Do equivalent Cayley Tables imply isomorphism?

Question

It works for $\Bbb{Z}_2$ under addition and $\{1,-1\}$ under multiplication, but I can't think of any counterexamples. I would think that it's enough, but are there exceptions to this?

Answer 1

Suppose the two Cayley tables are identical in the sense that whenever the first table has a symbol $a$ the second table has the symbol $a'$ (and conversely). Then the map $\phi: a \mapsto a'$ is a group isomorphism. For it is clear $\phi$ is bijective. And $\phi$ respects the group operation because if $ab=c$ in the first table, then $a'b'=c'$ in the second table, which can be rewritten as $\phi(a) \phi(b) = \phi(c)$.