Groups G such that g^2 = e for all g in G.

If G is a group and g^2 = e for all g in G then every element is it's own inverse.

Hence, xy = (xy)^-1 = y^-1x^-1 = yx so G is abelian.

This basically means, anytime we have a group such that g^2 = e, we know G is abelian.