Shortest Mixed Identities in Finite Groups

Definitions and mathematical statement

Let $G$ be a finite group. For a positive integer $r$, let $F_r=\langle x_1,\ldots,x_r\rangle$ be the free group on the displayed basis. For each $\mathbf a=(a_1,\ldots,a_r)\in G^r$, let \[ \operatorname{ev}_{\mathbf a}:G*F_r\longrightarrow G \] be the homomorphism that restricts to the identity on $G$ and sends $x_i$ to $a_i$. A nontrivial mixed identity for $G$ is an element $w\in G*F_r$ such that \[ w\ne 1\text{ in }G*F_r, \qquad \operatorname{ev}_{\mathbf a}(w)=1_G \quad\text{for every }\mathbf a\in G^r. \] Thus the coefficients from $G$ remain fixed under every substitution, and nontriviality refers to the free product before substitution.

For a specified length convention $\ell$, define \[ \mu_{\ell,r}(G)=\min\left\{\ell(w):w\in G*F_r\setminus\{1\},\ \operatorname{ev}_{\mathbf a}(w)=1_G\text{ for all }\mathbf a\in G^r\right\}. \] If the number of variables is allowed to vary, the corresponding quantity is \[ \mu_{\ell}(G)=\min_{r\geq 1}\mu_{\ell,r}(G). \] One possible convention is variable-letter length: sum the reduced word lengths, in the basis $x_1,\ldots,x_r$, of all $F_r$-syllables in the reduced free-product normal form of $w$; coefficients from $G$ contribute zero. Counting coefficient letters or counting free-product syllables gives different length conventions. The supplied excerpt does not define its convention or specify whether $r$ is fixed or variable. These choices therefore remain unresolved; the displayed quantities describe the possible precise targets rather than identifying a uniquely specified source invariant.

Problem. Determine the minimum length of a nontrivial mixed identity for a finite group $G$, with the source's intended length convention and choice of variable rank to be confirmed. Obtain exact values, or sharp asymptotic bounds, across the principal families of finite groups. The supplied record does not enumerate those families or specify the parameters and limiting regimes for the requested asymptotics.

Supplied partial result and qualification. The excerpt attributes to Tomanov, under its reference label [6], the assertion that $\operatorname{PSp}_{2m}(q)$, for $m\geq 2$, satisfies a mixed identity of length $8$. This is an assertion of existence, not an assertion that the shortest length equals $8$. Under the same length convention and with a compatible choice of variable rank, it would give an upper bound of $8$. Here $q$ is the finite-field parameter in the standard notation for the projective symplectic group; the excerpt supplies no explicit restrictions on $q$, exceptional cases, or variable rank. It also attributes to Tomanov the qualitative observation that projective symplectic and orthogonal groups have short mixed identities, in contrast to projective special linear groups. No orthogonal-group formula or quantitative meaning of this contrast is supplied, so no more precise assertion is made here.

Short English statement

A mixed identity is a nonidentity word with fixed coefficients from a finite group $G$ that evaluates to the identity for every substitution of its variables by elements of $G$. The problem asks for its shortest possible length and sharp results across families of finite groups. The source leaves the length convention and treatment of the number of variables unspecified. Its attributed length-$8$ symplectic-group result gives existence, not a claimed exact minimum.

Sources

• Local UnsolvedMath registry: supplied complete record for permanent ID $30006245$, title Shortest Mixed Identities in Finite Groups, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2025 and does not identify a proposer.
• Supplied workshop-report citation: Finite Groups, Fusion Systems and Applications (2025), Oberwolfach Reports, pp. 689--752, DOI 10.4171/owr/2025/16, https://doi.org/10.4171/owr/2025/16. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. This report is also the supplied literature source labeled "source question and partial results."
• Supplied attribution: Tomanov, reference [6] in the quoted source excerpt, for the short-identity observation and its Theorem 3 concerning $\operatorname{PSp}_{2m}(q)$, $m\geq 2$. The bibliographic entry corresponding to [6] is not included in the supplied record.
• Supplied literature source labeled "Bradford--Schneider--Thom, finite mixed identities": https://arxiv.org/abs/2306.14532.
• Supplied literature source labeled "Bradford--Ersoy--Schneider--Thom, 2026": https://arxiv.org/abs/2608.14537.
• Literature assessment checked at 2026-08-22: this is supplied dated curation, not independently verified here. It labels the problem "partially solved" and states that shortest mixed-identity lengths are known for several families and completely for some low-rank cases, while the general finite-group question and some family-level cases remain unresolved. The background explicitly describes this assessment as dated literature triage derived from the supplied OWR curation and calls for independent verification of time-sensitive status and cited claims.