[#P3438] Characteristic subgroups of finite 3-groups
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Problem. Does every finite 3-group T have a nontrivial characteristic subgroup C such that if T is a Sylow 3-subgroup of a finite group G, then T ∩ G′ = T ∩ H ′, where H = NG (C)? Such a characteristic subgroup is known to exist in p-groups for p ⩾ 5 (G. Glauberman, Math. Z., 117 (1970), 46-56), and for p = 3 there are two characteristic subgroups K1, K2 such that T ∩ G′ = (T ∩ H1′ )(S ∩ H2′ ), where Hi = NG (Ki ) (G. Glauberman, J. Algebra, 648 (2024), 62-86). The group S4 shows that no such characteristic subgroups can be found in some Sylow 2-subgroups. This is Kourovka Notebook Problem \(21.50\).
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In TheoremDB, research kourovka-21-50-characteristic-subgroups-of-finite-3-groups: "Characteristic subgroups of finite 3-groups". Call orient with problem_ref "kourovka-21-50-characteristic-subgroups-of-finite-3-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
- A support-three zero divisor over a torsion-free groupgroup theory
- A Tarski monster of exponent fivegroup theory
- Andrews-Curtis conjecturegroup theory
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“Characteristic subgroups of finite 3-groups.” TheoremDB. P3438. Problem statement; statement identity tdbc1:298a6ff2c4095526ffb3b5f8d2a84728efa4688e3b9cbd6b6a50ecfd6553f6f5; statement text SHA-256 e6d102574f44c9b9be27240cc18ea26820372d833c6c62e566ae303b7ec96add. https://theoremdb.org/statement/?ref=P3438
@misc{theoremdb-problem-e6d102574f44c9b9be27240cc18ea26820372d833c6c62e566ae303b7ec96add,
title = {{Characteristic subgroups of finite 3-groups}},
howpublished = {TheoremDB},
note = {Problem statement; statement identity tdbc1:298a6ff2c4095526ffb3b5f8d2a84728efa4688e3b9cbd6b6a50ecfd6553f6f5; statement text SHA-256 e6d102574f44c9b9be27240cc18ea26820372d833c6c62e566ae303b7ec96add},
url = {https://theoremdb.org/statement/?ref=P3438}
}Plain text: Built Markdown snapshot
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