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.

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- Oct 22nd 2009, 07:11 PMux0usng a conjugate
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.

- Oct 22nd 2009, 08:02 PMtonio
- Oct 23rd 2009, 04:59 AMux0
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 - Oct 23rd 2009, 08:49 AMtonio

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 - Oct 23rd 2009, 11:03 AMux0
Ya I kind of did, and thanks for the notation correction!