Kourovka Notebook Problem 13.4

Definitions and mathematical statement

For a positive integer $n$, let $\mathrm{GL}_n(2)=\mathrm{GL}_n({\mathbb{F}}_2)$, acting naturally on the column-vector space ${\mathbb{F}}_2^n$. The source considers a group $G$ with a normal "(pro-)$2$-subgroup" $N$. In the finite $2$-group interpretation, the Frattini subgroup $\Phi(N)$ is the intersection of the maximal proper subgroups of $N$. In the pro-$2$ interpretation, a pro-$2$ group is an inverse limit of finite $2$-groups, and $\Phi(N)$ is the intersection of its maximal proper open subgroups. The source does not fully specify the ambient topological category for $G$.

The hypotheses are that there exist isomorphisms \[ \alpha:G/N\longrightarrow\mathrm{GL}_n(2),\qquad \beta:N/\Phi(N)\longrightarrow({\mathbb{F}}_2^n,+) \] such that the action induced by conjugation is the natural module action, explicitly \[ \beta\left(gxg^{-1}\Phi(N)\right) =\alpha(gN)\,\beta\left(x\Phi(N)\right) \qquad(g\in G,\ x\in N). \] Here the action factors through $G/N$, since inner automorphisms of $N$ act trivially on its Frattini quotient in the finite and pro-$2$ settings.

Problem (B. Baumann, 1995). For $n=3$, determine $G$ satisfying these hypotheses such that $N$ is "as large as possible." The source does not define this comparison: in particular, it does not specify whether to maximize finite order, seek a universal pro-$2$ extension, or use another extremal criterion. Consequently, the extremal target remains unresolved at the level of formulation.

The source supplies the following contextual assertions: for $n>3$, it can be proved that $\Phi(N)=1$; for $n=2$, there exists such a group $G$ with $N$ the free pro-$2$ group on two generators. These are retained as source assertions, without independent verification. A free pro-$2$ group on two generators has the universal mapping property for assignments of its two generators to arbitrary pro-$2$ groups, with continuous homomorphisms.

Short English statement

Determine the groups $G$ whose normal $(\mathrm{pro}\text{-})2$-subgroup $N$ has quotient $G/N\cong\mathrm{GL}_3(2)$, acting naturally on $N/\Phi(N)$, with $N$ as large as possible. The meaning of "as large as possible" and the precise finite or topological category require clarification. The supplied registry labels the problem open; this is not an independently verified status assessment.

Sources

• Local UnsolvedMath registry, permanent ID $700922$, source code catalogue entry, title Kourovka Notebook Problem 13.4. The supplied record labels the problem open and gives upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• The Kourovka Notebook (Unsolved Problems in Group Theory), original problem 13.4, proposed by B. Baumann in the 13th issue (1995). The supplied background also identifies the 21st-issue edition (2026), arXiv identifier 1401.0300v48.
• The supplied collection description attributes editing of Kourovka Notebook, Issues 1--21 (1965--2026), to E. I. Khukhro and V. D. Mazurov. These bibliographic attributions and the mathematical contextual assertions are supplied source information; the cited edition has not been independently consulted here.