%0 Journal Article %T SOLVABILITY OF FREE PRODUCTS, CAYLEY GRAPHS AND COMPLEXES %J Journal of Algebraic Systems %I Shahrood University of Technology %Z 2345-5128 %A Mirebrahimi, Hanieh %A Ghanei, Fatemeh %D 2013 %\ 09/15/2013 %V 1 %N 1 %P 45-52 %! SOLVABILITY OF FREE PRODUCTS, CAYLEY GRAPHS AND COMPLEXES %K simplicial complex %K fundamental group %K covering space %K Caley graph %K solvable group %R 10.22044/jas.2013.165 %X 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 $m,n\geq2$ %U https://jas.shahroodut.ac.ir/article_165_76e847bc1b83709351833bc141c00f5a.pdf