Element orders in locally finite products of prime-power subgroups

Definitions and mathematical statement

A group is locally finite if every subgroup generated by finitely many elements is finite. For a prime $r$, an $r$-subgroup is a subgroup whose elements all have finite orders that are powers of $r$, including $1=r^0$. A $\{p,q\}$-group is a group in which every element has finite order of the form $p^a q^b$, where $a,b$ are nonnegative integers.

Let $p$ and $q$ be distinct primes, and let $G$ be a locally finite group. Suppose there exist a $p$-subgroup $P\leq G$ and a $q$-subgroup $Q\leq G$ such that \[ G=PQ=\{xy:x\in P,\ y\in Q\}. \] Neither subgroup is assumed to be normal. Must $G$ be a $\{p,q\}$-group? Explicitly, must every $g\in G$ satisfy \[ \operatorname{ord}(g)=p^a q^b \quad\text{for some nonnegative integers }a,b? \]

Short English statement

If a locally finite group is the setwise product of a $p$-subgroup and a $q$-subgroup for distinct primes $p,q$, must every element have order divisible only by these two primes?

Sources

• Local UnsolvedMath registry: permanent ID $700338$, exact source code catalogue entry, supplied complete source record.
• B. Hartley, Problem 5.59, The Kourovka Notebook (Unsolved Problems in Group Theory), originally in the 5th issue (1976). The supplied record cites the 21st issue edition (2026), arXiv:1401.0300v48, and identifies E. I. Khukhro and V. D. Mazurov as editors of the collection.
• The supplied registry labels this problem open. This is supplied source metadata, not an independently verified assessment of its current mathematical status.