If the group (G, ౦) is communitative show that, (a ౦ b)-1 = a-1 ౦ b-1, for all a, b ∈ G.
Let a-1 and b-1 be the inverse elements of a and b in the communitative group (G, ) with the identity element e, then,
a a-1 = e = a-1 a
b b-1 = e = b-1 b
(a b) (a-1 b-1) = (a b) (b-1 a-1)
= a (b b-1) a-1
= a e a-1
= e (a a-1)
= e e
= e
Similarly, (a-1 b-1) (a b ) = e
Hence, (a b)-1 = a-1 b-1