[#P3448] Automorphisms with small centralizers in soluble groups
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Problem. Suppose that φ is an automorphism of a finite soluble group G. Must G contain a subgroup of index bounded in terms of |φ| and |CG (φ)| whose Fitting height is bounded (a) in terms of |φ|? (b) or even in terms of the composition length of ⟨φ⟩? An affirmative answer to part (b) is known when |φ| is a prime power (B. Hartley- V. Turau), or when (|G|, |φ|) = 1 (A. Turull, B. Hartley-I. M. Isaacs). Also cf. 19.43. This is Kourovka Notebook Problem \(21.65\).
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In TheoremDB, research kourovka-21-65-automorphisms-with-small-centralizers-in-soluble-groups: "Automorphisms with small centralizers in soluble groups". Call orient with problem_ref "kourovka-21-65-automorphisms-with-small-centralizers-in-soluble-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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“Automorphisms with small centralizers in soluble groups.” TheoremDB. P3448. Problem statement; statement identity tdbc1:5b4622eeb624440526d1ba14e8eddc8438841b4b9709e540a0ee574e8aae94f4; statement text SHA-256 665a9831bf6d6a765382c67a0ebeb628d4f1f0c820c665c2b051c5bff61d7101. https://theoremdb.org/statement/?ref=P3448
@misc{theoremdb-problem-665a9831bf6d6a765382c67a0ebeb628d4f1f0c820c665c2b051c5bff61d7101,
title = {{Automorphisms with small centralizers in soluble groups}},
howpublished = {TheoremDB},
note = {Problem statement; statement identity tdbc1:5b4622eeb624440526d1ba14e8eddc8438841b4b9709e540a0ee574e8aae94f4; statement text SHA-256 665a9831bf6d6a765382c67a0ebeb628d4f1f0c820c665c2b051c5bff61d7101},
url = {https://theoremdb.org/statement/?ref=P3448}
}Plain text: Built Markdown snapshot
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