Kourovka Notebook Problem 19.71

Definitions and mathematical statement

For a topological group $G$, a compact open subgroup is a subgroup $U\leq G$ that is compact and open in the topology of $G$. An element $g\in G$ normalizes $U$ if $gUg^{-1}=U$; equivalently, $g\in N_G(U)$, where \[ N_G(U)=\{x\in G:xUx^{-1}=U\}. \] The source asks whether there exists a group $G$ in its class $\mathscr{S}$ such that \[ \forall g\in G\;\exists U_g\leq G:\quad U_g\text{ is compact and open in }G \quad\text{and}\quad gU_gg^{-1}=U_g. \] The subgroup $U_g$ may depend on $g$; a single compact open subgroup normalized by every element is not required.

Unresolved qualification. The supplied record does not define $\mathscr{S}$ or explicitly specify the topological hypotheses on its members. The displayed existence question therefore retains this undefined source class as a parameter. Its intended domain must be established before a complete mathematical formulation can be certified.

Short English statement

Does the source class $\mathscr{S}$ contain a group in which each element normalizes some compact open subgroup, possibly depending on that element? The supplied record leaves $\mathscr{S}$ undefined.

Sources

• Supplied local registry record: permanent UnsolvedMath ID $701619$, source code catalogue entry, title Kourovka Notebook Problem 19.71. Its status is supplied as open; this label has not been independently verified.
• Supplied source attribution: The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), problem 19.71; also identified with the 21st issue edition (2026), arXiv:1401.0300v48. The collection description names E. I. Khukhro and V. D. Mazurov as editors. These bibliographic details are supplied and have not been independently checked.
• The background and proposer field attribute the problem to C. Reid, with proposed year 2018. The statement itself begins with the attribution "(G. Willis)." Both attributions are retained; their respective roles are not explained in the record.