Kourovka Notebook Problem 12.85

Definitions and mathematical statement

For a group $K$, define its derived series by $K^{(0)}=K$ and $K^{(n+1)}=[K^{(n)},K^{(n)}]$ for $n\geq 0$, where $[L,L]$ is the subgroup generated by all commutators of elements of $L$. The group $K$ is soluble if $K^{(d)}=1$ for some integer $d\geq 0$; its derived length $\operatorname{dl}(K)$ is the least such integer. Thus the trivial group has derived length zero.

For a group $S$, let $\operatorname{Var}(S)$ denote the variety of groups generated by $S$, namely the class of all groups satisfying every group identity satisfied by $S$. Here a group is called two-generated if it has the form $\langle x,y\rangle$, allowing redundant generators.

The supplied question is whether, for every finite simple group $S$ subject to the source's qualification "(known)", and every integer $d\geq 0$, the following implication holds: \[ \left(\exists H\leq S\;[H\text{ is soluble and }\operatorname{dl}(H)=d]\right) \ \Longrightarrow\ \left(\exists G\in\operatorname{Var}(S)\;\exists x,y\in G\; [G=\langle x,y\rangle,\ G\text{ is soluble},\ \operatorname{dl}(G)=d]\right). \] The source does not define the class intended by "(known)", so the scope of the quantifier over $S$ remains unresolved. The displayed formulation uses the convention that two-generated means generated by at most two elements; whether the source instead requires minimum generating number exactly two is unspecified. No nonabelian restriction on $S$, and no finiteness requirement on $G$, is stated.

Short English statement

Must the variety generated by a finite simple group containing a soluble subgroup of derived length $d$ also contain a soluble group generated by two elements with derived length exactly $d$? The source qualifies the finite simple group as "(known)" without defining that restriction.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $700902$, exact source problem code catalogue entry, title Kourovka Notebook Problem 12.85; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Supplied attribution: S. A. Syskin, proposed in 1992. Source: The Kourovka Notebook (Unsolved Problems in Group Theory), 12th issue (1992), Problem 12.85; also identified in the record with the 21st issue edition (2026), arXiv:1401.0300v48.
• The supplied collection description attributes the notebook to editors E. I. Khukhro and V. D. Mazurov and describes coverage of Issues 1--21 (1965--2026). The registry labels this problem open; that label is supplied metadata, not an independently verified literature assessment. No source was independently consulted for this draft.