Noncoprime action of a cyclic group

Let \(A\) be a finite nilpotent group acting fixed point freely on the finite (solvable) group \(G\) by automorphisms. It is conjectured that the nilpotent length of \(G\) is bounded above by \(\ell(A)\), the number of primes dividing the order of \(A\) counted with multiplicities. In the present pa...

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Bibliographic Details
Published in:arXiv.org 2024-02
Main Authors: Ercan, Gülin, İsmail \c{S} Güloğlu
Format: Article
Language:English
Subjects:
Automorphisms
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Summary:Let \(A\) be a finite nilpotent group acting fixed point freely on the finite (solvable) group \(G\) by automorphisms. It is conjectured that the nilpotent length of \(G\) is bounded above by \(\ell(A)\), the number of primes dividing the order of \(A\) counted with multiplicities. In the present paper we consider the case \(A\) is cyclic and obtain that the nilpotent length of \(G\) is at most \(2\ell(A)\) if \(|G|\) is odd. More generally we prove that the nilpotent length of \(G\) is at most \(2\ell(A)+ \mathbf{c}(G;A)\) when \(G\) is of odd order and \(A\) normalizes a Sylow system of \(G\) where \(\mathbf{c}(G;A)\) denotes the number of trivial \(A\)-modules appearing in an \(A\)-composition series of \(G\).
ISSN:2331-8422