A general strategy to find isomorphisms using Cayley tables

Question

Suppose I have two finite groups, then how I can find an isomorphism between them. The reason I am asking is whenever I encounter a problem that requires this, the best I can do is guess and "hope" that I got the right answer. Consider, for example the group $(Z_4, +_4$) and the group $(U_{10}, \times_{10})$ then the Cayley tables of the groups are as below:

I also listed the mapping $\phi$ that is an isomorphism from the first group to the second. However, I really don't understand how would one find such mapping without "guessing", what would be the general strategy to obtain this result?

Answer 1

Isomorphisms must preserve element orders. In particular, they must map identity element to identity element, so in your example $0\mapsto1$. In your example, one element in each group has order $2$ ($2$ and $9$, respectively), so $2\mapsto9$. That leaves $1,3$ mapping to $3, 7$, and in fact either $1\mapsto3$ and $3\mapsto7$ or $1\mapsto7$ and $3\mapsto3$ could be an isomorphism.

Answer 2

The best strategy I can think of is by making observations about structure, for example you know the identity has to map to the identity, similarly elements of order n have to map elements of order n. Similar observations allow you to build up an isomorphism piece by piece.