$F:G\to B$ is an isomorphism between groups implies $F^{-1}$ is an isomorphism $F$ is an isomorphism from group $G$ onto group $B$. Prove $F^{-1}$ is an isomorphism from $B$ onto $G$.
I do not know which direction to go in.

Because $F$ is a bijection, $F^{-1}$ is a bijection as well. Now we know
$$F(a\cdot b)=F(a)\cdot F(b)$$
This means that 
$$F^{-1}(a\cdot b)=F^{-1}(F^{-1}(F(a\cdot b))) = F^{-1}(F^{-1}(F(a)))\cdot F^{-1}(F^{-1}(F(b))) = F^{-1}(a)\cdot F^{-1}(b)$$
Thus $F^{-1}$ is an isomorphism from $B$ to $G$.
