Kourovka Notebook Problem 14.97

Definitions and mathematical statement

For a group $G$, write $|g|$ for the order of an element $g\in G$. The group is periodic if every element has finite order. For distinct primes $p,q$, a periodic $\{p,q\}$-group is a group in which every element has order $p^a q^b$ for some integers $a,b\geq 0$, depending on the element. A periodic group is primary if there is a single prime $r$ such that every element has order a power of $r$. Thus, in a non-primary periodic $\{p,q\}$-group, elements of order $p$ and elements of order $q$ both occur.

A group $G$ is locally soluble if every subgroup generated by a finite subset of $G$ is soluble. Here a group $H$ is soluble if its derived series, defined by $H^{(0)}=H$ and $H^{(n+1)}=[H^{(n)},H^{(n)}]$, reaches the trivial subgroup after finitely many steps. The required number of steps may depend on the finitely generated subgroup. A $p$-subgroup means a subgroup all of whose elements have finite order a power of $p$; it need not be finite.

The question is whether, for every pair of distinct primes $p,q$, there exist a non-primary periodic locally soluble $\{p,q\}$-group $G$ and subgroups $A,B\leq G$ such that both $A$ and $B$ are $p$-subgroups and \[ G=AB=\{ab:a\in A,\ b\in B\}. \] The equality is an equality of sets: every element of $G$ must be a product of one element of $A$ followed by one element of $B$. Neither normality nor commutativity of the factors is assumed. No finiteness assumption is imposed on $G$, $A$, or $B$.

Short English statement

For every two distinct primes $p$ and $q$, does there exist a periodic locally soluble group whose element orders involve only these primes, with both primes occurring, that is the setwise product of two $p$-subgroups?

Sources

• Local UnsolvedMath registry, permanent ID $701081$, source problem code catalogue entry, title Kourovka Notebook Problem 14.97. The supplied record labels the problem open; that label is supplied curation, not an independently verified mathematical assessment.
• N. S. Chernikov, Problem 14.97, The Kourovka Notebook (Unsolved Problems in Group Theory), 14th issue (1999), 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. These bibliographic details are reproduced from the supplied record without independent source verification.