(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove that in any group the orders of [tex]ab[/tex] and [tex]ba[/tex] equal.

2. Relevant equations

n/a

3. The attempt at a solution

Let [tex](ab)^{x} = 1.[/tex]

Using associativity, we get

[tex](ab)^{x} = a(ba)^{x-1}b = 1.[/tex]

Because of the existence of inverses--namely [tex]a^{-1}[/tex] and [tex]b^{-1}[/tex]--this implies

[tex](ba)^{x-1} = a^{-1}b^{-1} = (ba)^{-1}.[/tex]

Multiplying both sides by [tex](ba) = ((ba)^{-1})^{-1}[/tex] yields

[tex](ba)^{x} = 1.[/tex]

So,

[tex](ab)^{x} = (ba)^{x} = 1[/tex],

and the orders [tex]ab[/tex] and [tex]ba[/tex] are the same.

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# Order of group elements ab and ba

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