2-Generation of finite simple groups and some related topics

Martino, L. Di; Tamburini, M. C.

doi:10.1007/978-94-011-3382-1_8

Cited by 29 publications

(12 citation statements)

References 66 publications

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“…It was also known that PSL 3 .q/ is a Hurwitz group only when q D 2, and that 13 of the other sporadic finite simple groups are not Hurwitz. Shortly afterwards, related information was provided also by Lino Di Martino and Chiara Tamburini in a nice survey paper about generating sets for finite simple groups [13].…”

Section: Simple and Linear Hurwitz Groupsmentioning

confidence: 99%

An update on Hurwitz groups

Conder¹

2010

Groups – Complexity – Cryptology

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A Hurwitz group is any non-trivial finite quotient of the .2; 3; 7/ triangle group, that is, any non-trivial finite group generated by elements x and y satisfying x 2 D y 3 D .xy/ 7 D 1. Every such group G is the conformal automorphism group of some compact Riemann surface of genus g > 1, with the property that jGj D 84.g 1/, which is the maximum possible order for given genus g. This paper provides an update on what is known about Hurwitz groups and related matters, following up the author's brief survey in Bull. Amer. Math. Soc. 23 (1990).

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Section: Simple and Linear Hurwitz Groupsmentioning

confidence: 99%

An update on Hurwitz groups

Conder¹

2010

Groups – Complexity – Cryptology

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“…In this section we compute the rank for each conjugacy class of O'N. It is well known that every sporadic simple group can be generated by three involutions (see [12]). In the following lemma we prove that the group O'N can be generated by three conjugate involutions a, b, c ∈ 2A such that abc ∈ 31A.…”

Section: Ranks Of the O'nan Group O'nmentioning

confidence: 99%

On the ranks of O’N and Ly

Ali

2007

Discrete Applied Mathematics

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Let G be a finite group and X a conjugacy class of G. We denote rank(G : X) to be the minimum number of elements of X generating G. In the present article, we determine the ranks for two sporadic simple groups, namely the O'Nan sporadic simple group O'N and the Lyons sporadic simple group, Ly. Computations were carried with the aid of the computer algebra system GAP [The GAP Group, GAP-Groups, Algorithms and Programming, Version 4.3, Aachen, St Andrews, 2003, http://www.gap-system.org ].

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“…For example, besides the previously raised question of whether we can do better than 6 involutions, we can also ask whether there exists a constant C, perhaps such that C = C(g), so that every element of Mod g,b can be written as a product of at most C torsion elements. 4 Furthermore, what kinds of relations exist amongst the torsion (or involution) generators? In particular, what kind of Coxeter groups arise in the context of Corollary 5?…”

Section: Final Remarksmentioning

confidence: 99%

“…The problem of finding small generating sets, torsion generating sets, and generating sets of involutions (i.e., elements of order two) is a classical one, and has been studied extensively, especially for finite groups (see, e.g., [4] for a survey). In 1971, Maclachlan [18] proved that Mod g,0 is generated by torsion elements and deduced from this that moduli space M g is simply-connected as a topological space.…”

Section: Introductionmentioning

confidence: 99%

Every mapping class group is generated by 6 involutions

Brendle

Farb

2004

Journal of Algebra

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Let Mod g,b denote the mapping class group of a surface of genus g with b punctures. Luo asked in [Torsion elements in the mapping class group of a surface, math.GT/0004048, v1 8Apr2000] if there is a universal upper bound, independent of genus, for the number of torsion elements needed to generate Mod g,b . We answer Luo's question by proving that 3 torsion elements suffice to generate Mod g,0 . We also prove the more delicate result that there is an upper bound, independent of genus, not only for the number of torsion elements needed to generate Mod g,b but also for the order of those elements. In particular, our main result is that 6 involutions (i.e., orientation-preserving diffeomorphisms of order two) suffice to generate Mod g,b for every genus g 3, b = 0 and g 4, b = 1.

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2-Generation of finite simple groups and some related topics

Cited by 29 publications

References 66 publications

An update on Hurwitz groups

An update on Hurwitz groups

On the ranks of O’N and Ly

Every mapping class group is generated by 6 involutions

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