## An Invitation to Computational Group Theory (1995)

Venue: | Groups' 93 -- Galway/St. Andrews, volume 212 of London Math. Soc. Lecture Note Ser |

Citations: | 3 - 0 self |

### BibTeX

@INPROCEEDINGS{Neubüser95aninvitation,

author = {J. Neubüser},

title = {An Invitation to Computational Group Theory},

booktitle = {Groups' 93 -- Galway/St. Andrews, volume 212 of London Math. Soc. Lecture Note Ser},

year = {1995},

pages = {457--475},

publisher = {Cambridge University Press}

}

### OpenURL

### Abstract

Algebra" in 1967 [Lee70]. Its proceedings contain a survey of what had been tried until then [Neu70] but also some papers that lead into the Decade of discoveries (1967--1977). At the Oxford conference some of those computational methods were presented for the first time that are now, in some cases varied and improved, work horses of CGT systems: Sims' methods for handling big permutation groups [Sim70], the Knuth-Bendix method for attempting to construct a rewrite system from a presentation [KB70], variations of the Todd-Coxeter method for the determination of presentations of subgroups [Men70]. Others, like J. D. Dixon's method for the determination of the character table [Dix67], the p-Nilpotent-Quotient method of I. D. Macdonald [Mac74] and the Reidemeister-Schreier method of G. Havas [Hav74] for subgroup presentations were published within a few years from that conference. However at least equally important for making group theorists aware of CGT were a number of applications of...

### Citations

494 |
Simple word problems in universal algebras
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- 1970
(Show Context)
Citation Context ...es varied and improved, work horses of CGT systems: Sims' methods for handling big permutation groups [Sim70], the Knuth-Bendix method for attempting to construct a rewrite system from a presentation =-=[KB70]-=-, variations of the Todd-Coxeter method for the determination of presentations of subgroups [Men70]. Others, like J. D. Dixon's method for the determination of the character table [Dix67], the p-Nilpo... |

100 |
On the maximal subgroups of the finite classical groups
- Aschbacher
- 1984
(Show Context)
Citation Context ...s [NPP84],[LP91]. A long neglected, but now very rapidly growing branch of CGT are methods for the study of matrix groups over finite fields [NP92], making strong use of the Aschbacher classification =-=[Asc84]-=-. Systems. A program system comprises various components: storage management, a problem-oriented language for both interactive use and for writing programs, that can call system functions (and possibl... |

61 |
An Introduction to Magma
- Cannon, Playoust
- 1994
(Show Context)
Citation Context ...ten in Fortran. This system developed into the Cayley System [BC91], the functions of which were semiautomatically translated to C about 1987. Cayley has been remodelled recently under the name MAGMA =-=[CP93]-=- and extended to a general Computer Algebra system reaching beyond CGT. Another general system for CGT called GAP (Groups, Algorithms, and Programming) was started in Aachen in 1986. Its design is inf... |

30 |
Über unendliche diskontinuierliche Gruppen
- Dehn
- 1911
(Show Context)
Citation Context ...termination of the groups of order pqr made up a good deal of last century's group theory. However Dehn's formulation of word, conjugacy, and isomorphism problem for finitely presented groups in 1911 =-=[Deh11]-=- (even though it had precursors ) may be thought of as the beginning of the prehistory proper of CGT, focussing attention on the request for group theoretical algorithms. Two points are worth noting: ... |

24 | Sylow’s theorem in polynomial time
- Kantor
- 1985
(Show Context)
Citation Context ...new methods for permutation groups, which have now become very practical, too, first were brought in through rather theoretical discussions of the complexity of permutation group algorithms. ([Neu86],=-=[Kan85]-=-,[KT88],[BLS88],[BCF + ar], [BCFS91],[BS92], to mention just a small selection of many papers on this subject.) A broad variety of methods, many interactive, are available for working with representat... |

21 | Fast management of permutation groups I
- Babai, Luks, et al.
- 1997
(Show Context)
Citation Context ... permutation groups, which have now become very practical, too, first were brought in through rather theoretical discussions of the complexity of permutation group algorithms. ([Neu86],[Kan85],[KT88],=-=[BLS88]-=-,[BCF + ar], [BCFS91],[BS92], to mention just a small selection of many papers on this subject.) A broad variety of methods, many interactive, are available for working with representations and charac... |

20 |
Topics in combinatorial group theory
- Baumslag
- 1993
(Show Context)
Citation Context ...theory could fulfil Dehn's request. Soon after came proofs of the non-existence of algorithms that could decide if a finitely presented group is trivial, finite, abelian, etc. (see G. Baumslag's book =-=[Bau93] for a viv-=-id description). Nevertheless in 1936 J. A. Todd and H.S.M. Coxeter [TC36] provided at least a systematic method of attempt for showing finiteness by "coset enumeration " and today CGT provi... |

17 | Luks: Computing in quotient groups
- Kantor, M
- 1990
(Show Context)
Citation Context ...90]. Working via homomorphic images has become the method of choice for handling polycyclicly presented finite soluble groups [LNS84] but also has become indispensable for handling permutation groups =-=[KL90]-=-. The methods for the investigation of permutation groups have almost undergone a revolution, bringing in structure theory such as the O'Nan-Scott classification of primitive groups or even the classi... |

15 |
Fundamental Algorithms for Permutation Groups, volume 559 of LNCS
- Butler
- 1991
(Show Context)
Citation Context ...991 [FK93] and 1994. In addition to a number of surveys (see in particular [CH92] for a recent one with many references) two monographs on parts of the field have appeared recently. While Greg Butler =-=[But91]-=- tries to introduce Computer science students with no preknowledge of group theory to computational methods for permutation groups, Charles Sims [Sim94] gives an authoritative account of methods for t... |

15 |
Some algorithms for computing with finite permutation groups
- Neumann
- 1986
(Show Context)
Citation Context ...f these new methods for permutation groups, which have now become very practical, too, first were brought in through rather theoretical discussions of the complexity of permutation group algorithms. (=-=[Neu86]-=-,[Kan85],[KT88],[BLS88],[BCF + ar], [BCFS91],[BS92], to mention just a small selection of many papers on this subject.) A broad variety of methods, many interactive, are available for working with rep... |

14 | Some remarks on the computation of complements and normalizers in soluble groups
- Celler, Neubüser, et al.
- 1990
(Show Context)
Citation Context ...1 ], [ 3, 9 ] ], [ [ 3, 9 ] ], [ [ 3, 1 ] ], [ [ 1, 1 ] ] ] This confirms some of the information about the structure of the group quoted above from the ATLAS. However using methods described e.g. in =-=[CNW90]-=- we can obtain further information about the splitting of extensions, e.g. we can determine representatives of the conjugacy classes of complements of the second last group in the derived series which... |

14 |
Brauer Trees of Sporadic Groups
- Hiss, Lux
- 1989
(Show Context)
Citation Context ...etical table, involved a great deal of implementation and use of group theoretical programs (in particular for working with characters), and the same holds for books listing Brauer trees of sporadics =-=[HL89]-=- or perfect groups [HP89] as well as for other listings, e. g. of primitive or transitive permutation groups. Methods. These and many more concrete computations were made possible by a large number of... |

13 |
Nearly Linear Time Algorithms for Permutation Groups with a Small Base
- Babai, Cooperman, et al.
- 1991
(Show Context)
Citation Context ... which have now become very practical, too, first were brought in through rather theoretical discussions of the complexity of permutation group algorithms. ([Neu86],[Kan85],[KT88],[BLS88],[BCF + ar], =-=[BCFS91]-=-,[BS92], to mention just a small selection of many papers on this subject.) A broad variety of methods, many interactive, are available for working with representations and characters [NPP84],[LP91]. ... |

13 |
Perfect groups
- Holt, Plesken
- 1989
(Show Context)
Citation Context ...great deal of implementation and use of group theoretical programs (in particular for working with characters), and the same holds for books listing Brauer trees of sporadics [HL89] or perfect groups =-=[HP89]-=- as well as for other listings, e. g. of primitive or transitive permutation groups. Methods. These and many more concrete computations were made possible by a large number of new methods and their in... |

11 |
MOLGEN, ein Computeralgebra-System für die Konstruktion molekularer Graphen
- Grund, Kerber, et al.
- 1992
(Show Context)
Citation Context ...rs complement the scopes of the two general systems, among them Quotpic [HR93], giving a very good visualisation for the steps in the calculation of factor groups of finitely presented groups, MOLGEN =-=[GKL92]-=- for the application of group theoretical methods to the construction of graphs, in particular representing the structure of organic compounds, and MOC [LP91] for the construction of modular character... |

11 |
A Reidemeister-Schreier program
- Havas
- 1974
(Show Context)
Citation Context ...0]. Others, like J. D. Dixon's method for the determination of the character table [Dix67], the p-Nilpotent-Quotient method of I. D. Macdonald [Mac74] and the Reidemeister-Schreier method of G. Havas =-=[Hav74]-=- for subgroup presentations were published within a few years from that conference. However at least equally important for making group theorists aware of CGT were a number of applications of computat... |

11 |
A graphics system for displaying finite quotients of finitely presented groups
- Holt, Rees
- 1991
(Show Context)
Citation Context ...he 80-ties a number of very specialized systems have been written which became absorbed or outdated by Cayley and GAP, some others complement the scopes of the two general systems, among them Quotpic =-=[HR93]-=-, giving a very good visualisation for the steps in the calculation of factor groups of finitely presented groups, MOLGEN [GKL92] for the application of group theoretical methods to the construction o... |

11 |
Coset Enumeration on Digital Computers
- Leech
- 1963
(Show Context)
Citation Context ... implementations of computing methods for groups. The first, about 1953, that I am aware of, are a partial implementation of the Todd-Coxeter method by B. Haselgrove on the EDSAC II in Cambridge (see =-=[Lee63]-=-) and of methods for the calculation of characters of symmetric groups by S. Comet on the BARK computer in Stockholm [Com54]. Other areas of group theory were tried soon after. E.T. Parker and P.J. Ni... |

11 |
Algorithms for finite soluble groups and the SOGOS system
- Laue, Neubüser, et al.
(Show Context)
Citation Context ...ration(N,orb[1]);; gap? MakeStabChainRandom(P);;time;Size(P); 73432 3265173504 However since we know S to be soluble, we prefer to work with a polycyclic presentation, called an AG-group by tradition =-=[LNS84]. We obtai-=-n this by gap? A:=AgGroup(P);;time;Size(A); 1101031 3265173504 gap? A.name:="Fi23M7";; in about 18 minutes. Again for safety (see section 3 of this paper) we have checked the order. We may n... |

8 |
Space groups and groups of prime-power order
- Leedham-Green, Newman
- 1980
(Show Context)
Citation Context ...hod, which builds on the p-Nilpotent Quotient Algorithm (pNQ) and links to the recent exploration of the possibility to classify families of p-groups of constant coclass with the help of space groups =-=[L-GN80]-=-. Another very recent offspring of the pNQ is a (proper) Nilpotent Quotient Algorithm stepping down the lower central series of a finitely presented group. A number of proposals have been made and a c... |

8 |
Computational aspects of representation theory of finite groups
- Lux, Pahlings
- 1991
(Show Context)
Citation Context ...[BCFS91],[BS92], to mention just a small selection of many papers on this subject.) A broad variety of methods, many interactive, are available for working with representations and characters [NPP84],=-=[LP91]-=-. A long neglected, but now very rapidly growing branch of CGT are methods for the study of matrix groups over finite fields [NP92], making strong use of the Aschbacher classification [Asc84]. Systems... |

8 |
A computer application to finite p-groups
- MACDONALD
- 1974
(Show Context)
Citation Context ...for the determination of presentations of subgroups [Men70]. Others, like J. D. Dixon's method for the determination of the character table [Dix67], the p-Nilpotent-Quotient method of I. D. Macdonald =-=[Mac74]-=- and the Reidemeister-Schreier method of G. Havas [Hav74] for subgroup presentations were published within a few years from that conference. However at least equally important for making group theoris... |

8 |
Determination of groups of prime-power order
- Newman
- 1975
(Show Context)
Citation Context ... p-Nilpotent Quotient techniques are now strong enough to show e. g. that the class 18 factor of the restricted Burnside group R(2; 7) has order 7 6366 [HNV-L90]. Following a proposal of M. F. Newman =-=[New77], E. -=-A. O'Brien implemented a p-group generation program sufficient to determine the (58760 isomorphism classes of) groups of order 2 n , ns8 [O'B91]. Making and checking the Cambridge "Atlas of Finit... |

7 | Polynomial-time versions of Sylow’s theorem
- Kantor, Taylor
- 1988
(Show Context)
Citation Context ...ods for permutation groups, which have now become very practical, too, first were brought in through rather theoretical discussions of the complexity of permutation group algorithms. ([Neu86],[Kan85],=-=[KT88]-=-,[BLS88],[BCF + ar], [BCFS91],[BS92], to mention just a small selection of many papers on this subject.) A broad variety of methods, many interactive, are available for working with representations an... |

7 |
Untersuchungen des Untergruppenverbandes endlicher Gruppen auf einer programmgesteuerten elektronischen Dualmaschine
- Neubüser
- 1960
(Show Context)
Citation Context ...ere tried soon after. E.T. Parker and P.J. Nicolai made an - unsuccessful - search for analogues of the Mathieu groups [PN58], 1959 programs for calculating the subgroup lattice of permutation groups =-=[Neu60]-=- and a little later for polycyclicly presented 2-groups were written. Other methods and special investigations followed, programs of this time were written mostly in machine code, use of all kinds of ... |

7 |
Calculating presentations for certain kinds of quotient groups, SYMSAC’76
- NEWMAN
- 1976
(Show Context)
Citation Context ...ng his permutation group methods [Sim73], the determination of the Burnside group B(4; 4) of order 2 422 by M. F. Newman and G. Havas using an extension of the original p-Nilpotent-Quotient method 10 =-=[New76]-=-, and the determination of the (4783 isomorphism classes of) space groups of 4-dimensional space [BBN + 78], using not only Zassenhaus' algorithm but also the possibility to find all subgroups of the ... |

6 |
A nilpotent quotient algorithm for graded Lie rings
- HAVAS, NEWMAN, et al.
- 1990
(Show Context)
Citation Context ... techniques for modular representations [Nor80]. p-Nilpotent Quotient techniques are now strong enough to show e. g. that the class 18 factor of the restricted Burnside group R(2; 7) has order 7 6366 =-=[HNV-L90]-=-. Following a proposal of M. F. Newman [New77], E. A. O'Brien implemented a p-group generation program sufficient to determine the (58760 isomorphism classes of) groups of order 2 n , ns8 [O'B91]. Mak... |

5 |
Some remarks on the computation of conjugacy classes of soluble groups
- Mecky, Neubüser
- 1989
(Show Context)
Citation Context ...ist([1..8],i-?Complementclasses(ds[i],ds[9]));; gap? List(compcl,Length); [ 3, 3, 1, 1, 9, 3, 1, 1 ] We can also determine (in less than a minute) the conjugacy classes of elements using methods from =-=[MN89]-=- gap? conjcl:=ConjugacyClasses(A);;time;Length(conjcl); 49309 181 We see that there are 181 classes. Finally we can determine the character table of this group, using Alexander's implementation of the... |

4 |
High speed computations of group characters
- Dixon
- 1967
(Show Context)
Citation Context ... and re-implementation in GAP of an algorithm for the determination of the character table of a finite group from its class multiplication coefficients, a method that had been proposed by J. D. Dixon =-=[Dix67]-=- and improved and implemented in Cayley by G. Schneider [Sch90]. Originally just to test this implementation, some hitherto unknown character tables of maximal subgroups of sporadic simple groups were... |

4 |
A soluble group algorithm
- LEEDHAM-GREEN
- 1984
(Show Context)
Citation Context ... been made and a couple of them implemented recently for finding soluble factorgroups of finitely presented groups, using concepts such as cohomology groups, modular representations, or Grobner bases =-=[L-G84]-=-,[Ple87],[Sim90]. Working via homomorphic images has become the method of choice for handling polycyclicly presented finite soluble groups [LNS84] but also has become indispensable for handling permut... |

3 |
A general purpose group theory program
- Cannon
- 1974
(Show Context)
Citation Context ...d attempts to design CGT systems in which various methods could be used without having to translate data from one program to the other. By 1974 a first "Aachen -- Sydney Group System" was op=-=erational [Can74], and in 1-=-976 John Cannon published "A Draft Description of a Group Theory Language Cayley" [Can76], which may be thought of as a turn to Modern Times. The claim that since about 1977 the development ... |

2 |
A draft description of the group theory language Cayley
- Cannon
- 1976
(Show Context)
Citation Context ... data from one program to the other. By 1974 a first "Aachen -- Sydney Group System" was operational [Can74], and in 1976 John Cannon published "A Draft Description of a Group Theory La=-=nguage Cayley" [Can76]-=-, which may be thought of as a turn to Modern Times. The claim that since about 1977 the development of CGT is speeding up rapidly can be justified by looking at four aspects: results, methods, system... |

2 |
Algorithms for groups, The Australian computer journal 27
- Cannon, Havas
- 1992
(Show Context)
Citation Context ... first meeting fully devoted to CGT was held 13 in Durham 1982 [Atk84], further ones were Oberwolfach 1988 and 1992, DIMACS 1991 [FK93] and 1994. In addition to a number of surveys (see in particular =-=[CH92]-=- for a recent one with many references) two monographs on parts of the field have appeared recently. While Greg Butler [But91] tries to introduce Computer science students with no preknowledge of grou... |

2 |
On the machine calculation of characters of the symmetric group
- Comet
- 1954
(Show Context)
Citation Context ... the Todd-Coxeter method by B. Haselgrove on the EDSAC II in Cambridge (see [Lee63]) and of methods for the calculation of characters of symmetric groups by S. Comet on the BARK computer in Stockholm =-=[Com54]-=-. Other areas of group theory were tried soon after. E.T. Parker and P.J. Nicolai made an - unsuccessful - search for analogues of the Mathieu groups [PN58], 1959 programs for calculating the subgroup... |

2 |
Alan Turing, The Enigma of Intelligence. Unwin Paperbacks
- Hodges
- 1985
(Show Context)
Citation Context ...er: "There will positively be no internal alterations to be made even if we wish suddenly to switch from calculating the energy levels of the neon atom to the enumeration of groups of order 720&q=-=uot; (see [Hod85]-=-, page 293). 9 Early history (1953--1967) of CGT may be thought of as starting with the first implementations of computing methods for groups. The first, about 1953, that I am aware of, are a partial ... |

1 |
A handbook of Cayley functions. Computer Algebra Group
- Bosma, Cannon
- 1991
(Show Context)
Citation Context ...owever, a good deal of the strength of CGT for application to problems in present day group theoretical research stems from the fact that in the presently available CGT program systems such as Cayley =-=[BC91]-=- or GAP [S + 93] a broad variety of methods is at hand - such as it is in a researcher's mind. I want to start by illustrating this point by a case study taken from the Aachen Diplom thesis of Alexand... |

1 |
Computing composition factors of small base groups in almost linear time
- Beals, Seress
- 1992
(Show Context)
Citation Context ...ve now become very practical, too, first were brought in through rather theoretical discussions of the complexity of permutation group algorithms. ([Neu86],[Kan85],[KT88],[BLS88],[BCF + ar], [BCFS91],=-=[BS92]-=-, to mention just a small selection of many papers on this subject.) A broad variety of methods, many interactive, are available for working with representations and characters [NPP84],[LP91]. A long ... |

1 |
A bibliography of Cayley citations, SIGSAM Bulletin 25
- Cannon
- 1991
(Show Context)
Citation Context ...ent in the last section. Both Cayley and GAP have found wide use. With Cayley more exact figures can be given, since it must be licensed for a certain fee, J. Cannon reports over 200 licenses by 1990 =-=[Can91]-=-. GAP can be obtained free of charge via ftp and handed on so that no full control of its spread exists. An indication are the about 240 members of the GAP-forum and more than 350 reports of installat... |

1 |
Zur Berechnung von Charaktertafeln
- Hulpke
- 1993
(Show Context)
Citation Context ... + 93] a broad variety of methods is at hand - such as it is in a researcher's mind. I want to start by illustrating this point by a case study taken from the Aachen Diplom thesis of Alexander Hulpke =-=[Hul93]-=- which obtains a rather detailed analysis of a fairly big group using just standard methods, in this case in GAP - and, of course, the mathematical insight into the relevant theory and the available t... |

1 |
Defining relations for subgroups of finite index of groups with a finite presentation
- Mendelsohn
(Show Context)
Citation Context ...ups [Sim70], the Knuth-Bendix method for attempting to construct a rewrite system from a presentation [KB70], variations of the Todd-Coxeter method for the determination of presentations of subgroups =-=[Men70]-=-. Others, like J. D. Dixon's method for the determination of the character table [Dix67], the p-Nilpotent-Quotient method of I. D. Macdonald [Mac74] and the Reidemeister-Schreier method of G. Havas [H... |

1 |
Investigations of groups on computers
- Neubuser
(Show Context)
Citation Context ...ly impress group theorists, is roughly marked by the Oxford conference "Computational Problems in Abstract Algebra" in 1967 [Lee70]. Its proceedings contain a survey of what had been tried u=-=ntil then [Neu70]-=- but also some papers that lead into the Decade of discoveries (1967--1977). At the Oxford conference some of those computational methods were presented for the first time that are now, in some cases ... |

1 |
The influence of automatic computers on mathematical methods
- Newman
- 1951
(Show Context)
Citation Context ...hm for the classification of space groups [Zas48] that much later has been put to very practical use. -- In 1951 M. H. A. Newman in a talk at the `Manchester University Computer Inaugural Conference' =-=[New51] proposed -=-to use probabilistic methods for getting some insight into the vast number of groups of order 256. The title of the talk: "The influence of automatic computers on mathematical methods" is re... |