Cayley Tables and Isomorphisms

Question

What exactly is the technique of seeing if there is a one-to-one correspondence between the sets of elements of two groups on their Cayley Tables? For instance, if the Cayley Table for group $\mathbb{Z}_8^* = \{1, 3, 5, 7\}$ (all elements that have a multiplicative inverse in $\mathbb{Z}_8$) and $\mathbb{Z}_{10}^* = \{1, 3, 7, 9\}$ "look different," what does that exactly imply? I'm just confused about the connection between Cayley Tables and isomorphisms.

Answer 1

We can just note that these are different groups. In $\mathbb{Z}^*_8$ each element has order two: $$\begin{cases}1^2\equiv_8 1\\3^2\equiv_8 1\\5^2\equiv_8 1\\7^2\equiv_8 1\\\end{cases}$$

Whereas in $\mathbb{Z}^*_{10}$ this is not the case: $$\begin{cases}1^2\equiv_{10} 1\\3^2\equiv_{10} 9\\7^2\equiv_{10} 9\\9^2\equiv_{10} 1\\\end{cases}$$

So these are two different groups. Since every element of $\mathbb{Z}^*_8$ has order two (besides $1$ which has order one), $\mathbb{Z}^*_8 \cong\mathbb{Z}_2\times \mathbb{Z}_2$. And since this is not the case in $\mathbb{Z}^*_{10}$, we must have that it is isomorphic to the other group of order four, $\mathbb{Z}_4$.