The question: Prove that each group of order 4 is isomorphic to Z/4Z or the Klein Group: (Z/2Z)x(Z/2Z). Attempt at solution: basically I think that a group of order 4 has e,a,b,c then this group can be characterise by the ordering 0,1,2,3 in the group Z/4Z or (0,0),(0,1),(1,0),(1,1) where addition on Klein Group is defined as (x,y)+(z,w)=(x+z,y+w), well it seems like a really easy question, I don't believe that I am asking this. (-: What I mean 0<->e, 1<->a, 2<->b,3<->c or (0,0)<->e, (0,1)<->a, (1,0)<->b, (1,1)<->c. ofcourse in one group 3+1=0 then ca=e but in the other ca=b. The real question is how do I show there aren't more options for this structure, I mean why only these two? Thanks in advance.