On Sims’ Presentation for Lyons’ Simple Group

Gollan, Holger W.; Havas, George

doi:10.1007/978-3-0348-8716-8_13

Cited by 4 publications

(4 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It was also demonstrated as a sequential coset enumeration by Havas and Sims [26]. That presentation was verified as producing Lyons's group by Gollan and Havas [25].…”

Section: Related Workmentioning

confidence: 83%

A parallel architecture for disk-based computing over the Baby Monster and other large finite simple groups

Robinson

Cooperman

2006

Proceedings of the 2006 International Symposium on Symbolic and Algebraic Computation

View full text Add to dashboard Cite

We outline a distributed, disk-based technique for computing over very large matrix groups. This technique is used to compute a permutation representation for the Baby Monster, a sporadic simple group that acts on 13,571,955,000 points. Its group order is approximately 4 × 10 33 . This is a landmark because it is 100 times larger than any previous construction of a permutation representation. By using the computed on-disk data structures, computation over the Baby Monster is now feasible using the distributed disks of a cluster. Our work allows researchers to use either a matrix, a permutation, or a word representation for computing over the Baby Monster where previously only a matrix representation was available. The methodology is demonstrated by using as a signature the image of a vector that is stabilized by the maximal subgroup. The technique extends to finite simple groups and to other groups, through other signatures.

show abstract

“…It was also demonstrated as a sequential coset enumeration by Havas and Sims [26]. That presentation was verified as producing Lyons's group by Gollan and Havas [25].…”

Section: Related Workmentioning

confidence: 83%

A parallel architecture for disk-based computing over the Baby Monster and other large finite simple groups

Robinson

Cooperman

2006

Proceedings of the 2006 International Symposium on Symbolic and Algebraic Computation

View full text Add to dashboard Cite

show abstract

“…It was also demonstrated as a sequential coset enumeration by Havas and Sims [27]. That presentation was verified as producing Lyons' group by Gollan and Havas [26].…”

Section: Previous Literaturementioning

confidence: 92%

Memory-based and disk-based algorithms for very high degree permutation groups

Cooperman

Robinson

2003

Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation

View full text Add to dashboard Cite

Group membership is a fundamental algorithm, upon which most other algorithms of computational group theory depend. Until now, group membership for permutation groups has been limited to ten million points or less. We extend the applicability of group membership algorithms to permutation groups acting on more than 100,000,000 points. As an example, we experimentally construct a group membership data structure for Thompson's group, acting on 143,127,000 points, in 36 minutes. More significantly, we require approximately 10 GB of RAM for the computation -even though a single permutation of Thompson's group already requires half a gigabyte of storage.In addition, we propose a disk-based group membership algorithm with the promise of extending group membership to well over one billion (1,000,000,000) points. Such a diskbased algorithm has formerly been impossible, due in part to the lack of a practical disk-based algorithm for multiplying and taking inverses of such large permutations. Random access to disk is prohibitively expensive. We demonstrate the first practical disk-based implementation of the basic permutation operations. We also propose a disk-based architecture for group membership data structures.

show abstract

“…Other existence proofs for Lyons' simple group can be found in Gollan and Havas (1999), using the original presentation of Sims (1972) and results from this paper, and in the unpublished manuscript that was announced in Sims (1972).…”

Section: Introductionmentioning

confidence: 93%

A New Existence Proof for Ly, the Sporadic Simple Group of R. Lyons

Gollan¹

2001

Journal of Symbolic Computation

Self Cite

View full text Add to dashboard Cite

This paper reports on a new and independent existence proof for the sporadic simple group Ly of Lyons, using only two permutations of degree 9 606 125, computed by Cooperman, Finkelstein, Tselman, and York. We will show that these two permutations generate a group G Ly, by first computing a base and strong generating set for G, and then checking the two hypotheses for Ly from Lyons' original paper. Moreover, this produces a new presentation for Ly.

show abstract

On Sims’ Presentation for Lyons’ Simple Group

Cited by 4 publications

References 8 publications

A parallel architecture for disk-based computing over the Baby Monster and other large finite simple groups

A parallel architecture for disk-based computing over the Baby Monster and other large finite simple groups

Memory-based and disk-based algorithms for very high degree permutation groups

A New Existence Proof for Ly, the Sporadic Simple Group of R. Lyons

Contact Info

Product

Resources

About