Kourovka Notebook Problem 9.1

Definitions and mathematical statement

For a group $G$ and $x\in G$, let $\operatorname{ord}_G(x)$ denote the order of $x$, with value $\infty$ when $x$ has infinite order. Adopt the convention that every positive integer divides $\infty$. The group $G$ is potent if, for every $x\in G$ and every positive integer $n$ satisfying $n\mid\operatorname{ord}_G(x)$, there exist a finite group $Q$ and a surjective homomorphism $\varphi:G\to Q$ such that \[ \operatorname{ord}_Q(\varphi(x))=n. \] The group $Q$ and the homomorphism $\varphi$ may depend on both $x$ and $n$.

For groups $A$ and $B$, write $A*B$ for their free product, the group equipped with canonical homomorphisms from $A$ and $B$ such that every pair of homomorphisms from these groups to a common group extends uniquely to a homomorphism from $A*B$.

Is it true that, for every pair of potent groups $A$ and $B$, their free product $A*B$ is potent? No finiteness or finite generation assumption on $A$ or $B$ is imposed in the supplied statement.

Short English statement

A group is potent if each element can have any prescribed positive divisor of its order realized as its exact order in a finite quotient; for an element of infinite order, every positive integer is allowed. Must the free product of any two potent groups also be potent?

Sources

• Supplied local UnsolvedMath registry record, permanent ID $700536$, exact source problem code catalogue entry; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. The supplied registry labels the problem open; this label has not been independently verified.
• The Kourovka Notebook (Unsolved Problems in Group Theory), original problem 9.1, 9th issue (1984), proposed by R. B. J. T. Allenby. The supplied background identifies 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 attribution, not independently verified source consultation.