TITLE: Is a bijective map between two groups always an isomorphism?
QUESTION [5 upvotes]: Does a bijective map between two groups always produce an isomorphism?
I am trying to find a bijective map between two groups which does not preserve the group operations.
I have found a bijection $f(g)=g^3$, where $f:\mathbb{R}\rightarrow\mathbb{R}$, 
defined by $f(g+h)=(g+h)^3$. 
But $f(g + h) \neq     $
$ f(g)+f(h) \\=g^3+h^3$

REPLY [2 votes]: A maybe more "natural" example is the following map: if $G$ is not trivial, pick $g\in G,g\neq 1_G$.
Then the map $$\ell_g:x\in G\mapsto gx\in G$$ is bijective (with inverse $\ell_{g^{-1}}$), but not a group homomorphism, since $g=f(1_G)=f(1_G ^2)\neq f(1_G)^2=g^2$ (otherwise, by simplification by $g$, we would have $g=1_G$)