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$
Best Answer
Interestingly, there are a few groups where any bijective map from the group to itself which preserves the identity element are automorphisms (ie, isomorphisms of the group with itself).
These groups are precisely $\{1\}$, $\mathbb{Z}/2\mathbb{Z}$, $\mathbb{Z}/3\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$.
It is easy to check that these groups satisfy the above property, and that these are the only ones follows from my answer at Is a Bijection From a Group to Itself Automatically an Isomorphism If It Maps the Identity to Itself?