gistfile1.txt

Let a, b ∈ G, and let b be a right inverse of a, i.e. a*b = e.

Then we can show that b is also a left inverse of a:

b*a
=b*a*((b*a)*(b*a)^-1)
=((b*a)*(b*a))*(b*a)^-1 (by associativity)
=(b*(a*b)*a))*(b*a)^-1 (by associativity)
=(b*a)*(b*a)^-1 (by assumption)
=e

Note that we don't even require G to be Abelian for this proof to work.