Document Type : Research Note


Department of pure Mathematics, Ferdowsi University of Mashhad, P.O.Box 1159-91775 Mashhad, Iran


In this paper, we verify the solvability of free product of finite cyclic groups with topological methods. We use Cayley graphs and Everitt methods to construct suitable 2-complexes corresponding to the presentations of groups and their commutator subgroups. In particular, using these methods, we prove that the commutator subgroup of ${Z}_{m}*{Z}_{n}$ is free of rank $(m-1)(n-1)$ for all $m,n\geq2$