Hello All,

I'm stuck on part (b)

Problem:

(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,

Now,

QED

(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 .

Now,

...

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