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Nice Efficient Presentions for all Small Simple Groups and their Covers

Colin M. Campbell, George Havas, Colin Ramsay, Edmund F. Robertson
2004 LMS Journal of Computation and Mathematics  
Substantial use is made of systems for computational group theory and, in partic-ular, of computer implementations of coset enumeration.  ...  The paper concludes with a selection of unsolved problems about efficient presentations for simple groups and their covers.  ...  In this case, since there is only one canonic presentation of this length, we find the best coset enumeration for a shortest presentation using the method of [17] applied to all different two-generator  ... 
doi:10.1112/s1461157000001121 fatcat:6iex26f5ovbdledkvbzvs6aacu

A Reidemeister-Schreier Program [chapter]

George Havas
1974 Lecture notes in mathematics  
The Reidemeister-Schreier method yields a presentation for a subgroup H of a group G when H is of finite index in G and G is finitely presented.  ...  In particular, this is done by using a canonical form for the relators, by eliminating redundant generators and by attempting relator simplification.  ...  The reason for this is that the coset It is also interesting to consider the nature of the presentation yielded by the original Reidemeister-Schreier method and the nature of the presentation obtained  ... 
doi:10.1007/978-3-662-21571-5_32 fatcat:toihirwkbjbqjkf52seeuu4xea

On the Efficiency of the Simple Groups of Order Less Than a Million and Their Covers

Colin M. Campbell, George Havas, Colin Ramsay, Edmund F. Robertson
2007 Experimental Mathematics  
Our results are based on major amounts of computation. We make substantial use of systems for computational group theory and, in particular, of computer implementations of coset enumeration.  ...  In a previous paper we produced nice efficient presentations for all small simple groups and for their covering groups.  ...  Acknowledgements The second and third authors were partially supported by the Australian Research Council.  ... 
doi:10.1080/10586458.2007.10129006 fatcat:l43stkh2gjhfrfwms3tofcqc7a

All simple groups with order from 1 million to 5 million are efficient

Edmund F. Robertson, George Havas, Colin Ramsay, Colin M. Campbell
2014 International Journal of Group Theory  
We make substantial use of systems for computational group theory and, in particular, of computer implementations of coset enumeration to find and verify our presentations.  ...  Here we show that all simple groups with order between 1 million and 5 million are efficient by giving efficient presentations for all of them.  ...  Acknowledgments The second and third authors were partially supported by the Australian Research Council.  ... 
doaj:23c9f13de3ff4d9ea77aed5793fe89c9 fatcat:sb5jlzjpmfba3mluyti4nfrv7i

Applications of computational tools for finitely presented groups [article]

George Havas, Edmund F. Robertson
1994 arXiv   pre-print
A general computational approach for investigating finitely presented groups by way of quotients and subgroups is described and examples are presented.  ...  Computer based techniques for recognizing finitely presented groups are quite powerful. Tools available for this purpose are outlined.  ...  For finitely presented abelian groups a canonical form can be obtained by using algorithms for Smith normal form computation for integer matrices.  ... 
arXiv:math/9406207v1 fatcat:rothr6tagrdtpa3v65mjbcafpy

Double-coset enumeration algorithm for symmetrically generated groups

Mohamed Sayed
2005 International Journal of Mathematics and Mathematical Sciences  
A double-coset enumeration algorithm for groups generated by symmetric sets of involutions together with its computer implementation is described.  ...  In this paper, we present a double-coset enumeration algorithm which is specially developed for groups symmetrically generated by involutions.  ...  ( 0) 1( 0) 1( 0) 1( 0) Conclusion The present double-coset enumeration algorithm is different from the existing coset enumeration techniques in the way it handles the elements of the group and for telling  ... 
doi:10.1155/ijmms.2005.699 fatcat:t7dkd5b2jfcktoxktaakjf7cee

Algorithms for groups [article]

John Cannon, George Havas
1994 arXiv   pre-print
Group theory is a particularly fertile field for the design of practical algorithms. Algorithms have been developed across the various branches of the subject and they find wide application.  ...  This paper examines the basic ideas behind some of the more important algorithms for finitely presented groups and permutation groups, and surveys recent developments in these fields.  ...  ACKNOWLEDGEMENTS Both authors were partially supported by Australian Research Council grants. The second author was partially supported by DIMACS/Rutgers-NSFSTC88-09648.  ... 
arXiv:math/9406203v1 fatcat:f2ywn3ibjvhltcbxnudit44fba

Page 1686 of Mathematical Reviews Vol. , Issue 2002C [page]

2002 Mathematical Reviews  
Moreover, they also give a construction of a standard presentation for an abelian group B given a standard presentation for a subgroup of B and the corresponding quotient group This paper also appeared  ...  The authors actually proceed by assuming that such a presentation is in a canonical form which can easily be derived from the Smith normal form algorithm.  ... 

Double coset enumeration

Stephen A. Linton
1991 Journal of symbolic computation  
Coset enumeration is the principal method for solving the word problem in finitely presented groups.  ...  Present applications are limited by the space available to store the coset table in computer memory. Enumerating double cosets offers a substantial saving of spaee in suitable cases.  ...  As for single coset enumeration, performance varies wildly between groups of similar size or even between presentations of the same group.  ... 
doi:10.1016/s0747-7171(08)80094-6 fatcat:4dm4zzyh65exzeeshhjvl5w4di

Page 60 of Journal of Research and Practice in Information Technology Vol. 24, Issue 2 [page]

1992 Journal of Research and Practice in Information Technology  
SIMS, C.C. (1991): The Knuth-Bendix procedure for strings as a substi- tute for coset enumeration, J Symbolic Comput. 12, pp. 439-442. SIMS, C.C.  ...  A NN RTM. GE CRO \ a a ea LEON, J.S. (1980): On an algorithm for finding a base and strong generating set for a group given by generating permutations, Math. Comp. 35, pp. 941-974.  ... 

Harnessing parallel disks to solve Rubik's cube

Daniel Kunkle, Gene Cooperman
2009 Journal of symbolic computation  
This enables the product of a symmetrized coset by a generator at a rate of 10 million moves per second.  ...  The number of moves required to solve any configuration of Rubik's cube has held a fascination for over 25 years. A new upper bound of 26 is produced.  ...  Robinson et al. (2007a) presents a comparative analysis of several different methods for parallel disk-based search and enumeration.  ... 
doi:10.1016/j.jsc.2008.04.013 fatcat:i6ppbm6byjey7noz3qahfs56sm

Nested symmetric representation of elements of the Suzuki chain groups

Mohamed Sayed
2003 International Journal of Mathematics and Mathematical Sciences  
We demonstrate an economic and concise method for representing the elements of groups involved in the Suzuki chain.  ...  For example, we represent each element ofSuz:2by a permutation on 14 letters fromL3(2):2followed by four words, each of length at most two, in 14, 36, 100, and 416involutory symmetric generators, respectively  ...  A canonical presentation for this progenitor is which is centralized by the involution π 00 = (0, 0)(1, 1)(2, 2)(3, 3)(4, 4)(5, 5)(6, 6).  ... 
doi:10.1155/s0161171203212023 fatcat:ealsp4oqybfjvkd4itdppyp57y

Translates of linear codes over Z/sub 4/

A. Bonnecaze, I.M. Duursma
1997 IEEE Transactions on Information Theory  
For the particular case of quaternary Preparata codes, we obtain that the number of distinct complete weights for the dual Preparata codes and the number of distinct complete coset weight enumerators for  ...  We give a method to compute the complete weight distribution of translates of linear codes over Z Z Z4.  ...  Solé for their interest in this work and for stimulating discussions. They also wish to thank the two anonymous referees for many constructive comments.  ... 
doi:10.1109/18.605585 fatcat:qs2w4ubf4ffpfguvsimlz4k4q4

Parabolic double cosets in Coxeter groups [article]

Sara C. Billey, Matjaž Konvalinka, T. Kyle Petersen, William Slofstra, Bridget E. Tenner
2017 arXiv   pre-print
In this article we look at a less studied object: the set of all double cosets W_I w W_J for I, J ⊆ S. Double coset are not uniquely presented by triples (I,w,J).  ...  Lex-minimal presentations are then used to enumerate double cosets via a finite automaton depending on the Coxeter graph for (W,S).  ...  Acknowledgments Many thanks to Edward Richmond, Milen Yakimov, Tewodros Amdeberhan, Victor Reiner, Richard Stanley, and Joshua Swanson for helpful discussions.  ... 
arXiv:1612.00736v2 fatcat:2a5wwuyexrc75npsoejjc4ilqq

Effective computation with group homomorphisms

Gregory Butler
1985 Journal of symbolic computation  
A discussion of coset enumeration in permutation groups is included .  ...  Particular emphasis is given to those homomorphisms induced by invariant subsets and systems of imprimitivity of a permutation group, and by the right cosets of a subgroup in an arbitrarily represented  ...  Coset Tables of Permutation Groups Before discussing the homomorphism induced by the action of a group G on the right cosets of a subgroup H, we wish to present some techniques for computing coset tables  ... 
doi:10.1016/s0747-7171(85)80011-0 fatcat:krymumdqkrgrdo3ecanlnhn2fi
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