Kourovka Notebook Problem 18.46

Definitions and mathematical statement

For a finite group $G$, define \[ m(G)=\min\left\{|H|:\ H\text{ is a finite group and there exists an injective homomorphism }\iota:G\to H\text{ such that }(\forall g\in G)(\exists h\in H)\ h^2=\iota(g)\right\}. \] The square root may depend on $g$. The supplied statement asserts that such an overgroup always exists and that \[ (\forall\text{ finite groups }G)\qquad m(G)\le 2|G|^2. \] The problem asks whether this estimate is sharp.

The intended meaning of sharpness is not explicitly specified. If it means optimality of the constant $2$ among uniform quadratic bounds, the question is whether \[ (\forall c\in{\mathbb{R}},\ 0<c<2)(\exists\text{ a finite group }G)\qquad m(G)>c|G|^2. \] If it means attainment of the stated bound, the question instead asks whether some finite group satisfies $m(G)=2|G|^2$. These interpretations are not asserted to be equivalent.

The source additionally reports, attributing the result to D. V. Baranov and Ant. A. Klyachko, that the best possible estimate cannot be better than $|H|\le |G|^2$. The supplied record does not state the precise lower-bound quantifiers or the examples supporting this qualification; it is therefore retained as a reported restriction on possible improvements, without imposing an additional quantified theorem.

Short English statement

For each finite group, consider the smallest finite overgroup in which every original element has a square root. The supplied bound is twice the square of the original group order. Is this bound sharp? The source leaves the precise sense of sharpness and the quantifiers of its reported lower-bound obstruction unspecified.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $701475$, exact source problem code catalogue entry, title "Kourovka Notebook Problem 18.46." The registry labels the problem open; this is supplied source status, not an independently verified assessment.
• The Kourovka Notebook (Unsolved Problems in Group Theory), Problem 18.46, proposed by Ant. A. Klyachko in 2014. The supplied background identifies the 18th issue (2014) and the 21st issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors. These bibliographic details have not been independently verified.
• D. V. Baranov and Ant. A. Klyachko, Siber. Math. J., 53, no. 2 (2012), 201--206. The supplied statement cites this reference for its qualification that the best possible estimate cannot be better than $|H|\le |G|^2$; no article title or link is supplied.