order of products

If $a$ and $b$ are elements of a group, then both $ab$ and $ba$ have always the same order.

Proof.  Let $e$ be the indentity element of the group.  For  $n>1$,  we have theequivalent (http://planetmath.org/Equivalent3) conditions

$$e={(ab)}^n=\underset{n}{\underbrace{(ab)(ab)\mathrm{\cdots }(ab)}}=a{(ba)}^{n-1}b,$$

$$a^{-1}{b}^{-1}={(ba)}^{n-1},$$

$${(ba)}^{-1}={(ba)}^{n-1},$$

$$e={(ba)}^n.$$

As for the infinite order, it makes the conditions false.

Note.  More generally, all elements of any conjugacy class have the same order.