Is it possible to draw a homomorphism between all groups of the same finite order

Question

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 $\phi(n)=m$

Answer 1

There exist a homomorphism between two any group (at least the trivial one), but you will not have a priori an isomorphism between two group of same cardinal. For example, $\mathbb Z/6\mathbb Z$ and $\mathfrak S_3$ has same cardinal but they are not isomorphe.

Answer 2

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.