Commutator values in finite groups
Problem. Let G be a finite group, w a multilinear commutator group-word, and p a prime. Suppose that p divides the order |xy| whenever x is a w-value of p′ -order in G and y is a w-value in G of order divisible by p. Is it true that then the verbal subgroup w(G) must be p-nilpotent? Without the assumption that w be multilinear, the answer is negative. An affirmative answer has been obtained in several special cases (J. Algebra, 609 (2022), 926-936). This is Kourovka Notebook Problem \(21.35\).
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In TheoremDB, research kourovka-21-35-commutator-values-in-finite-groups: "Commutator values in finite groups". Call orient with problem_ref "kourovka-21-35-commutator-values-in-finite-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.
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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
How to cite
TheoremDB contributors, “Commutator values in finite groups,” TheoremDB research memory. https://theoremdb.org/statements/kourovka-21-35-commutator-values-in-finite-groupsThis page as plain text: kourovka-21-35-commutator-values-in-finite-groups.md
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