Kourovka Notebook Problem 13.30

Definitions and mathematical statement

For a group $G$, let $\operatorname{Sym}(G)$ be the group of all permutations of its underlying set, and let $\lambda:G\to\operatorname{Sym}(G)$ be its left regular representation, defined by $\lambda(g)(x)=gx$. A permutation group $H\leq\operatorname{Sym}(G)$ is primitive if it is transitive and the only $H$-invariant equivalence relations on $G$ are equality and the universal relation. For $|G|\geq2$, it is doubly transitive if, for all $x,y,u,v\in G$ with $x\ne y$ and $u\ne v$, there exists $h\in H$ such that $h(x)=u$ and $h(y)=v$.

For $|G|\geq2$, the supplied definition of a $B$-group is therefore \[ \forall H\leq\operatorname{Sym}(G),\qquad \left(\lambda(G)\leq H\ \text{and }H\text{ is primitive}\right) \Longrightarrow H\text{ is doubly transitive}. \] The source asks: does there exist a countable $B$-group?

The meaning of "countable" is unresolved in the supplied record. If finite groups are included, the cardinality condition is $|G|\leq\aleph_0$; if countably infinite groups are intended, it is $|G|=\aleph_0$. These are distinct formulations, and the source does not select between them. The source also gives no convention for primitivity or double transitivity in degree one; the explicit definitions above apply to $|G|\geq2$, leaving the treatment of the trivial group unresolved.

Short English statement

P. J. Cameron asks whether a countable group can have the property that every primitive permutation group containing its regular representation is doubly transitive. The source does not clarify whether "countable" includes finite groups or how the trivial group is treated.

Sources

• Local UnsolvedMath registry, permanent ID $700948$, exact source code catalogue entry, title Kourovka Notebook Problem 13.30. The supplied registry labels the problem open; this label is not an independently verified mathematical or literature assessment.
• P. J. Cameron, problem 13.30 in The Kourovka Notebook (Unsolved Problems in Group Theory), 13th issue (1995), as attributed in the supplied record.
• The supplied background also identifies the 21st issue edition (2026), arXiv:1401.0300v48. The collection description names E. I. Khukhro and V. D. Mazurov as editors. This reference information is supplied attribution and has not been independently checked.