Kourovka Notebook Problem 21.102

Definitions and mathematical statement

Fix a finite group $A$. A group identity is an equation $w(x_1,\ldots,x_r)=1$ required to hold under every substitution. Let $\mathcal V(A)$ be the variety of all groups satisfying every identity valid in $A$. Its free group on $n$ generators, denoted $F_{\mathcal V(A)}(n)$, is generated by $n$ distinguished elements with the property that any assignment of these elements into any group in $\mathcal V(A)$ extends uniquely to a homomorphism.

A concrete realization is useful. In the direct product $A^{A^n}$ of functions from $A^n$ to $A$ with pointwise multiplication, let $\pi_i(a_1,\ldots,a_n)=a_i$. Then $F_{\mathcal V(A)}(n)$ is the subgroup generated by the $n$ coordinate functions. In particular it is finite. Set \[ f_A(n)=|F_{\mathcal V(A)}(n)|,\qquad b_A(n)=\left(\log f_A(n)\right)^{1/n},\quad n\ge1, \] where $\log$ is the natural logarithm and $0^{1/n}=0$.

Must $\lim_{n\to\infty}b_A(n)$ exist and be an integer for every finite group $A$? Both convergence and integrality are required. The concrete realization yields $f_A(n)\le|A|^{|A|^n}$, giving a finite upper bound on possible growth bases. Changing the fixed logarithm base only multiplies $b_A(n)$ by a factor tending to one. If $A$ is trivial, the sequence is constantly zero.

Short English statement

For any variety generated by a finite group, does the exponential growth base of the logarithm of its finite free-group orders converge to an integer?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2611 (KOU-21.102), Kourovka Notebook - New Problems, Issue 21. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 The Kourovka Notebook, 21st edition, version 47 (30 September 2026), new problems, Problem 21.102. Expanded formulation of the supplied catalogue statement. https://arxiv.org/pdf/1401.0300v47.