If G is a group and $\displaystyle a,b \in G$, prove that ab and ba have the same order. I think i need to use the conjugate to prove this, because conjugate elements have the same order.
its just that simple?
$\displaystyle ba=ba$
$\displaystyle ba = a^{-1}a(ba)$ ---------Multiplication of the identity
$\displaystyle ba=a^{-1}(ab)a$ ---------- Associativity
$\displaystyle <ba>=<ab>$ --------------- Conjugate elements have the same order
As painfully and embarrasingly simple as that. Why, did you expect something very difficult?
And what did you mean in your last line $\displaystyle <ba>=<ab>$? This is not true, not even close, in general. What you sure meant is that either $\displaystyle |ab|=|ba|\,\,\, or \,\,\,ord (ab)=ord (ba) $
Tonio