Suppose I have $A_1, A_2, ..., A_n$ algebraic structures (groups, rings, modules, etc...). Naïvely, in order to prove that all of these structures are distinct, I need to show all pairs of these structures are nonisomorphic. But that requires ${n \choose 2}$ proofs which leads to a combinatorial explosion. In a research paper I might need to prove that a large number of algebraic structures are distinct, so this method doesn't scale.
Is there a way to prove this without combinatorial explosion?
Invariants
If you can find some object, which is invariant under isomorphism, and is pairwise distinct for your objects, you just proved that the structures are distinct.
Even if some of the invariants repeat, you can either find another invariant which distinguishes between those particular objects, or reduce (typically greatly) the number of proofs.