Kourovka Notebook Problem 21.47

Definitions and mathematical statement

A group is finitely presented if it has finitely many generators and finitely many defining relators. For groups $G,H$, their direct product $G\times H$ consists of pairs, with componentwise multiplication $(g,h)(g',h')=(gg',hh')$. An isomorphism $G\cong G\times H$ is a bijective group homomorphism, not just an embedding or an epimorphism.

Does there exist a finitely presented group $G$ and a nontrivial group $H$ such that \[ G\cong G\times H? \] The factor $H$ is to be chosen along with $G$ and is only required to have an element other than its identity. The isomorphism is not required to identify any prescribed elements or subgroups. The question asks whether the entire direct factor can be absorbed without changing the abstract isomorphism type of a finitely presented group.

Equivalently, such a $G$ must have commuting normal subgroups $A,B$ with $G=AB$, $A\cap B=1$, $A\cong G$, and $B\ne1$. These conditions express an internal direct product; mere commuting subgroups without the trivial-intersection and generation conditions do not suffice.

No finite group can qualify, since the equality of finite orders $|G|=|G||H|$ would force $|H|=1$. Thus infinitude follows from the requirements. A finitely generated example without a finite presentation, or a finitely presented group merely surjecting onto its direct square, is weaker than the requested isomorphism.

Short English statement

Can a finitely presented group be isomorphic to its direct product with a nontrivial group?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2556 (KOU-21.47), Kourovka Notebook - New Problems, Issue 21. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 The Kourovka Notebook, 21st edition, new problems, Problem 21.47. Expanded formulation of the supplied catalogue statement. https://arxiv.org/pdf/1401.0300v47.