Forum Problem Lists Definitions FAQ

20.104 (2022)

Open

(Well-known questions). Suppose that a group $G$ is finitely generated and decomposable into a direct product $G = G_1 \times G_2$.
$\qquad$ a) Is it true that the elementary theory of $G$ is decidable if and only if the elementary theories of the groups $G_1$ and $G_2$ are decidable?
$\qquad$ b) Is it true that the universal theory of $G$ is decidable if and only if the universal theories of the groups $G_1$ and $G_2$ are decidable?

Contributor: E. I. Timoshenko

Progress

a) An affirmative answer for almost soluble groups follows from (G. A. Noskov, Izv. Akad. Nauk SSSR, Ser. Mat., 47, no. 3 (1983), 498–517).

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.