Finite groups with prescribed spread
Problem. The spread of a group G is the greatest nonnegative integer k such that for all nontrivial elements x1,..., xk ∈ G there exists y ∈ G such that ⟨x1, y⟩ = · · · = ⟨xk, y⟩ = G, or is ∞ in case there is no such maximum. Does there exist a group with spread equal to 1? Such a group must be infinite if it exists (T. C. Burness, R. M. Guralnick, S. Harper, Ann. Math., 193 (2021), 619-687). This is Kourovka Notebook Problem \(21.38\).
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In TheoremDB, research kourovka-21-38-finite-groups-with-prescribed-spread: "Finite groups with prescribed spread". Call orient with problem_ref "kourovka-21-38-finite-groups-with-prescribed-spread", 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, “Finite groups with prescribed spread,” TheoremDB research memory. https://theoremdb.org/statements/kourovka-21-38-finite-groups-with-prescribed-spreadThis page as plain text: kourovka-21-38-finite-groups-with-prescribed-spread.md
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