On the Construction of the Finite Simple Groups with a Given Centralizer of a 2-Central Involution

Michler, Gerhard O.

doi:10.1006/jabr.2000.8549

Cited by 7 publications

(19 citation statements)

References 12 publications

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“…As results we get J 2 as a subgroup of GL 14 (11) and the triple cover 3 J 3 as a subgroup of GL 18 (31).…”

Section: Mathematics Subject Classificationmentioning

confidence: 87%

“…• It is uniform -not only because we prove the existence of J 2 and J 3 simultaneously but also because we employ a quite general construction method [11] which is not restricted to sporadic simple groups. (2000): Primary 20D08; Secondary 20C34.…”

Section: (J1) the Centre Z(s) Of A Sylow-2-subgroup S Of G Is Cyclicmentioning

confidence: 99%

“…Let mfchar C (H ) denote the set of all multiplicity-free faithful characters of H, and let fchar C (E) denote the set of all faithful characters of E. Going back to a definition given in [11] the elements of the finite set In order to keep computations as short as possible (and, by the way, results as nice as possible) we have to deviate a little bit from the original Step 4 of Michler's algorithm now: For each compatible pair ( χ , ϑ) to process we choose the smallest prime p such that p does not divide the product |H| · |E| and such that the prime field GF( p) is a splitting field for all irreducible constituents occurring in χ and ϑ. A quick look at the Character Tables 4.1 and 4.2 tells us to work over GF (11) for the compatible pairs (i), (ii) of Proposition 4.4(b) and to select GF(31) for the rest.…”

Section: Classmentioning

confidence: 99%

“…

In this article both sporadic Janko-groups J 2 and J 3 are constructed from their common involution centralizer H ∼ = 2 1+4 : A 5 within one single run of Michler's deterministic algorithm described in [11]. At the beginning we choose the -in some sense -most natural point to start from, and in the end we realize that Michler's algorithm does not necessary lead us to a simple group but sometimes to a covering group of a simple group.

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confidence: 99%

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Uniform and natural existence proofs for Janko‚s sporadic groups J2 and J3

Kratzer¹

2002

Arch. Math.

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“…As results we get J 2 as a subgroup of GL 14 (11) and the triple cover 3 J 3 as a subgroup of GL 18 (31).…”

Section: Mathematics Subject Classificationmentioning

confidence: 87%

Section: (J1) the Centre Z(s) Of A Sylow-2-subgroup S Of G Is Cyclicmentioning

confidence: 99%

Section: Classmentioning

confidence: 99%

“…

…”

mentioning

confidence: 99%

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Uniform and natural existence proofs for Janko‚s sporadic groups J2 and J3

Kratzer¹

2002

Arch. Math.

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“…The approach described and advocated by Michler in this book is remarkable in that it offers a uniform method that can, in principle, be used to prove computationally the existence and uniqueness of almost all of the sporadic groups. Indeed, according to one of his earlier papers [15], the smallest Mathieu group M 11 is the only sporadic group for which this method could not work. Furthermore, if, as seems likely at present, a few of the results of this type are going to remain dependent on computer calculations, then it is highly desirable for them to be checked independently using a variety of different approaches.…”

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confidence: 99%

Book Review: Theory of finite simple groups

Holt¹

2008

Bull. Amer. Math. Soc.

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A group is called simple if it has more than one element and if its only normal subgroups are the trivial subgroup and the whole group. In other words, it is a group with precisely two normal subgroups. It is an elementary exercise to show that the abelian simple groups are precisely the finite cyclic groups of prime order. The nonabelian simple groups, on the other hand, are far more complicated.There are many interesting examples of infinite simple groups, including examples with finite presentations, and the so-called "Tarski Monsters", which have the highly counterintuitive property that their only proper nontrivial subgroups have order p for a fixed (large) prime number p. But there is little prospect of any kind of complete description of the infinite simple groups.The finite simple groups are of more immediate interest to many mathematicians, firstly because they are more tractable, and secondly because, in their role as composition factors, they form the building blocks of all finite groups. The possibility that they might eventually be completely classified was first raised by Otto Hölder in 1892 and, around the turn of the 20th century, significant progress in the study of finite simple groups was made by William Burnside and Georg Frobenius in particular.There followed a relatively inactive period, and work on this topic got underway again immediately after the Second World War, with the descriptions by Chevalley, Steinberg, Ree, and others of the simple groups of Lie type, and the classification theorems of Brauer, Suzuki, and Wall. But the contemporary study of finite simple groups really began in earnest in the early 1960's, with the proof by Walter Feit and John Thompson that all finite nonabelian simple groups have even order [11], a result that had been conjectured much earlier by Burnside. Its exceptionally long and difficult proof set the scene for the following two decades of intense activity, culminating with the announcement in 1981 of the final outcome:

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Thompson's sporadic group uniquely determined by the centralizer of a 2-central involution

Weller¹,

Michler²,

Previtali³

2006

Journal of Algebra

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In this article we give a self contained existence and uniqueness proof for that sporadic simple group which was discovered by J.G. Thompson [J.G. Thompson, A simple subgroup of E 8 (3), in: N. Iwahori (Ed.), Finite Groups, Japan Soc. Promotion Science, Tokyo, 1976, pp. 113-116]. The centralizer H of a 2-central involution of that group has been described in terms of generators and relations by Havas, Soicher and Wilson in [G. Havas, L.H. Soicher, R.A. Wilson, A presentation for the Thompson sporadic simple group, in: W.M. Kantor, A. Seress (Eds.), Groups and Computation III, de Gruyter, Berlin, 2001, pp. 192-200]. Taking this presentation as the input of the second author's algorithm [G.O. Michler, On the construction of the finite simple groups with a given centralizer of a 2-central involution, J. Algebra 234 (2000) 668-693] we construct a simple subgroup G of GL 248 (11) which has a 2-central involution z whose centralizer is isomorphic to H . In order to prove that the order of G is 2 15 · 3 10 · 5 3 · 7 2 · 13 · 19 · 31, a faithful 143,127,000-dimensional permutation representation of this matrix group has been constructed on a supercomputer. In the second part of this article it is shown that any simple group G having a 2-central involution z with centralizer C G (z) ∼ = H is isomorphic to G. We construct its concrete character table as well.

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On the Construction of the Finite Simple Groups with a Given Centralizer of a 2-Central Involution

Cited by 7 publications

References 12 publications

Uniform and natural existence proofs for Janko‚s sporadic groups J2 and J3

Uniform and natural existence proofs for Janko‚s sporadic groups J2 and J3

Book Review: Theory of finite simple groups

Thompson's sporadic group uniquely determined by the centralizer of a 2-central involution

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