5
$\begingroup$

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 $f(G \times H)=f(G)f(H)$.
$\endgroup$

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.