order of productsOrderOfProducts2009-05-25 16:26:532009-05-25 20:50:23Theoremorder (of a group)% this is the default PlanetMath preamble. as your knowledge
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If $a$ and $b$ are elements of a group, then both $ab$ and $ba$ have always the same order.\\
\emph{Proof.}\, Let $e$ be the indentity element of the group.\, For\, $n > 1$,\, we have the
\PMlinkname{equivalent}{Equivalent3} conditions
$$e \;=\; (ab)^n \;=\; \underbrace{(ab)(ab)\cdots(ab)}_{n} \;=\; a(ba)^{n-1}b,$$
$$a^{-1}b^{-1} \;=\; (ba)^{n-1},$$
$$(ba)^{-1} \;=\; (ba)^{n-1},$$
$$e \;=\; (ba)^n.$$
As for the infinite order, it makes the conditions false.\\
\textbf{Note.}\, More generally, all elements of any conjugacy class have the same order.