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.