Bulletin of the
Korean Mathematical Society BKMS

We consider a congruence $\rho$ on a group $G$ as a subsemigroup of the direct product $G\times G$. It is well known that a relation $\rho$ on $G$ is a congruence if and only if there exists a normal subgroup $N$ of $G$ such that $\rho =\left\{(s,t):st^{-1}\in N \right\}$. 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 $\rho=\left\{(s,t):st^{-1}\in N \right\}$ on $G$ is finitely presented.

Keywords: congruence, normal subgroup, semigroup presentation

MSC numbers: 20M05