Exact factorizations of countable groups
We give an elementary affirmative answer to Problem~20.73 of the Kourovka Notebook. More precisely, we show that every countably infinite group admits a factorization into two infinite subsets in which each element has a unique representation. The main ingredient is an inductive extension lemma for groups in which every nonidentity element has only finitely many square roots. The general case follows by combining this lemma with elementary reductions involving abelian subgroups, infinite-index subgroups, and the FC-centre.
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Theorem 4.2
Every countably infinite group $G$ admits an exact factorization $G=AB$ with $A$ and $B$ infinite.
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- External referenceG. Gardam, Answer to Existence of $m$ infinite subsets in an arbitrary group such that all products of one element from each (in order) are distinct, MathOverflow, July 2022, https://mathoverflow.net/questions/425005/.LaTeX source\bibitem{Gardam} G.~Gardam, Answer to \emph{Existence of $m$ infinite subsets in an arbitrary group such that all products of one element from each (in order) are distinct}, MathOverflow, July 2022, \url{https://mathoverflow.net/questions/425005/}.
- External referenceE. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory. The Kourovka Notebook, No. 21, Sobolev Institute of Mathematics, Novosibirsk, 2026, Problem 20.73, p. 166 (October 2026 update).LaTeX source\bibitem{Kourovka} E.~I.~Khukhro and V.~D.~Mazurov (eds.), \emph{Unsolved Problems in Group Theory. The Kourovka Notebook}, No.~21, Sobolev Institute of Mathematics, Novosibirsk, 2026, Problem~20.73, p.~166 (October 2026 update).
- External referenceI. V. Protasov, On factoring groups into thin subsets, Axioms 10 (2021), no. 2, article 89. https://doi.org/10.3390/axioms10020089.LaTeX source\bibitem{Protasov} I.~V.~Protasov, On factoring groups into thin subsets, \emph{Axioms} \textbf{10} (2021), no.~2, article~89. \url{https://doi.org/10.3390/axioms10020089}.
- External referenceS. Szab\'o and A. D. Sands, Factoring Groups into Subsets, CRC Press, Boca Raton, 2009.LaTeX source\bibitem{SzaboSands} S.~Szab\'o and A.~D.~Sands, \emph{Factoring Groups into Subsets}, CRC Press, Boca Raton, 2009.
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Machine-readable theorem index · 8 statements
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- PF:2026.000023/v1/THM-1.1
Every countably infinite group $G$ possesses infinite subsets $A,B$ such that \[ G=AB \quad\text{and}\quad ab=a'b'\ \Longrightarrow\ (a,b)=(a',b') \qquad(a,a'\in A,\ b,b'\in B). \]
- PF:2026.000023/v1/LEM-2.1
Let $H\leq G$. If $H=XY$ is an exact factorization with $X$ and $Y$ infinite, then $G$ has an exact factorization into two infinite subsets.
- PF:2026.000023/v1/LEM-2.2
Let $\pi:G\twoheadrightarrow Q$ be an epimorphism. If $Q=XY$ is an exact factorization with $X$ and $Y$ infinite, then $G$ has an exact factorization into two infinite subsets.
- PF:2026.000023/v1/COR-2.3
If $G$ has an infinite subgroup of infinite index, then $G$ has an exact factorization into two infinite subsets.
- PF:2026.000023/v1/LEM-2.4
Every countably infinite abelian group admits an exact factorization into two infinite subsets.
- PF:2026.000023/v1/LEM-3.1
Let $G$ be a countably infinite group such that \bigl|\{x\in G:x^2=h\}\bigr|<\infty \qquad\text{for every }h\in G\setminus\{1\}. Then $G$ admits an exact factorization into two infinite subsets.
- PF:2026.000023/v1/LEM-4.1
Every infinite group $G$ has either an infinite abelian subgroup or an infinite quotient in which every nonidentity element has an infinite conjugacy class.
- PF:2026.000023/v1/THM-4.2
Every countably infinite group $G$ admits an exact factorization $G=AB$ with $A$ and $B$ infinite.