# Order of an element in a group

Hey guys!

I'm having some trouble trying to solve this question.. Any advice/help is appreciated!

## Homework Statement

Suppose that a and b belong to a group such that:
|a| = 4, |b| = 2, and suppose a3b=ba
Find the order of ab.

## The Attempt at a Solution

So I am unsure of which theorems I should be looking at... but anyways...... my horrible attempt:
so a,b are in G.
|ab| = (ab)n = 1

Let n = order(ab), anbn = 1 --> an=b-n

I don't know if I should continue from this point because I don't know how exactly I would go about finding the order.

I think |ab| = |ba|, yes or no?
If it does then I'm guessing I can use that to find the order of a3b?
Any hints or suggestions on how?

Thank you.

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Dick
Homework Helper
(ab)^n and a^n*b^n are not the same thing. Clearly your group isn't abelian. Hint: (ab)(ab)=a(ba)b.

So is this a matter of manipulating a and b ?

For your hint, I'm just wondering why is it (ab)(ab)?
I see that a(ba)b = a^4b^2, which we have as the original orders, but why (ab)^2 and not some other number n?

Dick