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

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.

Another Existence and Uniqueness Proof of the Tits Group

2008 ◽  
Vol 15 (02) ◽  
pp. 241-278
Author(s):  
Gerhard O. Michler ◽  
Lizhong Wang
Keyword(s):  
Finite Groups ◽  
Existence And Uniqueness ◽  
Frobenius Group ◽  
Nilpotency Class ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Uniqueness Proof ◽  
Groups Of Lie Type ◽  
Uniqueness Criteria ◽  
Tits Group

In this article we give a self-contained existence and uniqueness proof for the Tits simple group T. Parrott gave the first uniqueness proof. Whereas Tits' and Parrott's results employ the theory of finite groups of Lie type, our existence and uniqueness proof follows from the general algorithms and uniqueness criteria for abstract finite simple groups described in the first author's book [11]. All we need from the previous papers is the fact that the centralizer H of the Tits group T is an extension of a 2-group J with order 29 and nilpotency class 3 by a Frobenius group F of order 20 such that the center Z(H) has order 2 and any Sylow 5-subgroup Q of H has a centralizer CJ(Q) ≤ Z(H).

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).

A Note on the Existence of a Solution of the Falkner-Skan Equation

1970 ◽  
Vol 13 (1) ◽  
pp. 125-127 ◽  
Author(s):  
K. Kuen Tam
Keyword(s):  
Fixed Point ◽  
Phase Space ◽  
Fixed Point Theorem ◽  
Existence And Uniqueness ◽  
Three Dimensional ◽  
Existence Proof ◽  
Existence Of A Solution ◽  
Dimensional Phase Space ◽  
Image Position ◽  
Uniqueness Proof

We are concerned with the existence proof of solution of the Falkner-Skan equation1subject to boundary conditions2The first existence and uniqueness proof based on a fixed point theorem was given by Weyl [4] in 1942, with the added assumption that f' > 0. In 1960, Coppel [1] proved the existence (and uniqueness with the assumption 0 < f' < 1) by considering trajectories in the three-dimensional phase space.

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