# On group varieties and numbers

Suppose $$\mathfrak{U}$$ is a group variety. Let’s define $$N_{\mathfrak{U}} \subset \mathbb{N}$$ as a such set of numbers, that for any finite group $$G$$, $$|G| \in N_{\mathfrak{U}}$$ implies $$G \in \mathfrak{U}$$.

Examples:

If $$\mathfrak{O}$$ is the variety of all groups, then $$N_{\mathfrak{O}} = \mathbb{N}$$.

If $$\mathfrak{B}_m$$ is the variety of all groups of exponent $$m$$, then $$N_{\mathfrak{B}_m}$$ is the set of all divisors of $$m$$

If $$\mathfrak{N}_c$$ is the variety of all groups of nilpotency class $$c$$, then $$N_{\mathfrak{N}_c}$$ is the set of all numbers $$n=p_1^{e_1}\cdots p_m^{e_m}$$ with $$p_i^k\not\equiv 1(\mod p_j)$$ for $$i,j\in\{1,\ldots,m\}$$ and $$1\leqslant k\leqslant e_i$$, and $$e_i \leq c + 1$$ for $$i\in\{1,\ldots,m\}$$.

If $$\mathfrak{U}$$ and $$\mathfrak{V}$$ are two varieties, then $$N_{\mathfrak{U}\cap\mathfrak{V}} = N_{\mathfrak{U}} \cap N_{\mathfrak{V}}$$

My question is:

Does there exist some number-theoretic characterisation of all such subsets $$N \subset \mathbb{N}$$, such that $$N = N_{\mathfrak{U}}$$ for some variety $$\mathfrak{U}$$?

Any $$N_{\mathfrak{U}}$$ satisfies the property:

If $$a \in N_{\mathfrak{U}}$$ and $$b | a$$, then $$b \in N_{\mathfrak{U}}$$

Suppose $$|G| = b$$ and $$G \notin \mathfrak{U}$$. Then $$G \times C_{\frac{a}{b}} \notin \mathfrak{U}$$.

If $$\exists n \in \mathbb{N}$$, such that $$\forall k \in \mathbb{N}$$ $$n^k \in N_{\mathfrak{U}}$$, then $$\mathfrak{U} = \mathfrak{O}$$.

By previous lemma, we can assume without loss of generality, that $$n = p$$ is prime. The only variety, that contains all $$p$$-groups for a fixed prime $$p$$ is $$\mathfrak{O}$$

However, I am not sure, whether those two conditions are sufficient to characterise all such sets or not.

This question was inspired by this MO question