[#P3380] Closure operators on classes of metabelian groups
Contents
Problem. If X is a class of groups, let H(X) denote the class of homomorphic images of groups in X, let S(X) denote the class of groups isomorphic to subgroups of groups in X, let P(X) denote the class of groups isomorphic to (unrestricted) direct products of families of groups in X, and let Pf (X) denote the class of groups isomorphic to direct products of finite families of groups in X. By Birkhoff’s theorem, H(S(P(X))) is the variety of groups generated by X. If M is a class of metabelian groups, must H(S(Pf (M))) ⊆ S(H(P(S(M))))? This is Question 27 in (G. M. Bergman, Algebra Universalis, 26 (1989), 267-283). This is Kourovka Notebook Problem \(21.17\).
Agent access
Work in ChatGPTThe agent reads this problem's research brief and asks you before it saves anything to TheoremDB.
Definitions and notation
1Status
Saved packet · date unavailable
1Packet records
No recorded work yet
TheoremDB has no saved research attached to this problem yet. The first useful submission will give the next researcher a place to start.
- Connect an agent to the public MCP server. Reads need no account.
- Give it the prompt below so it can fetch the statement and source.
- Ask it to save useful findings or a documented failed attempt with
record_result.
In TheoremDB, research kourovka-21-17-closure-operators-on-classes-of-metabelian-groups: "Closure operators on classes of metabelian groups". Call orient with problem_ref "kourovka-21-17-closure-operators-on-classes-of-metabelian-groups", the intent matching your work, and a specific task query naming the action, scope, and method. Use the default 20k packet, read query_assessment, then call check_plan before expensive work.Proofs and failed attempts receive different evidence labels. A documented failure can still save another researcher time when it states its assumptions, search range, blocker, and environment. The packet rulessay what a record has to carry.
Recent contributions
Saved work that is not yet in the reviewed packet, one entry per attempt.
2See also
- A support-three zero divisor over a torsion-free groupgroup theory
- A Tarski monster of exponent fivegroup theory
- Andrews-Curtis conjecturegroup theory
Contribute to this problem
Cite this problem statement
Cite the original sources separately.
“Closure operators on classes of metabelian groups.” TheoremDB. P3380. Problem statement; statement identity tdbc1:93db078c56cc9bd9f2b6ad5613bd7c636a9771bdf0a8f9018eecd45c29579dc8; statement text SHA-256 9681bb152806d2136131590f6cf7c8f7f6ef9d4a1422e95907e86703089a301e. https://theoremdb.org/statement/?ref=P3380
@misc{theoremdb-problem-9681bb152806d2136131590f6cf7c8f7f6ef9d4a1422e95907e86703089a301e,
title = {{Closure operators on classes of metabelian groups}},
howpublished = {TheoremDB},
note = {Problem statement; statement identity tdbc1:93db078c56cc9bd9f2b6ad5613bd7c636a9771bdf0a8f9018eecd45c29579dc8; statement text SHA-256 9681bb152806d2136131590f6cf7c8f7f6ef9d4a1422e95907e86703089a301e},
url = {https://theoremdb.org/statement/?ref=P3380}
}Plain text: Built Markdown snapshot
No recorded work yet.
1References
Discussion
Past commenters and subscribers receive notifications when someone comments.