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.

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).

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

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.

A NOTE ON THE NUMBER OF ZEROS IN THE COLUMNS OF THE CHARACTER TABLE OF A GROUP

2012 ◽  
Vol 12 (02) ◽  
pp. 1250150 ◽  
Author(s):  
JINSHAN ZHANG ◽  
ZHENCAI SHEN ◽  
SHULIN WU
Keyword(s):  
Finite Groups ◽  
Irreducible Character ◽  
Conjugacy Classes ◽  
Character Table ◽  
Number Of Zeros

The finite groups in which every irreducible character vanishes on at most three conjugacy classes were characterized [J. Group Theory13 (2010) 799–819]. Dually, we investigate the finite groups whose columns contain a small number of zeros in the character table.

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.

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