TY - JOUR ID - 165 TI - SOLVABILITY OF FREE PRODUCTS, CAYLEY GRAPHS AND COMPLEXES JO - Journal of Algebraic Systems JA - JAS LA - en SN - 2345-5128 AU - Mirebrahimi, Hanieh AU - Ghanei, Fatemeh AD - Department of pure Mathematics, Ferdowsi University of Mashhad, P.O.Box 1159-91775 Mashhad, Iran Y1 - 2013 PY - 2013 VL - 1 IS - 1 SP - 45 EP - 52 KW - simplicial complex KW - fundamental group KW - covering space KW - Caley graph KW - solvable group DO - 10.22044/jas.2013.165 N2 - 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 $m,n\geq2$ UR - https://jas.shahroodut.ac.ir/article_165.html L1 - https://jas.shahroodut.ac.ir/article_165_76e847bc1b83709351833bc141c00f5a.pdf ER -