Kourovka Notebook Problem 16.14

Definitions and mathematical statement

Let $G$ be a finite $2$-group, meaning that $|G|=2^n$ for some integer $n\geq 0$. Define \[ Z(G)=\{z\in G:zg=gz\text{ for every }g\in G\},\qquad \Omega_1(G)=\langle g\in G:g^2=1\rangle,\qquad G^2=\langle g^2:g\in G\rangle. \] Here $G^2$ denotes the subgroup generated by squares. For a finite abelian $2$-group $A$, use the convention \[ \operatorname{rk}(A)=\dim_{{\mathbb{F}}_2}(A/A^2), \] which is its minimum number of generators. The supplied source does not explicitly define its rank convention; this draft uses the preceding standard convention for the two abelian groups in the question. The quotient $G/G^2$ is elementary abelian.

The question attributed to Ya. G. Berkovich is whether every finite $2$-group $G$ satisfying $\Omega_1(G)\leq Z(G)$ satisfies \[ \operatorname{rk}(G/G^2)\leq 2\operatorname{rk}(Z(G)), \] or, equivalently, \[ \dim_{{\mathbb{F}}_2}(G/G^2)\leq 2\dim_{{\mathbb{F}}_2}\left(Z(G)/Z(G)^2\right). \] The supplied registry labels this problem open; that label is not an independently verified assessment of its current status.

Short English statement

Must the minimum number of generators of a finite $2$-group whose involutions are all central be at most twice the minimum number of generators of its center? This uses the rank convention specified above.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $701206$, exact source problem code catalogue entry, title Kourovka Notebook Problem 16.14; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. The record was created and updated on $2026\text{-}10\text{-}16$ and labels the problem open.
• The Kourovka Notebook (Unsolved Problems in Group Theory), Problem 16.14, proposed by Ya. G. Berkovich, 16th issue (2006). The supplied background also identifies the 21st issue edition (2026), arXiv:1401.0300v48. These bibliographic details are supplied attribution and have not been independently verified.
• The supplied collection description attributes editing of Issues 1--21 (1965--2026) to E. I. Khukhro and V. D. Mazurov. No source excerpt beyond the supplied record, explicit reference URL, or dated literature-verification assessment is provided.