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A variety of groups is a class of groups satisfying a specified set of equations. Equivalently, it is a class of groups that is closed under homomorphic images, subgroups, and direct products. A typical example is the variety of groups satisfying $x^n = 1$, for a fixed $n$.

The 1967 paper Varieties of groups by Bernhard Neumann mentions that the number of varieties of groups is between $\aleph_0$ and $2^{\aleph_0}$, but that the precise number is not known. (The paper also mentions several other open problems about the lattice of varieties of groups.)

Question. Is the precise number of varieties of groups still unknown (as of 2024)? And if so, what partial results have been obtained so far?

Notice that the varieties of groups correspond to the fully invariant subgroups of the free group $F_{\omega}$ of rank $\omega$ (see Theorem 14.31 in Hanna Neumann's Varieties of groups), so the question is equivalent to enumerating these.

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    $\begingroup$ Actually for any universal algebra structure with finitely (or countably) many laws, it can't be strictly between $\aleph_0$ and $c$. For instance, for groups this is because the set of varieties is in natural bijection with the set of fully invariants subgroups of $F_\omega$, which is a closed subsets of the power set $2^{F_\omega}$. In general given the free object on countably many generators $F$, it ought to be in bijection with a suitable closed subset of $2^{F\times F}$. $\endgroup$
    – YCor
    Commented Oct 21 at 11:06

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The number of varieties of groups is $2^{\aleph_0}$ as shown by Olshanskii in

Ol’shanskiĭ, A. Yu., On the finite basis property of identities in groups, Izv. Akad. Nauk SSSR, Ser. Mat. 34, 376-384 (1970). ZBL0215.10504; English version, doi:10.1070/IM1970v004n02ABEH000911; Russian version.

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    $\begingroup$ An English version of the paper is available on Math-Net.Ru $\endgroup$
    – Wojowu
    Commented Oct 21 at 10:33
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    $\begingroup$ See also Vaughan-Lee's paper "Uncountably many varieties of groups" zbmath.org/0216.08401 $\endgroup$
    – Giles Gardam
    Commented Oct 21 at 10:34
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    $\begingroup$ DOI link of the link given by Wojowu, with English version. $\endgroup$
    – YCor
    Commented Oct 21 at 11:07

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