemigroup of the direct product G × G. It is well known that a relation ? on G is a congruence if and only if there exists a normal subgroup N of G such that ? = {(s, t): st-1 {small element of} N}. In this paper we prove that if G is a finitely presented group, and if N is a normal subgroup of G with finite index, then the congruence ? = {(s, t): st-1 ? N} on G is finitely presented. © 2013 The Korean Mathematical Society