Arithmetical conditions on invariant Sylow numbers
type: Item
instance of: scholarly article
title: Arithmetical conditions on invariant Sylow numbers
zbMATH Open document ID: 1414.20009
author: Antonio Beltrán
author: Changguo Shao
published in: Mediterranean Journal of Mathematics
publication date: 2018-06-01T00:00:00Z
review text: Let $G$ be a finite group and let $A$ be a subgroup of the automorphism group of $G$. Let $\pi(G)$ be the set of all prime divisors of $|G|$. For every $p\in \pi(G)$, define $ \nu_p^A(G)$ to be the number of Sylow $p$-subgroups of $G$ which are invariant under the action of $A$, and let $\mathrm{sn}_A(G)$ be the set of all values of $\nu_p^A(G)$ as $p$ runs through all primes in $\pi(G)$. The main result of the paper consists of results characterizing when $G$ is solvable under the condition that $(|A|,|G|)=1$ and $\mathrm{sn}_A(G)$ contains only prime powers, or only squarefree numbers, or both.
reviewed by: Thomas Michael Keller
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D15
zbMATH DE Number: 6879889
zbMATH Keywords: finite groups
zbMATH Keywords: Sylow numbers
zbMATH Keywords: solvable groups
zbMATH Keywords: coprime action
MaRDI profile type: Publication
full work available at URL: s00009-018-1094-z
OpenAlex ID: W2791037036
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cites work: Actions with nilpotent fixed point subgroup
cites work: Number of Sylow subgroups and \(p\)-nilpotence of finite groups
cites work: Q3684278
cites work: Endliche Gruppen I
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cites work: The theory of finite groups. An introduction.
cites work: Groups with two Sylow numbers are the product of two nilpotent Hall subgroups.
cites work: Sylow numbers of finite groups
DOI: 10.1007/S00009-018-1094-Z
Wikidata QID: Q130140450
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description: scientific article; zbMATH DE number 6879889