Can two groups be the same if their digraphs/cayley graphs) are different?

Question

Are the groups $\Bbb Z /4\Bbb Z$ of integers$\mod 4$ (addition) and the set of all iterated compositions of $a$ and $b$ where $a,b$: $\Bbb R^2 → \Bbb R^2$ be defined by $a(x,y)=(-x,y)$ and $b(x,y)=(x,-y)$ for all $(x,y)$ in $\Bbb R^2$ the same? I created digraphs for both groups and they are different. I believe that means they are not the same but I don't know why?

Answer 1

First thing first. The group you are refering to is isomorphic to $\mathbb{Z}_2\oplus\mathbb{Z}_2$. The isomorphism is given by $a\mapsto (1,0)$, $b\mapsto (0, 1)$. And $\mathbb{Z}_2\oplus\mathbb{Z}_2$ is not isomorphic to $\mathbb{Z}_4$.

As for Cayley graphs: the group $G$ can be recovered uniquely from the Cayley graph $\Gamma(G, S)$. See wiki: https://en.wikipedia.org/wiki/Cayley_graph#Characterization

Thus different groups cannot give "the same" (i.e. isomorphic) Cayley graph.