Originally Posted by

**Zorae** a) Prove that if a and b are elements of an Abelian group G, with o(a) = m and o(b) = n, then $\displaystyle (ab)^{mn}= e$ .

b) With G, a, and b as in part (a), prove that o(ab) divides o(a)o(b).

c) Give an example of an Abelian group G and elements a and b in G such that o(ab) $\displaystyle \neq\$o(a)o(b). Compare part (b)

I suspect I could figure out, or at least know where to start, part b if I knew part a (If t is an integer, then $\displaystyle a^t=e$ iff n is a divisor of t)

But, I just don't know what to do with part a...

From what I understand, $\displaystyle a^m=e$ and $\displaystyle b^n=e$

I have done all sorts of random inserts of $\displaystyle a^m$, $\displaystyle a^m * a^-m$ , etc. Nothing seems to give me what I need though... erm, I was wondering, since $\displaystyle a^m=e$ does $\displaystyle a^{-m}=e$?