On the orders of primitive groups
type: Item
instance of: scholarly article
title: On the orders of primitive groups
zbMATH Open document ID: 1018.20002
DOI: 10.1016/S0021-8693(02)00646-4
author: Attila Maróti
published in: Journal of Algebra
publication date: 2003-02-12T00:00:00Z
review text: The author tries to find bounds to the order of a primitive permutation group in function of its degree. His main theorem states that, with exceptions the four Mathieu groups, a primitive permutation group of degree \(n\) has order at most \[ n\cdot\prod^{[\log_2n]-1}_{i=0}(n-2^i)<n^{[\log_2n]+1} \] or has a socle isomorphic to a direct power of some alternating group. This bound is sharp. The proof of this theorem uses the classification of finite simple groups. The author establishes several corollaries to this result. He proves that a primitive permutation group of degree \(n\) has order less than \(n^{\sqrt n}\) if it is not 3-transitive and has order less than \(50\cdot n^{\sqrt n}\) if it does not contain \(A_n\). This is a sharp version of a result of \textit{P. J. Cameron} [Bull. Lond. Math. Soc. 13, 1-22 (1981; Zbl 0463.20003)]. He also improves a result of \textit{C. E. Praeger} and \textit{J. Saxl} [Bull. Lond. Math. Soc. 12, 303-307 (1980; Zbl 0443.20001)] by showing that a primitive permutation group of degree \(n\) not containing \(A_n\) has order less than \(3^n\), or even \(2^n\) if \(n>24\). The author also gives a list of the 24 primitive permutation groups of degree \(n\) whose order is greater than \(2^{n-1}\).
reviewed by: Alice Devillers
Mathematics Subject Classification ID: 20B15
zbMATH DE Number: 1868105
zbMATH Keywords: order bounds
zbMATH Keywords: primitive permutation groups
zbMATH Keywords: Mathieu groups
zbMATH Keywords: alternating groups
MaRDI profile type: Publication
cites work: On the order of uniprimitive permutation groups
cites work: On the order of doubly transitive permutation groups
cites work: On the orders of primitive groups with restricted nonabelian composition factors
cites work: Finite Permutation Groups and Finite Simple Groups
cites work: Elementary methods in the study of the distribution of prime numbers
cites work: The Fitting subgroup of a linear solvable group
cites work: The primitive permutation groups of degree less than 1000
cites work: Q4882944
cites work: Representing the quotient groups of a finite permutation group
cites work: The projective characters of the symmetric groups that remain irreducible on subgroups
cites work: Q3996618
cites work: On minimal degrees and base sizes of primitive permutation groups
cites work: Upper bounds for the number of conjugacy classes of a finite group
cites work: Subgroup growth.
cites work: On the orders of Primitive Permutation Groups
cites work: On the orders of doubly transitive permutation groups, elementary estimates
cites work: Q4273611
cites work: Q5512231
full work available at URL: s0021-8693(02)00646-4
OpenAlex ID: W2090590511
Recommended article: On the orders of primitive linear P'-groups
Recommended article: Q3774055
Recommended article: Q4383924
Recommended article: On q-orders in primitive modular groups
Recommended article: On groups generated by elements of prime order
Recommended article: Q4544915
description: scientific article; zbMATH DE number 1868105