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Show every finite group of even order contains an element of order 2.

I have tried using Lagrange's theorem, but I am unsure if this is the right path.

Tom Gatward
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    Lagrange's Theorem does not tell you that if $n|N$, then there is an element of order $n$, where $|G| = N$. (ie. there is no element of order $6$ in $A_4$) – Matthew Levy Dec 24 '14 at 07:13

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Hint:

Every element of a group has an inverse. So given such a group $G$, try pairing every element with its inverse. What happens? What must therefore be the case?

Side-note: This is a special case of Cauchy's theorem.

Kaj Hansen
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