Is there a way to show that two groups are isomorphic by visual representation(Cayley diagram)?

Question

I got a question asking me to prove that $V_4$ and $C_2 \times C_2$ are isomorphic.

I can do this algebraically. However, I am curious if there is there a way to explain this using the diagram?

Thanks.

Edit: If the Cayley graph looks the same, are they isomorphic? And if not, then how? Also, I am stuck on how to draw $C_2 \times C_2$ Cayley graph.

Answer 1

Yes, if two groups have the same Cayley graph*, then they are isomorphic. But note that for a given group, by choosing different generating sets, you can produce non-isomorphic Cayley graphs.

For example, the Cayley graph of $\mathbb{Z},$ with the usual generateors $\{+1,-1\}$, is an infinite line with vertices at the integers. Notice there are no cycles. But if we look at the Cayley graph with respect to the generating set $\{2,3,-2,-3\}$, then you will get cycles; hence the graphs are note isomorphic. Nevertheless, if you zoom out and ignore the small scale structure, it looks a lot like the first graph. If this example is interesting, you can check out Wikipedia's article on Quasi-isometrys.

*Edit: Note the comment by user1729. Yes, the graph does need to be labeled. One way would be as in the definition found on Wikipedia.