We look at functions $f: \text{FinGrp} \rightarrow \mathbb{N}$ such that $f$ is constant on isomorphism classes.

Let's say that $f$ is representable if there is a (possibly infinite) group $K$ such that $f(G)=\left | Hom(K,G) \right|$ for any finite group $G$.

Let's also say that $f$ is semirepresentable if it is the (pointwise) quotient of representable functions.

Note that both classes of representable functions and semirepresentable functions are closed by pointwise multiplication.

My question is, given a function $f$, when we can say that it is representable? or semirepresentable?

Here are some clearly necessary assumptions:

  1. If $f$ is semirepresentable, then for any $p$ prime and $k\geq1$ natural, it happens that $f(C_{p^k})=p^l$ for some $l$.
  2. If $f$ is representable, then $G \subset H$ implies $f(G)\leq f(H)$
  3. If $f$ is semirepresentable, then $f(G \times H)=f(G)f(H)$.

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