The rank 3 permutation representations of finite groups of type \(G_ 2\)
type: Item
instance of: scholarly article
title: The rank 3 permutation representations of finite groups of type \(G_ 2\)
zbMATH Open document ID: 0534.20002
DOI: 10.1016/0021-8693(83)90145-X
author: Shengming Shi
published in: Journal of Algebra
publication date: 1983-01-01T00:00:00Z
review text: A group \(G^*\) is said to be of type \(G_ 2\) provided \(G_ 2(q)\leq G^*\leq Aut(G_ 2(q)),\) where \(q=p^ f\), p is a prime. In this paper the author proves the following Main Theorem: There exists no faithful primitive permutation representation of rank 3 of \(G^*\) with \(p\geq 3\), except when \(G^*=G_ 2(3)\) and the degrees of such permutation representations are equal to 351. The last section of the paper contains two examples which are inequivalent primitive permutation representations of rank 3 of \(G_ 2(3)\). The proof of the main theorem depends on some properties of complex characters of finite groups of Lie type \(G_ 2(q)\).
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20D06
zbMATH DE Number: 3847640
zbMATH Keywords: inequivalent representations
zbMATH Keywords: faithful primitive permutation representation of rank 3
zbMATH Keywords: complex characters
zbMATH Keywords: finite groups of Lie type
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3847640