Kourovka Notebook Problem 20.113

Definitions and mathematical statement

The source uses $c$ both as a generator and as the nilpotency-class parameter. Here the generator is renamed $u$, and the parameter is renamed $r$. Fix the commutator convention $[x,y]=x^{-1}y^{-1}xy$.

The following makes the presentation precise under the usual universal-quotient interpretation of its exponent and nilpotency conditions. For $(q,r)\in\{(8,12),(7,13)\}$, let $F$ be the free group on $a,b,u,d$. Define $\gamma_1(F)=F$ and $\gamma_{j+1}(F)=[\gamma_j(F),F]$, and let $F^q$ be the normal subgroup generated by all $q$-th powers of elements of $F$. Put \[ H(q,r)=F\big/\left(F^q\,\gamma_{r+1}(F)\,\langle\!\langle[b,a][d,u]^{-1}\rangle\!\rangle_F\right), \] where $\langle\!\langle w\rangle\!\rangle_F$ denotes the normal closure of $w$ in $F$. Thus this quotient is generated by the images of $a,b,u,d$, satisfies $[b,a]=[d,u]$, has exponent dividing $q$, and has nilpotency class at most $r$.

The source says "of exponent $q$" and "nilpotent of class $c$". It does not explicitly distinguish the universal quotient above from requirements of exact exponent and exact nilpotency class. Those exactness requirements, and the intended universal interpretation, require confirmation; they are not established here.

For a group $G$, its Schur multiplier is $M(G)=H_2(G,{\mathbb{Z}})$, with trivial integral coefficients. For an abelian group $A$, its exponent is the least positive integer $n$, if one exists, such that $nx=0$ for every $x\in A$.

Under the stated interpretation of $H(q,r)$, the two questions attributed to M. R. Vaughan-Lee are:

1. (a) Is $\operatorname{exp} M(H(8,12))=32$?
2. (b) Is $\operatorname{exp} M(H(7,13))=49$?

Short English statement

Vaughan-Lee asks whether the Schur multipliers of the two specified four-generator groups have exponents exactly $32$ and $49$, respectively. The draft uses an explicitly qualified universal-quotient interpretation of the presentation; the intended exponent and nilpotency-class conventions require review.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $701771$, exact source code catalogue entry, title Kourovka Notebook Problem 20.113; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. The record supplies the status label open; this is not an independently verified assessment.
• M. R. Vaughan-Lee, proposer, 2022. The supplied attribution identifies The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), Problem 20.113, and the 21st issue edition (2026), identified as arXiv:1401.0300v48. These bibliographic details are reproduced from the record without independent source consultation.
• The supplied statement reports that the groups were too large for the versions of the $p$-Quotient Algorithm described there as using 32-bit arithmetic, and suggests that a version using 64-bit arithmetic would help. This is a source-specific computational observation, not a verified assessment of present software capabilities. The registry record was updated on 2026-10-16; that timestamp is not an independent literature-check date.