I'm stuck on part (b)
(a) Prove that if and are elements of an Abelian group , with and , then ( denotes the identity element in my text). Indicate where you use the condition that is Abelian.
(b) With and as in part (a), prove that divides .
(c) Give an example of an Abelian group and elements and in such that . Compare part (b)
Things that might come in Handy:
Let be a group and .
We define the order of , denoted , to be the smallest, positive integer such that . If no such integer exists, we say has infinite order.
of course means operating on itself times. that is,
Tell me if you guys need anymore information.
What I have tried:
(a) Proof: Let be as in part (a). Since is Abelian,
(b) This is where I'm stuck. Here's what I did so far.
Let be as in part (a). And let .
Then are all the smallest, positive integers such that: , and . We want to show that , that is, for some .
that's it, I'm not sure where to go from there. I've tried several things, but they all end up doing nothing.
(c) My example was with and
Thanks guys and gals