Relation between finite topological spaces and finitely presentable groups

In this paper it is shown how to construct a finite topological space \(X\) for a given finitely presentable group \(G\) such that \(\pi_1(X)\cong G\). Our construction is not optimal in the sense that the cardinality of the space \(X\) might not be the smallest possible. Our main result applies to a large class of interesting groups, including all finite groups.