This question is a softer question: I'd like some reference for any information on the following if possible.
Say $0 \notin \mathbb{N}$. Define (for convenience of expressing statements in this question) the function $\mathcal{G}: \mathbb{N} \to \mathbb{N}$ where $\mathcal{G}(n)$ is the number of groups of order $n$ up to isomorphism, for example, for $p$ a prime, $\mathcal{G}(p)=1$, $\mathcal{G}(p^2)=2$, $\mathcal{G}(p^3)= 5$.
Some results about the values of $\mathcal{G}$ for particular natural numbers are not currently known, for example the value of $\mathcal{G}(2048)$.
But, can we say anything more generally about the possible values of $\mathcal{G}(n)$? That is,
- Do we know that any natural number can be the number of groups up to isomorphism of some natural number, that is, $\mathcal{G}(\mathbb{N}) = \mathbb{N}$? If so, are all values taken on an infinite number of times? (Of course some are, as given in the examples with prime powers)
- Or do we know the inverse, that $\exists n \in \mathbb{N}$ such that $n \notin \mathcal{G}(\mathbb{N})$? If so, do we know such an $n$? Indeed, the smallest $n$? Are there any conditions on $n$ that classify when we can say for certain that $n \in \mathcal{G}(\mathbb{N})$ or $n \notin \mathcal{G}(\mathbb{N})$? Have we completely classified this for all $n \in \mathbb{N}$ or are some unknown?
Or is nothing here known?
Note a similar question (asked here) for abelian groups is well understood since the number of abelian groups can be worked out using partitions. But non-abelian groups aren't nearly as easy to handle.
