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

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

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

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

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

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

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

Title	order of products
Canonical name	OrderOfProducts
Date of creation	2013-03-22 18:56:43
Last modified on	2013-03-22 18:56:43
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	5
Author	pahio (2872)
Entry type	Theorem
Classification	msc 20A05
Related topic	InverseFormingInProportionToGroupOperation