Minimal permutation representation of the simple group \(F_ 5\)
type: Item
instance of: scholarly article
title: Minimal permutation representation of the simple group \(F_ 5\)
zbMATH Open document ID: 0659.20015
DOI: 10.1007/BF02009329
author: N. P. Mazurova
author: V. D. Mazurov
published in: Algebra and Logic
publication date: 1987-01-01T00:00:00Z
review text: Let \(F_ 5\) be the finite simple group of Harada of order \(2^{14}\cdot 3^ 6\cdot 5^ 6\cdot 7\cdot 11\cdot 19\). It contains a subgroup H which is isomorphic to the alternating group \(A_{12}\). The authors prove that H is of minimal possible index in \(F_ 5\) and study the permutation representation of \(F_ 5\) on \(F_ 5/H\). Namely, three theorems are proved: Theorem 1. The minimal index of a proper subgroup in \(F_ 5\) is \(1,140,000=| F_ 5:H|\). Each subgroup of this index in \(F_ 5\) is conjugate to H. - The permutation representation of \(F_ 5\) on cosets \(F_ 5/H\) has rank 12 (Theorem 2; it also gives subdegrees and stabilizers of two points). Theorem 3. \(F_ 5\neq HA\) for every proper subgroup A in \(F_ 5\). In particular, \(F_ 5\) does not contain wide subgroups.
Mathematics Subject Classification ID: 20D08
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 4077536
zbMATH Keywords: \(F_ 5\)
zbMATH Keywords: finite simple group of Harada
zbMATH Keywords: permutation representation
zbMATH Keywords: minimal index of a proper subgroup
reviewed by: S. A. Syskin
MaRDI profile type: Publication
cites work: Finite simple non-Abelian groups
cites work: Subgroups of large finite groups and the linear optimization problem
cites work: The Nonabelian Simple Groups G, | G | < 10 6 -Maximal Subgroups
cites work: The non-abelian simple groups g, |g| < 106— character tables
cites work: Maximal subgroups of the Harada-Norton group
cites work: Q3684278
full work available at URL: https://doi.org/10.1007/bf02009329
OpenAlex ID: W2069572098
description: scientific article; zbMATH DE number 4077536