Let G {\displaystyle G} be a group and let a , b ∈ G {\displaystyle a,b\in G} . Then ( a ∗ b ) ∗ ( b − 1 ∗ a − 1 ) = a ∗ ( b ∗ b − 1 ) ∗ a − 1 = a ∗ e ∗ a − 1 = a ∗ a − 1 = e {\displaystyle (a*b)*(b^{-1}*a^{-1})=a*(b*b^{-1})*a^{-1}=a*e*a^{-1}=a*a^{-1}=e} . Also ( b − 1 ∗ a − 1 ) ∗ ( a ∗ b ) = b − 1 ∗ ( a − 1 ∗ a ) ∗ b = b − 1 ∗ e ∗ b = b − 1 ∗ b = e {\displaystyle (b^{-1}*a^{-1})*(a*b)=b^{-1}*(a^{-1}*a)*b=b^{-1}*e*b=b^{-1}*b=e} . Thus ( a b ) − 1 = b − 1 a − 1 {\displaystyle (ab)^{-1}=b^{-1}a^{-1}} by definition of inverse.
