Kourovka Notebook Problem 12.8

Definitions and mathematical statement

Let $\mathcal V$ be a non-trivial variety of groups, meaning a class defined by group identities that contains a group with more than one element. Write $F_m(\mathcal V)$ for the relatively free group of rank $m$ in $\mathcal V$. Fix an integer $r>0$ and a free basis $a_1,\ldots,a_r$ of $F_r(\mathcal V)$.

The source does not define ordinary discrimination. Using its usual simultaneous homomorphism-separation meaning, say that $F_r(\mathcal V)$ discriminates $\mathcal V$ if, for every positive integer $m$ and every finite nonempty set $S\subseteq F_m(\mathcal V)\setminus\{1\}$, there exists a homomorphism $h:F_m(\mathcal V)\to F_r(\mathcal V)$ such that \[ h(u)\ne1\qquad\text{for every }u\in S. \] This interpretation of the undefined source term requires review.

An inclusion of $F_r(\mathcal V)$ as a varietally free factor means that its displayed basis extends to a free basis of the containing relatively free group. For a nonnegative integer $t$, identify $F_r(\mathcal V)$ with the subgroup generated by $a_1,\ldots,a_r$ in the relatively free group \[ F_{r+t}(\mathcal V)=F_{\mathcal V}(a_1,\ldots,a_r,b_1,\ldots,b_t). \] Following the source, $F_r(\mathcal V)$ strongly discriminates $\mathcal V$ if the following holds for every integer $k\ge0$, every integer $n\ge1$, and every family of group words $w_1,\ldots,w_n$ in the displayed parameters and variables: whenever, for some $t\ge0$, there exists $\mathbf c\in F_{r+t}(\mathcal V)^k$ satisfying \[ w_i(a_1,\ldots,a_r,\mathbf c)\ne1\qquad(1\le i\le n), \] there exists $\mathbf d\in F_r(\mathcal V)^k$ satisfying \[ w_i(a_1,\ldots,a_r,\mathbf d)\ne1\qquad(1\le i\le n). \] The parameters $a_j$ remain fixed throughout. Although the source does not specify whether the containing free group may have infinite rank, finite ranks suffice here: a finite solution tuple involves only finitely many additional basis elements.

The group-theoretic question is whether there exist a non-trivial variety $\mathcal V$ and an integer $r>0$ such that \[ F_r(\mathcal V)\text{ discriminates }\mathcal V \quad\text{but does not strongly discriminate }\mathcal V. \] Thus failure of strong discrimination must be witnessed by a finite system of inequalities with these fixed free parameters that is solvable in a relatively free extension but has no solution in $F_r(\mathcal V)$.

The source also asks the analogous existence question for general algebras in universal algebra. A usual interpretation replaces nonidentity group elements by pairs of distinct elements of relatively free algebras, discrimination by simultaneous separation of finitely many such pairs by a homomorphism, and the strong condition by transfer of finite systems of term inequalities with fixed free parameters from a free extension to the original free algebra. The source supplies neither a signature nor an explicit definition of this algebraic analogue; this description records a candidate interpretation, and the exact intended scope remains unresolved.

Short English statement

Does some non-trivial group variety have a relatively free group of positive finite rank that discriminates the variety, yet fails to realize a finite system of inequalities with fixed free-generator parameters that is realizable in a relatively free extension? The source also asks for an analogue for general algebras, without specifying its precise scope. Ordinary discrimination is interpreted in its usual simultaneous separation sense.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $700828$, exact source code catalogue entry, title Kourovka Notebook Problem 12.8. The supplied status is open; this is a registry label, not an independently verified mathematical or literature assessment.
• The Kourovka Notebook (Unsolved Problems in Group Theory), Problem 12.8, 12th issue (1992); the supplied background also identifies the 21st issue edition (2026), arXiv:1401.0300v48. Proposed by A. M. Gaglione and D. Spellman in 1992. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The Notebook attribution and version identifier are supplied evidence. No external source was visited or independently checked, and no source-verification date or external link was supplied.