Abstract
In this survey an attempt is made to give some impression of the capabilities of currently available programs for computations with finitely generated groups and their representations.
This paper is a modified version of the paper presented at the 10th International Colloquium on “Group-Theoretical Methods in Physics” and published in Physica A 114A, 493–506 (1982).
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References
Aasman, M., Janißen, W., Lammers, H., Neubüser, J., Pahlings, H., Plesken, W.: The CAS System, User Manual (provisional version). RWTH Aachen: Lehrstuhl D f. Math., Mimeographed Rep., 83 p., 1981.
Aasman, M., Janißen, W., Lammers, H., Neubüser, J., Pahlings, H., Plesken, W.: Das CAS System, Handbuch (vorläufige Version). RWTH Aachen: Lehrstuhl D f. Math., Mimeographed Rep., 555 p., 1981.
Atkinson, M. D.: An Algorithm for Finding the Blocks of a Permutation Group. Math. Comput. 29, 911–913 (1975).
Beetham, M. J., Campbell, C. M.: A Note on the Todd-Coxeter Coset Enumeration Algorithm. Proc. Edinburgh Math. Soc. (2) 20, 73–79 (1976).
Birkhoff, G., Hall, M., Jr. (eds.): Computers in Algebra and Number Theory. (SIAM-AMS Proc. 4.) Providence, R. I.: Amer. Math. Soc. 1971.
Brott, C., Neubüser, J.: A Programme for the Calculation of Characters and Representations of Finite Groups. In: Proc. of the Conference on Computational Problems in Abstract Algebra, Oxford, 1967 (Leech, J., ed.), pp. 101–110. Pergamon Press 1970.
Butler, G.: The Schreier Algorithm for Matrix Groups. SYMSAC 1976, 167–170.
Butler, G.: Computational Approaches to Certain Problems in the Theory of Finite Groups. Ph.D. Thesis, Univ. of Sydney, 316 p., 2 appendices on microfiche, 1979.
Cannon, J. J.: Computers in Group Theory: A Survey. Commun. ACM 12, 3–12 (1969).
Cannon, J. J.: Computing Local Structure of Large Finite Groups. In Birkhoff, G., Hall, M., Jr. (eds.): Computers in Algebra and Number Theory. (SIAM-AMS Proc. 4.) Providence, R. I.: Amer. Math. Soc. 1971, 161–176.
Cannon, J. J.: Construction of Defining Relators for Finite Groups. Discrete Math. 5, 105–129 (1973).
Cannon, J. J.: A General Purpose Group Theory Program. In Newman, M. F. (ed.): Proc. 2nd Internat. Conf. on the Theory of Groups (Canberra: Austral. Nat. Univ. 1973). Lecture Notes in Mathematics, Vol. 372. Berlin-Heidelberg-New York: Springer 1974, 204–217.
Cannon, J. J.: A Draft Description of the Group Theory Language Cayley. SYMSAC 1976, 66–84.
Cannon, J. J.: Effective Procedures for the Recognition of Primitive Groups. In Cooperstein, B., Mason, G. (eds.): The Santa Cruz Conference on Finite Groups. (Proc. Sympos. Pure Math. 37.) Providence, R. I.: Amer. Math. Soc. 1980, 487–493.
Cannon, J. J.: Software Tools for Group Theory. In Cooperstein, B., Mason, G. (eds.): The Santa Cruz Conference on Finite Groups. (Proc. Sympos. Pure Math. 37.) Providence, R. I.: Amer. Math. Soc. 1980, 495–502.
Cannon, J. J., Dimino, L. A., Havas, G., Watson, J. M.: Implementation and Analysis of the Todd-Coxeter Algorithm. Math. Comput. 27, 463–490 (1973).
Cooperstein, B., Mason, G. (eds.): The Santa Cruz Conference on Finite Groups. (Proc. Sympos. Pure Math. 37.) Providence, R. I.: Amer. Math. Soc. 1980.
Dietze, A., Schaps, M.: Determining Subgroups of Given Finite Index in a Finitely Presented Group. Can. J. Math. 26, 769–782 (1974).
Dixon, J. D.: High Speed Computation of Group Characters. Numer. Math. 10, 446–450 (1967).
Dixon, J. D.: Computing Irreducible Representations of Groups. Math. Comput. 24, 707–712 (1970).
Felsch, V.: A Machine Independent Implementation of a Collection Algorithm for the Multiplication of Group Elements. SYMSAC 1976, 159–166.
Felsch, V.: A Bibliography on the Use of Computers in Group Theory and Related Topics: Algorithms, Implementations, and Applications. Kept current and obtainable from Lehrstuhl D für Mathematik, RWTH Aachen, D-5100 Aachen, Federal Republic of Germany.
Felsch, V., Neubüser, J.: An Algorithm for the Computation of Conjugacy Classes and Centralizers in p-Groups. EUROSAM 1979, 452–465.
Flodmark, S., Jansson, P.-O.: Irreducible Representations of Finite Groups (Proc. 10th Internat. Coll. on Group Theoret. Methods in Phys.). (To appear in Physica A, 1982.)
Frame, J. S.: Recursive Computation of Tensor Power Components. Bayreuther Math. Schriften (to appear 1982).
Gabriel, J. R.: Numerical Methods for Reduction of Group Representations. SYMSAM 1971, 180–182.
Havas, G.: A Reidemeister-Schreier Program. In Newman, M. F. (ed.): Proc. 2nd Internat. Conf. on the Theory of Groups (Canberra: Austral. Nat. Univ. 1973). Lecture Notes in Mathematics, Vol. 372. Berlin-Heidelberg-New York: Springer 1974, 347–356.
Havas, G., Newman, M. F.: Application of Computers to Questions like Those of Burnside. In: Burnside Groups Proc., Bielefeld, Germany: 1977 (Mennicke, J. L., ed.). Lecture Notes in Mathematics, Vol. 806, pp. 211–230. Berlin-Heidelberg-New York: Springer 1980.
Havas, G., Nicholson, T.: Collection. SYMSAC 1976, 9–14.
Havas, G., Richardson, J. S., Sterling, L. S.: The Last of the Fibonacci Groups. Proc. Roy. Soc. (Edinburgh) A83, 199–203 (1979).
Havas, G., Sterling, L. S.: Integer Matrices and Abelian Groups. EUROSAM 1979, 431–451.
Hunt, D. C.: A Computer-Based Atlas of Finite Simple Groups. In Cooperstein, B., Mason, G. (eds.): The Santa Cruz Conference on Finite Groups. (Proc. Sympos. Pure Math. 37.) Providence, R. I.: Amer. Math. Soc. 1980, 507–510.
Huppert, B.: Endliche Gruppen I. Berlin-Heidelberg-New York: Springer 1967.
Isaacs, I. M.: Character Theory of Finite Groups. New York-San Francisco-London: Academic Press 1976.
Johnson, D. L.: Topics in the Theory of Group Presentations. London Math. Soc. Lect. Note Series, Vol. 42. Cambridge: Cambridge University Press 1980.
Leech, J.: Computer Proof of Relations in Groups. In: Topics in Group Theory and Comput. (Proc., Galway, 1973), (Curran, M. P. J., ed.), pp. 38–61. London-New York-San Francisco: Academic Press 1977.
Leon, J. S.: On an Algorithm for Finding a Base and a Strong Generating Set for a Group Given by Generating Permutations. Math. Comput. 35, 941–974 (1980).
Leon, J. S.: Finding the Order of a Permutation Group. In Cooperstein, B., Mason, G. (eds.): The Santa Cruz Conference on Finite Groups. (Proc. Sympos. Pure Math. 37.) Providence, R. I.: Amer. Math. Soc. 1980, 511–517.
Leon, J. S., Pless, V.: CAMAC 1979. EUROSAM 1979, 249–257.
Loos, R.: Term Reduction Systems and Algebraic Algorithms. Informatik Fachberichte 47, 214–234 (1981).
Magnus, W., Karrass, A., Solitar, D.: Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations. New York-London-Sydney: Interscience 1966.
McLain, D. H.: An Algorithm for Determining Defining Relations of a Subgroup. Glasg. Math. J. 18, 51–56 (1977).
Neubüser, J.: Investigations of Groups on Computers. In: Proc. of the Conference on Computational Problems in Abstract Algebra, Oxford, 1967 (Leech, J., ed.), pp. 1–19. Pergamon Press 1970.
Neubüser, J.: Computing Moderately Large Groups: Some Methods and Applications. In Birkhoff, G., Hall, M., Jr. (eds.): Computers in Algebra and Number Theory. (SIAM-AMS Proc. 4.) Providence, R. I.: Amer. Math. Soc. 1971, 183–190.
Neubüser, J.: Some Computational Methods in Group Theory. In: 3rd Internat. Coll. Adv. Comput. Methods in Theoret. Phys., B-II-1–B-II-35. Marseille: Centre de Physique Théorique CNRS 1973.
Neubüser, J.: An Elementary Introduction to Coset Table Methods in Computational Group Theory. (To appear in Proc. St. Andrews 1981 Conf. on Groups.)
Newman, M. F. (ed.): Proc. 2nd Internat. Conf. on the Theory of Groups (Canberra: Austral. Nat. Univ. 1973). Lecture Notes in Mathematics, Vol. 372. Berlin-Heidelberg-New York: Springer 1974.
Newman, M. F.: Calculating Presentations for Certain Kinds of Quotient Groups. SYMSAC 1976, 2–8.
Newman, M. F. (ed.): Topics in Algebra. (Canberra: Proc. 1978). Lecture Notes in Mathematics, Vol. 697. Berlin-Heidelberg-New York: Springer 1978.
Sandlöbes, G.: Perfect Groups of Order Less than 104. Commun. Algebra 9, 477–490 (1981).
Sims, C. C.: Determining the Conjugacy Classes of a Permutation Group. In [4] Birkhoff, G., Hall, M., Jr. (eds.): Computers in Algebra and Number Theory. (SIAM-AMS Proc. 4.) Providence, R. I.: Amer. Math. Soc. 1971, 191–195.
Sims, C. C.: Computation with Permutation Groups. SYMSAM 1971, 23–28.
Sims, C. C.: The Role of Algorithms in the Teaching of Algebra. In Newman, M. F. (ed.): Topics in Algebra. (Canberra: Proc. 1978). Lecture Notes in Mathematics, Vol. 697. Berlin-Heidelberg-New York: Springer 1978, 95–107.
Sims, C. C.: Some Group-Theoretic Algorithms. In Newman, M. F. (ed.): Topics in Algebra. (Canberra: Proc. 1978). Lecture Notes in Mathematics, Vol. 697. Berlin-Heidelberg-New York: Springer 1978, 108–104.
Sims, C. C.: Group-Theoretic Algorithms, a Survey. Proc. Internat. Congress of Mathematicians 2 (Helsinki 1978). (Lehto, O., ed.), pp. 979–985. Helsinki: Acad. Sci. Fennica 1980.
Sims, C. C.: Computational Group Theory. Manuscript, distributed at the Amer. Math. Soc. Short Course on Comput. Algebra–Symb. Math. Comput., Ann Arbor, Michigan, 15 p., 1980.
Stanley, R. P.: Invariants of Finite Groups and Their Applications to Combinatorics. Bull. AMS 1, 475–511 (1979).
Thackray, J. G.: Reduction of Modules in Non-Zero Characteristics. Lect. Notes, distributed at the Amer. Math. Soc. Summer Inst. on Finite Group Theory, 4 p., Santa Cruz: Univ. of Calif. 1979.
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Neubüser, J. (1982). Computing with Groups and Their Character Tables. In: Buchberger, B., Collins, G.E., Loos, R. (eds) Computer Algebra. Computing Supplementum, vol 4. Springer, Vienna. https://doi.org/10.1007/978-3-7091-3406-1_3
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