Forum Problem Lists Definitions FAQ

17.94 (2010)

Open

(Well-known problem). Can the free product $\mathbb{Z} \ast G$ of the infinite cyclic group $\mathbb{Z}$ and a nontrivial group $G$ be the normal closure of a single element?

Contributor: J. Mycielski

Progress

Editors’ comments: the negation is also known as Kervaire’s conjecture. It was proved for torsion-free groups (A. Klyachko, Commun. Algebra, 21, no. 7 (1993), 2555–2575) and for residually finite groups (M. Gerstenhaber, O. S. Rothaus, Proc. Nat. Acad. Sci. U.S.A., 48 (1962) 1531–1533).

Comments

All comments are the responsibility of the user. Comments appearing on this page are not verified for correctness. Please keep posts mathematical and on topic. If you want to submit a proof (or a partial proof), please use the dedicated proof submission form rather than posting it in the comments.
Order by newest first or oldest first.

No comments yet. Be the first to comment.

Proof claims

Proof claims are the responsibility of the submitter. Appearance here does not mean the claim has been checked for mathematical correctness. Moderators only screen for spam, abuse, and obviously low-effort submissions.

No proof claims yet.