Another element that squares to the identity?

[Genady]

Prove that groups of even order contain at least one element (which is not the identity) that squares to the identity. 

In case of a cyclic group, it is easy. Such group consists of {e, g, g², g³, ..., g^n-1} and contains element h = g^n/2. This element, h² = (g^n/2)² = gⁿ = e.

But how to prove it when the group is not cyclic?

P.S. Oh, got it. Just count the pairs, element and its inverse.