I have this homework problem. It states:
By constructing the possible Cayley tables, list all groups where the underlying set has at most 4 elements, up to isomorphisms. That is, no two groups in your list should be isomorphic.
I understand the basic use of Cayley tables in the sense that they are used to see what the operation on two of the elements becomes. One group that I know that would fit this would be $\mathbb Z_4$. I'm not sure how to list all of them and if there is a way to know if it will be isomorphic or not.
Either such a group has an element of order $4$, and it is cyclic, isomorphic to $\mathbf Z/4\mathbf Z$. Or the maximal order of its elements is $2$, by Lagrange's theorem. Let $a$ be such an element,; the group must have another one, say $b$, since it has order $4$. Note we have $\langle a\rangle\cap\langle b\rangle=\{e\}$ and $ab=ba$, otherwise the group would have a fifth element.
Hence the group is commutative, made up of $\{e, a, b, ab\}$ and it is the direct sum $\langle a\rangle\oplus\langle b\rangle$, isomorphic to $\mathbf Z/2\mathbf Z\times\mathbf Z/2\mathbf Z$ (Klein's Vierergruppe).