# Homework Help: Group theory question

1. Oct 4, 2011

### eddyski3

1. The problem statement, all variables and given/known data

If a and b are in a group, show that if (ab)^n=e then (ba)^n=e.

2. Relevant equations

3. The attempt at a solution

I'm not sure how one would prove this. The question is obviously for non-abelian groups.

2. Oct 4, 2011

### Dick

If (ab)^2=e then (ab)(ab)=e. So b(ab)(ab)a=bea=ba. Now use associativity and 'cancellation'. Do you see how to do the same trick for (ab)^n?

3. Oct 4, 2011

### eddyski3

I'm not sure how this helps us show that (ba)^2=e? When generalizing to (ab)^n I see we'll get a similar result but I'm not sure how this shows that (ba)^n=e.

4. Oct 4, 2011

### Dick

b(ab)(ab)a=ba, yes? That's the same as (ba)(ba)(ba)=(ba). Do you see it now?

5. Oct 4, 2011

### Bacle

abab=e , so ab=b-1a-1

ababab=e , so baba=a-1b-1

Play around with these until you can figure out one, then , if you don't have a general
argument for all n, maybe induction on n will help.

6. Oct 4, 2011

### eddyski3

Oh, ok. Now I understand the argument. Thank you.