# Suppose G  is a group where all non-identity elements are of order 2. Is G abelian ?

Let $a , b \in G$
$a b = {b}^{2} a b {a}^{2} = \left(b b\right) a b \left(a a\right) = b \left(b a b a\right) a = b {\left(b a\right)}^{2} a = b a$