Kourovka Notebook Problem 19.72

Definitions and mathematical statement

The supplied record does not define the class $\mathscr{S}$ or specify a topology on its groups. A contraction group requires topological data, so the question cannot be formulated unconditionally from the supplied record alone.

For a topological group $G$ with identity $e_G$, use the convention \[ \operatorname{con}_G(g)=\{x\in G: g^n x g^{-n}\longrightarrow e_G\text{ as }n\to\infty\},\qquad g\in G. \] The contraction group of $g$ is called trivial when $\operatorname{con}_G(g)=\{e_G\}$.

Subject to recovering the intended definition of $\mathscr{S}$ and its topological hypotheses, the apparent existence question is: \[ \text{Does there exist }G\in\mathscr{S}\text{ such that } \forall g\in G,\quad \operatorname{con}_G(g)=\{e_G\}\,? \] This is a conditional reconstruction: the missing definition of $\mathscr{S}$ must supply all restrictions on the admissible groups and their topologies. The supplied wording does not explicitly distinguish this existence question from a question about an arbitrarily specified member of $\mathscr{S}$.

Short English statement

The apparent question asks whether some group in the intended class $\mathscr{S}$ can have trivial contraction group for every element. The record leaves $\mathscr{S}$ and its topological hypotheses undefined, so the formulation remains conditional.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $701620$, exact source problem code catalogue entry, title Kourovka Notebook Problem 19.72.
• The record attributes Problem 19.72 to C. Reid, proposed in 2018, and cites The Kourovka Notebook (Unsolved Problems in Group Theory), 19th issue (2018), with the 21st issue edition (2026), arXiv:1401.0300v48. The supplied collection description identifies E. I. Khukhro and V. D. Mazurov as editors.
• The registry labels the problem open. This is supplied source metadata, not an independently verified mathematical or literature assessment. The cited source was not independently consulted for this draft.