Let's say a cardinal-valued function $r$ of cardinal numbers (which is eventually zero) is conjugal if there exists a group $G$ in which the number of conjugacy classes of order $\alpha$ is given by $r(\alpha)$ for each cardinal number $\alpha$. In the case that $r$ is strictly bounded by $\aleph_0$ and has finite support, $G$ must be finite and for each natural $n$, $r(n)>0$ only if $n$ divides $|G|=\sum_k r(k)$. Otherwise, what do we know about when such functions may be conjugal or not? And what may be said about explicit constructions of a $G$ associated to various sorts of $r$ through, say, extensions and limits?
What sort of conjugacy class data is realizable?
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abstract-algebra
group-theory