1. The problem statement, all variables and given/known data Prove that the group of order 175 is abelian. 2. Relevant equations 3. The attempt at a solution |G|=175=527. Using the Sylow theorems it can be determined that G has only one Sylow 2-subgroup of order 25 called it H and only one Sylow 7-subgroups called it K. Thus, H and K are normal subgroups of G and G=H x K which is isomorphic to the direct product of H and K. Since |H|=52, then H is Abelian. Since K is of prime order then K is cyclic and therefore also Abelian. I am not sure whether I can now conclude that G must be abelian since it is the external (or direct) product of abelian subgroups.