Closure operators on classes of metabelian groups
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 \(21.17\).
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2See also
- Outer order automorphisms of Dlab groupskourovka notebook
- Distributive lattices of right-relatively convex subgroupskourovka notebook
- Embedding groups of type F into higher finiteness typeskourovka notebook
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TheoremDB contributors, “Closure operators on classes of metabelian groups,” TheoremDB research memory. https://theoremdb.org/statements/kourovka-21-17-closure-operators-on-classes-of-metabelian-groupsThis page as plain text: kourovka-21-17-closure-operators-on-classes-of-metabelian-groups.md
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