Another Existence and Uniqueness Proof for the Higman–Sims Simple Group

2005 ◽  
Vol 12 (03) ◽  
pp. 369-398
Author(s):  
Gerhard O. Michler ◽  
Andrea Previtali
Keyword(s):  
Automorphism Group ◽  
Simple Group ◽  
Existence And Uniqueness ◽  
Short Proof ◽  
Conjugacy Classes ◽  
Character Table ◽  
Sporadic Simple Group ◽  
Uniqueness Criterion ◽  
Generators And Relations ◽  
Uniqueness Proof

In this article, we give a short proof for the existence and uniqueness of the Higman–Sims sporadic simple group 𝖧𝖲 by means of the first author's algorithm [17] and uniqueness criterion [18], respectively. We realize 𝖧𝖲 as a subgroup of GL 22(11), and determine its automorphism group Aut (𝖧𝖲). We also give a presentation for Aut (𝖧𝖲) in terms of generators and relations. Furthermore, the character table of 𝖧𝖲 is determined and representatives of its conjugacy classes are given as short words in its generating matrices inside GL 22(11).

O'NAN GROUP UNIQUELY DETERMINED BY THE CENTRALIZER OF A 2-CENTRAL INVOLUTION

2007 ◽  
Vol 06 (01) ◽  
pp. 135-171 ◽  
Author(s):  
GERHARD O. MICHLER ◽  
ANDREA PREVITALI
Keyword(s):  
Simple Group ◽  
Existence And Uniqueness ◽  
Conjugacy Classes ◽  
Character Table ◽  
Permutation Representation ◽  
Defining Relations ◽  
New Methods ◽  
Uniqueness Proof ◽  
Janko Group

In this paper we give a self-contained existence and uniqueness proof for the sporadic O'Nan group ON by showing that it is uniquely determined up to isomorphism by the centralizer H of a 2-central involution z. We establish for such a simple group G a presentation in terms of generators and defining relations and a faithful permutation representation of degree 2.624.832 with a uniquely determined stabilizer isomorphic to the small sporadic Janko group J1. We also calculate its character table by new methods and determine a system of representatives of the conjugacy classes of G.

NATURAL EXISTENCE PROOF FOR LYONS SIMPLE GROUP

2003 ◽  
Vol 02 (03) ◽  
pp. 277-315
Author(s):  
GERHARD O. MICHLER ◽  
MICHAEL WELLER ◽  
KATSUSHI WAKI
Keyword(s):  
Automorphism Group ◽  
Simple Group ◽  
Conjugacy Classes ◽  
Existence Proof ◽  
Sporadic Simple Group ◽  
Character Tables ◽  
The Given ◽  
Fold Cover

In this article we give a self-contained existence proof for Lyons' sporadic simple group G by application of the first author's algorithm [18] to the given centralizer H ≅ 2A11 of a 2-central involution of G. It also yields four matrix generators of G inside GL 111 (5) which are given in Appendix A. From the subgroup U ≅ (3 × 2A8) : 2 of H ≅ 2A11, we construct a subgroup E of G which is isomorphic to the 3-fold cover 3McL: 2 of the automorphism group of the McLaughlin group McL. Furthermore, the character tables of E ≅ 3McL : 2 and G are determined and representatives of their conjugacy classes are given as short words in their generating matrices.

The Fischer–Clifford Matrices of a Maximal Subgroup of the Sporadic Simple Group of Held

2007 ◽  
Vol 14 (01) ◽  
pp. 135-142 ◽  
Author(s):  
Faryad Ali
Keyword(s):  
Simple Group ◽  
Computer Algebra ◽  
Local Structure ◽  
Computer Algebra System ◽  
Conjugacy Classes ◽  
Character Table ◽  
Maximal Subgroups ◽  
Split Extension ◽  
Sporadic Simple Group ◽  
Held Group

The Held group He discovered by Held [10] is a sporadic simple group of order 4030387200 = 210.33.52.73.17. The group He has 11 conjugacy classes of maximal subgroups as determined by Butler [5] and listed in the 𝔸𝕋𝕃𝔸𝕊. Held himself determined much of the local structure of He as well as the conjugacy classes of its elements. Thompson calculated the character table of He . In the present paper, we determine the Fischer–Clifford matrices and hence compute the character table of the non-split extension 3·S7, which is a maximal subgroups of He of index 226560 using the technique of Fischer–Clifford matrices. Most of the computations were carried out with the aid of the computer algebra system 𝔾𝔸ℙ.

Another existence and uniqueness proof for McLaughlin's simple group

2003 ◽  
Vol 6 (4) ◽  
Author(s):  
Mathias Kratzer ◽  
Wolfgang Lempken ◽  
Gerhard O. Michler ◽  
Katsushi Waki
Keyword(s):  
Simple Group ◽  
Existence And Uniqueness ◽  
Uniqueness Proof

scholarly journals Some subgroups of the Thompson group

1988 ◽  
Vol 44 (1) ◽  
pp. 17-32 ◽  
Author(s):  
Robert A. Wilson
Keyword(s):  
Simple Group ◽  
Conjugacy Classes ◽  
Simple Groups ◽  
Sporadic Simple Group ◽  
Non Local ◽  
Thompson Group

AbstractWe determine all conjugacy classes of maximal local subgroups of Thompson's sporadic simple group, and all maximal non-local subgroups except those with socle isomorphic to one of five particular small simple groups.

Another Uniqueness Proof of the Dickson Simple Group G2(3)

2011 ◽  
Vol 18 (02) ◽  
pp. 181-210
Author(s):  
Gerhard O. Michler ◽  
Lizhong Wang
Keyword(s):  
Simple Group ◽  
Cyclic Group ◽  
Unique Extension ◽  
Uniqueness Criterion ◽  
Central Product ◽  
Uniqueness Proof

In this article, we give a self-contained uniqueness proof for the Dickson simple group G=G2(3) using the first author's uniqueness criterion. The uniqueness proof for G2(3) was first given by Janko. His proof depends on Thompson's deep and technical characterization of G2(3). Let H′ be the amalgamated central product of SL 2(3) with itself. Then there is a unique extension H of H′ by a cyclic group of order 2 such that H has a center of order 2 and both factors SL 2(3) are normal in H. We prove that any simple group G having a 2-central involution z with centralizer CG(z)≅ H is isomorphic to G2(3).

scholarly journals Corrigendum

1988 ◽  
Vol 103 (2) ◽  
pp. 383-383
Author(s):  
Peter B. Kleidman ◽  
Robert A. Wilson
Keyword(s):  
Automorphism Group ◽  
Simple Group ◽  
Maximal Subgroups ◽  
Sporadic Simple Group ◽  
Computer Calculations

Volume 102 (1987), 17–23‘The maximal subgroups of Fi22’We reported on some computer calculations which we used to complete the enumeration of the maximal subgroups of the sporadic simple group Fi22 and of its automorphism group Fi22:2. Unfortunately there was an error in these calculations. We have therefore repeated all the calculations, incorporating much more thorough checking routines.

scholarly journals Local Fusion Graphs and Sporadic Simple Groups

10.37236/4298 ◽  
2015 ◽  
Vol 22 (3) ◽  
Author(s):  
John Ballantyne ◽  
Peter Rowley
Keyword(s):  
Automorphism Group ◽  
Conjugacy Class ◽  
Simple Group ◽  
Simple Groups ◽  
Sporadic Simple Group ◽  
Odd Order ◽  
Vertex Set ◽  
Sporadic Simple Groups

For a group $G$ with $G$-conjugacy class of involutions $X$, the local fusion graph $\mathcal{F}(G,X)$ has $X$ as its vertex set, with distinct vertices $x$ and $y$ joined by an edge if, and only if, the product $xy$ has odd order. Here we show that, with only three possible exceptions, for all pairs $(G,X)$ with $G$ a sporadic simple group or the automorphism group of a sporadic simple group, $\mathcal{F}(G,X)$ has diameter $2$.

ABSTRACT REGULAR POLYTOPES FOR THE O'NAN GROUP

2014 ◽  
Vol 24 (01) ◽  
pp. 59-68 ◽  
Author(s):  
THOMAS CONNOR ◽  
DIMITRI LEEMANS ◽  
MARK MIXER
Keyword(s):  
Automorphism Group ◽  
Simple Group ◽  
Regular Polytopes ◽  
Sporadic Simple Group ◽  
Regular Polytope ◽  
Abstract Regular Polytope

In this paper, we consider how the O'Nan sporadic simple group acts as the automorphism group of an abstract regular polytope. In particular, we prove that there is no regular polytope of rank at least five with automorphism group isomorphic to O′N. Moreover, we classify all rank four regular polytopes having O′N as their automorphism group.

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