Is it possible to draw a homomorphism between all groups of the same finite order For example, if I have $2$ groups of order $n$, then could I label their elements $1,2,...,n$ and $1,2,...,m$ and say that $\phi(n)=m$

Suppose $n$ is 2. We apply your procedure as follows:


*

*For the first group we label the identity as 1 and the other element as 2

*For the second group we label the identity as 2 and the other element as 1


Then $\phi$ is a well-defined function, but it's not a group homomorphism, since it doesn't map the identity to the identity.
