Kourovka Notebook Problem 20.50

Definitions and mathematical statement

For a group $G$, write \[ I(G)=\{x\in G:x^2=1_G,\ x\ne 1_G\} \] for its set of involutions. For each positive integer $m$, the source defines $G_m$ as the "largest" group generated by $m$ involutions and satisfying \[ \forall x,y\in I(G_m),\qquad (xy)^4=1_{G_m}. \] Here "largest" is interpreted in the universal sense: $G_m$ has distinguished involutory generators $s_1,\ldots,s_m$, and for every group $H=\langle t_1,\ldots,t_m\rangle$ with $t_i\in I(H)$ and \[ \forall u,v\in I(H),\qquad (uv)^4=1_H, \] there is a surjective homomorphism $G_m\to H$ sending $s_i$ to $t_i$ for every $i$. This makes explicit an interpretation of "largest" that the supplied record does not itself define. The condition concerns all involutions in the group, including those other than the distinguished generators.

Problem. Determine $\lvert G_4\rvert$. No finiteness assumption is imposed here.

The source reports the known value $\lvert G_3\rvert=2^{11}$ and refers additionally to Kourovka Notebook Problem 18.58; that referenced problem's contents are not supplied.

Short English statement

Determine the order of the universal group generated by four involutions in which the product of any two involutions has fourth power equal to the identity. The source reports $\lvert G_3\rvert=2^{11}$; its term "largest" is interpreted universally and requires review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $701708$, exact source code catalogue entry, titled Kourovka Notebook Problem 20.50. The registry labels the problem open; this is supplied metadata, not an independently verified status assessment.
• The Kourovka Notebook (Unsolved Problems in Group Theory), Problem 20.50, proposed by D. V. Lytkina in 2022. The supplied background identifies the 20th issue (2022) and the 21st issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The supplied statement reports $\lvert G_3\rvert=2^{11}$ and includes the cross-reference "Also cf. 18.58." No separate reference supporting that numerical assertion or text of Problem 18.58 is supplied.