1. The problem statement, all variables and given/known data My question is about the rules behind the method of finding the solution not necessarily the method itself. (I am prepping for the GRE subject) How many mutually nonisomorphic Abelian groups of order 50? 3. The attempt at a solution so, if understand this correctly, we have 2 of these groups: 1) Z2 + Z5 + Z5 2) Z2 + Z25 However, what I don't understand is why these are mutually nonisomorphic. The theorems presented before this problem state: The direct sum Zm + Zn is cyclic iff gcd(m,n) = 1. If this is the case, then, since Zm + Zn has order mn, Zm +Zn is isomorphic to Zmn, So are groups 1 and 2 both isomorphic to Z50 but not isomorphic to each other? which dosen't make sense to me.