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 A Characterization of the Finite Simple Groups PSp(4, q), G2(q), I

1969 ◽  
Vol 36 ◽  
pp. 143-184 ◽  
Author(s):  
Paul Fong ◽  
W.J. Wong
Keyword(s):  
Simple Group ◽  
Finite Simple Group ◽  
Symplectic Group ◽  
Prime Power ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Central Product ◽  
Twisted Group ◽  
Index 2

Suppose that G is the projective symplectic group PSp(4, q), the Dickson group G2(q)> or the Steinberg “triality-twisted” group where q is an odd prime power. Then G is a finite simple group, and G contains an involution j such that the centralizer C(j) in G has a subgroup of index 2 which contains j and which is the central product of two groups isomorphic with SL(2,q1) and SL(2,q2) for suitable ql q2. We wish to show that conversely the only finite simple groups containing an involution with this property are the groups PSp(4,q), G2(q)9. In this first paper we shall prove the following result.

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).

scholarly journals A characterization of the simple group 2D4(2)

1981 ◽  
Vol 69 (2) ◽  
pp. 467-482 ◽  
Author(s):  
A.R Prince
Keyword(s):  
Simple Group

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 A CHARACTERIZATION OF PSL(4,p) BY SOME CHARACTER DEGREE

2019 ◽  
pp. 679
Author(s):  
Younes Rezayi ◽  
Ali Iranmanesh
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Irreducible Character ◽  
Character Degree ◽  
A Finite Group ◽  
Irreducible Character Degree

‎Let G be a finite group and cd(G) be the set of irreducible character degree of G‎. ‎In this paper we prove that if  p is a prime number‎, ‎then the simple group PSL(4,p) is uniquely determined by its order and some its character degrees‎. 

scholarly journals Face Numbers and Nongeneric Initial Ideals

10.37236/1882 ◽  
2006 ◽  
Vol 11 (2) ◽  
Author(s):  
Eric Babson ◽  
Isabella Novik
Keyword(s):  
Cyclic Group ◽  
Simple Proof ◽  
Prime Order ◽  
Necessary Conditions ◽  
Betti Numbers ◽  
Proper Action ◽  
Initial Ideals ◽  
The Face ◽  
Algebraic Shifting

Certain necessary conditions on the face numbers and Betti numbers of simplicial complexes endowed with a proper action of a prime order cyclic group are established. A notion of colored algebraic shifting is defined and its properties are studied. As an application a new simple proof of the characterization of the flag face numbers of balanced Cohen-Macaulay complexes originally due to Stanley (necessity) and Björner, Frankl, and Stanley (sufficiency) is given. The necessity portion of their result is generalized to certain conditions on the face numbers and Betti numbers of balanced Buchsbaum complexes.

scholarly journals Automorphisms of Regular Wreath Product -Groups

2009 ◽  
Vol 2009 ◽  
pp. 1-12 ◽  
Author(s):  
Jeffrey M. Riedl
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Representation Theory ◽  
Wreath Product ◽  
Cyclic Group ◽  
Product Group ◽  
Finite Cyclic Group ◽  
Arbitrary Finite Group

We present a useful new characterization of the automorphisms of the regular wreath product group of a finite cyclic -group by a finite cyclic -group, for any prime , and we discuss an application. We also present a short new proof, based on representation theory, for determining the order of the automorphism group Aut(), where is the regular wreath product of a finite cyclic -group by an arbitrary finite -group.

Quasirecognition of E6(q) by the orders of maximal abelian subgroups

2018 ◽  
Vol 17 (07) ◽  
pp. 1850122 ◽  
Author(s):  
Zahra Momen ◽  
Behrooz Khosravi
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Simple Group ◽  
Outer Automorphism ◽  
Composition Factor ◽  
Outer Automorphism Group ◽  
Abelian Subgroups ◽  
A Finite Group ◽  
Nonabelian Composition Factor

In [Li and Chen, A new characterization of the simple group [Formula: see text], Sib. Math. J. 53(2) (2012) 213–247.], it is proved that the simple group [Formula: see text] is uniquely determined by the set of orders of its maximal abelian subgroups. Also in [Momen and Khosravi, Groups with the same orders of maximal abelian subgroups as [Formula: see text], Monatsh. Math. 174 (2013) 285–303], the authors proved that if [Formula: see text], where [Formula: see text] is not a Mersenne prime, then every finite group with the same orders of maximal abelian subgroups as [Formula: see text], is isomorphic to [Formula: see text] or an extension of [Formula: see text] by a subgroup of the outer automorphism group of [Formula: see text]. In this paper, we prove that if [Formula: see text] is a finite group with the same orders of maximal abelian subgroups as [Formula: see text], then [Formula: see text] has a unique nonabelian composition factor which is isomorphic to [Formula: see text].

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