Finding all groups of order $7$ up to isomorphism? I'm learning group theory but I didn't learn any concepts of building groups.
I know that there exists the identity group $\{e\}$, and the group with 2 elements: $\{e,a\}$.
If I try to create a group with 3 elements, let's say: $\{e,a,b\}$ then we would have:
$ea = a, eb = b, aa = ?$, and what about $ab$?
Am I supposed to try this for the $7$ elements?
What does the statement ``up to isomorphism" mean? In particular, how many groups of order $7$ are there up to isomorphism?
I'm really confused, and my book says nothing about it.

Up to isomorphism, there is only one group of prime order $p$.
Any group of prime order $p$ isomorphic to $C_p$: the cyclic group of order $p$.
