# 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 the equivalent (http://planetmath.org/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.

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

Title order of products OrderOfProducts 2013-03-22 18:56:43 2013-03-22 18:56:43 pahio (2872) pahio (2872) 5 pahio (2872) Theorem msc 20A05 InverseFormingInProportionToGroupOperation