On the uniqueness of the Harada-Norton group
type: Item
instance of: scholarly article
title: On the uniqueness of the Harada-Norton group
zbMATH Open document ID: 0780.20010
DOI: 10.1016/0021-8693(92)90115-3
author: Yoav Segev
published in: Journal of Algebra
publication date: 1993-05-16T00:00:00Z
review text: By definition, \(2\cdot HS\) denotes the non-split extension of the group of order 2 by the Higman-Sims sporadic finite simple group and \((2\cdot HS):2\) denotes the split extension of \((2\cdot HS)\) by a group of order 2 with center of order 2. Also \(2^{1+8}_ +\) denotes an extraspecial group of order \(2^ 9\) and of type + and a group \(H\) is of \(2^{1+8}_ +\cdot(A_ 5\text{ wr }2)\)-type if \(H\) is 2-constrained and a nonsplit extension of \(2^{1+8}_ +\) by \((A_ 5\text{ wr }2)\). A finite group \(G\) is said to be of \(F_ 5\) type (where \(F_ 5\) denotes a Harada- Norton simple group) if \(G\) contains involutions \(a\), \(z\) such that \(C_ G(a)\cong (2\cdot HS):2\) and \(C_ G(z)\) is of \(2^{1+8}_ +\cdot(A_ 5\text{ wr }2)\)-type. The existence of the Monster implies the existence of a group of \(F_ 5\)-type. The main result of this paper establishes the uniqueness of the Harada-Norten simple group \(F_ 5\): Main Theorem: Let \(G\) be a finite group of \(F_ 5\)-type. Then \(G\) is unique up to isomorphism. Let \(G\) be a finite group of \(F_ 5\)-type. Consider the graph \(\Gamma\): vertex set is \(a^ G\) and \(x,y\in a^ G\) form an edge if \(xy \in a^ G\). The main theorem is established by demonstrating the uniqueness of \(\Gamma\) and that \(G=[\text{Aut}(\Gamma),\text{Aut}(\Gamma)]\).
Mathematics Subject Classification ID: 20D08
Mathematics Subject Classification ID: 05C25
zbMATH DE Number: 165838
zbMATH Keywords: Higman-Sims sporadic finite simple group
zbMATH Keywords: split extension
zbMATH Keywords: extraspecial group
zbMATH Keywords: Harada-Norton simple group
zbMATH Keywords: uniqueness
zbMATH Keywords: finite group of \(F_ 5\)- type
zbMATH Keywords: graph
reviewed by: Morton E. Harris
MaRDI profile type: Publication
full work available at URL: 0021-8693(92)90115-3
OpenAlex ID: W2009008394
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description: scientific article; zbMATH DE number 165838