Suppose we are given two multiplication tables of two groups. one corresponds to $\mathbb{Z}_{15}$ and other corresponds to $ \mathbb{Z}_5 \times \mathbb{Z}_3$. I know that these two groups are isomorphic. Is it possible to find out which table corresponds to $\mathbb{Z}_{15}$ out of these two given tables?
If yes how otherwise state explicitly what other information is needed to differenitate these two table.
There is no way to differentiate the two groups, up to isomorphism.
Since $\gcd(5, 3) = 1$, we know that
$$\mathbb Z_{5} \times \mathbb Z_{3} \cong \mathbb Z_{15}$$
The set of elements are ordered pairs in $$\mathbb Z_5 \times \mathbb Z_3 = \{(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), \ldots (2, 0), \ldots (5, 0), (5, 1), (5, 2)\}$$ You'd need to make a list 15 elements along the column headers, and the same along the row headers, just as you get when making a table for $\mathbb Z_{15}$. The element $(0,0)$ is the identity, and recall that the operation on $\mathbb Z_{5} \times \mathbb Z_{3}$ is component-wise addition, modulo 5 for the first term, mod 3 for the second term.
For example: $(1, 3) + (2, 2) = (3_{\text{ mod }5}, 2_{\text{ mod }3}) = (3, 2).$
Every of the 15 elements in $\mathbb Z_5\times \mathbb Z_3$ can be mapped to a unique element in $\mathbb Z_{15}$. So the Cayley tables will be identical, save for the name of each element.
