Kourovka Notebook Problem 12.86

Definitions and mathematical statement

For a nontrivial finite simple group $G$ and an integer $n\geq 1$, let $G^n=\prod_{i=1}^{n}G$, with coordinatewise multiplication. A group $H$ is called 2-generator if there exist $x,y\in H$ such that $H=\langle x,y\rangle$; this means that at most two generators suffice.

The problem asks, for each known finite simple group $G$, to determine the largest positive integer $n$ such that $G^n$ is 2-generator. Equivalently, determine $m\geq 1$ satisfying \[ \left(\exists x,y\in G^m\right)\;\langle x,y\rangle=G^m, \qquad \left(\forall n>m\right)\left(\forall u,v\in G^n\right)\;\langle u,v\rangle\neq G^n. \] Here the source's word "known" does not specify a precise collection or a reference date. Under the usual meaning of finite simple group, cyclic groups of prime order are included; the supplied statement gives no restriction to nonabelian groups. The interpretation of "2-generator" above is the usual at-most-two convention; the source does not explicitly discuss whether exactly two generators must be necessary.

Short English statement

For every known finite simple group, determine the greatest number of copies whose direct product can be generated by two elements. The scope of "known" remains unspecified in the supplied source.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $700903$, exact source problem code catalogue entry, title Kourovka Notebook Problem 12.86. Upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Attributed to S. A. Syskin, proposed in 1992. The supplied background identifies The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), Problem 12.86, and the 21st issue edition (2026), arXiv:1401.0300v48. The registry describes the collection as edited by E. I. Khukhro and V. D. Mazurov.
• The registry labels this problem open and records an update timestamp of 2026-10-16T14:00:00Z. This is supplied registry metadata, not an independently verified literature assessment. The cited Notebook text has not been independently inspected for this draft.