For those of you with "A First Course in Abstract Algebra 7th ed.," I'm on chapter 15 and in particular, the first 12 questions. Basically, I understand very well how to classify any factor groups of finite groups, but it's these factor groups of infinite groups that just seem so arbitrarily classified. Here's a few examples that exemplify my problem: 8. (Z x Z x Z) / <(1,1,1)>. That is, Z x Z x Z "over" the subgroups generated by (1,1,1). Now the answer according to the book is that this factor group is isomorphic to Z x Z since every coset of the factor group contains a unique element of the form (0,m,n) where m and n are integers. This reasoning makes sense but then why can't I use the same reasoning and say that each coset also has a unique element of the form (0,0,m) or (m,n,q) thus making the factor group isomorphic to Z or Z x Z x Z respectively? 11. (Z x Z) / <(2,2)>. According to the book, this factor group is isomorphic to Z2 x Z. If I'm given the answer I can come up with a reason why it's true: the factor group contains an element of order 2 and an element of infinite order... but not only is this logic flimsy at best, I also would have no idea how to come up with the answer in the first place. 12. (Z x Z x Z) / <(3,3,3)> The answer here is Z3 x Z x Z with much of the same logic/problems as number 11. above. Any help is appreciated, these problems have been bugging me.