It's still not obvious to me (although I believe it) why all representations of finite groups are defined over cyclotomic fields. Could you sketch that argument?

In other contexts, maybe, but I don't see what this has to do with the properties of, say, Z/pZ.

You mean eventually zero. Now you know that the residue mod 5^m and the residue mod 2^m uniquely determines the residue mod 10^m by CRT, and you know that one is periodic and the other is eventually constant. What can you conclude?

Yes, any particular subgroup.

You know that the powers of two have a certain period mod 5^m. What is their period mod 2^m?

No; all you know that it divides 30.

A more succinct way to put this is that Z/2Z doesn't have any non-identity elements; in other words, 2 is "degenerate." But for some reason I don't find this very satisfying.

Yes, that's more or less what I was trying to say in my parenthetical comment. But I can't decide whether this is deep or whether it's just because historically mathematicians happen to like additive inverses.

I'm told that Kummer actually didn't care about Fermat's last theorem; it just happened that the techniques he developed were applicable.

It's exactly the Chinese Remainder Theorem.

Right. The techniques used to make GIMPS successful could also be applied to other distributed problems, like Folding@home.

I'm not sure the premise of this question is valid any more than the core question of group theory is to figure out whether an isomorphism exists between two groups.

Okay. What if we also require that the group of automorphisms acts transitively on each rank?