# The GAP 4 Reference Manual - References

[AMW82]
  D[avid] G. Arrell, S[anjiv] Manrai, and M[ichael] F. Worboys.
  A procedure for obtaining simplified defining relations for a subgroup.
  In Campbell and Robertson GrpsStAndrews81, pages 155--159.
[AR84]
  D[avid] G. Arrell and E[dmund] F. Robertson.
  A modified Todd-Coxeter algorithm.
  In Atkinson Durham1982, pages 27--32.
[Art68]
  E[mil] Artin.
  Galoissche Theorie.
  Verlag Harri Deutsch, Frankfurt/Main, 1968.
[Durham1982]
  Michael D. Atkinson, editor.
  Computational Group Theory, Proceedings LMS Symposium on Computational Group Theory, Durham 1982. Academic Press, 1984.
[Baker84]
  Alan Baker.
  A concise introduction to the theory of numbers.
  Cambridge University Press, 1984.
[BC76]
  M[ichael] J. Beetham and C[olin] M. Campbell.
  A note on the Todd-Coxeter coset enumeration algorithm.
  Proc. Edinburgh Math. Soc. Edinburgh Math. Notes, 20:73--79, 1976.
[BC89]
  Richard P. Brent and Graeme L. Cohen.
  A new lower bound for odd perfect numbers.
  Math. Comp., 53:431--437, 1989.
[BC94]
  Ulrich Baum and Michael Clausen.
  Computing irreducible representations of supersolvable groups.
  Math. Comput., 207:351--359, 1994.
[BCFS91]
  L[ászló] Babai, G[ene] Cooperman, L[arry] Finkelstein, and Á[kos] Seress.
  Nearly linear time algorithms for permutation groups with a small base.
  In Proceedings of the International Symposium on Symbolic and Algebraic Computation (ISSAC'91), Bonn 1991, pages 200--209. ACM Press, 1991.
[BescheEick98]
  Hans Ulrich Besche and Bettina Eick.
  Construction of finite groups.
  J. Symbolic Comput., 27(4):387--404, 1999.
[BescheEick1000]
  Hans Ulrich Besche and Bettina Eick.
  The groups of order at most 1000 except 512 and 768.
  J. Symbolic Comput., 27(4):405--413, 1999.
[BescheEick768]
  Hans Ulrich Besche and Bettina Eick.
  The groups of order qn ·p.
  Comm. Alg., 29(4):1759--1772, 2001.
[BEO01]
  Hans Ulrich Besche, Bettina Eick, and E. A. O'Brien.
  A millennium project: constructing small groups, In preparation.
[Ber76]
  T. R. Berger.
  Characters and derived length in groups of odd order.
  J. Algebra, 39:199--207, 1976.
[BuekenhoutLeemans96]
  Francis Buekenhout and Dimitri Leemans.
  On the list of finite primitive permutation groups of degree £ 50.
  J. Symbolic Comput., 22(2):215--225, 1996.
[BreuerLinton98]
  Thomas Breuer and Steve Linton.
  The GAP 4 type system. organizing algebraic algorithms.
  In Gloor ISSAC98, pages 38--45.
[BM83]
  Gregory Butler and John McKay.
  The transitive groups of degree up to 11.
  Comm. Algebra, 11:863--911, 1983.
[Bourbaki70]
  N. Bourbaki.
  Éléments de Mathématique, Algèbre I, volume 1.
  Hermann, Paris, 1970.
[BP98]
  Thomas Breuer and Götz Pfeiffer.
  Finding Possible Permutation Characters.
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[Bre91]
  Thomas Breuer.
  Potenzabbildungen, Untergruppenfusionen, Tafel-Automorphismen.
  Diplomarbeit, Lehrstuhl D für Mathematik, Rheinisch Westfälische Technische Hochschule, Aachen, Germany, 1991.
[Bre97]
  Thomas Breuer.
  Integral bases for subfields of cyclotomic fields.
  AAECC, 8:279--289, 1997.
[Bre99]
  Thomas Breuer.
  Computing Possible Class Fusions from Character Tables.
  Comm. Algebra, 27(6):2733--2748, 1999.
[BTW93]
  Bernhard Beauzamy, Vilmar Trevisan, and Paul S. Wang.
  Polynomial factorization: Sharp bounds, Efficient algorithms.
  J. Symbolic Comput., 15:393--413, 1993.
[Bur55]
  W[illiam S.] Burnside.
  Theory of Groups of Finite Order.
  Dover Publications, New York, 1955.
  Unabridged republication of the second edition, published in 1911.
[Butler93]
  Gregory Butler.
  The transitive groups of degree fourteen and fifteen.
  J. Symbolic Comput., pages 413--422, 1993.
[Can73]
  John J. Cannon.
  Construction of defining relators for finite groups.
  Discrete Math., pages 105--129, 1973.
[Car72a]
  R. W. Carter.
  Simple groups of Lie type, volume 28 of Pure and Applied Mathematics.
  John Wiley and Sons, 1972.
[CCN85]
  J[ohn] H. Conway, R[obert] T. Curtis, S[imon] P. Norton, R[ichard] A. Parker, and R[obert] A. Wilson.
  Atlas of finite groups.
  Oxford University Press, 1985.
[ConwayHulpkeMcKay98]
  John H. Conway, Alexander Hulpke, and John McKay.
  On transitive permutation groups.
  LMS J. Comput. Math., 1:1--8, 1998.
[Coh93]
  Henri Cohen.
  A Course in Computational Algebraic Number Theory, volume 138 of Graduate Texts in Mathematics.
  Springer, Berlin, Heidelberg and New York, 1993.
[Con90a]
  S[am] B. Conlon.
  Calculating characters of p-groups.
  J. Symbolic Comput., 9(5 & 6):535--550, 1990.
[Con90b]
  S[am] B. Conlon.
  Computing modular and projective character degrees of soluble groups.
  J. Symbolic Comput., 9(5 & 6):551--570, 1990.
[GrpsStAndrews81]
  Colin M. Campbell and Edmund F. Robertson, editors.
  Groups-St.Andrews 1981, Proceedings of a conference, St.Andrews 1981, volume 71 of London Math. Soc. Lecture Note Series. Cambridge University Press, 1982.
[Dix67]
  J[ohn] D. Dixon.
  High speed computations of group characters.
  Numer. Math., 10:446--450, 1967.
[Dix93]
  John D. Dixon.
  Constructing representations of finite groups.
  In Larry Finkelstein and William M. Kantor, editors, Proceedings of the 1991 DIMACS Workshop on Groups and Computation, volume 11 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, pages 105--112. American Mathematical Society, 1993.
[DixonMortimer88]
  John D. Dixon and Brian Mortimer.
  The primitive permutation groups of degree less than 1000.
  Math. Proc. Cambridge Philos. Soc., 103:213--238, 1988.
[Dre69]
  Andreas [W. M.] Dress.
  A characterization of solvable groups.
  Math. Z., 110:213--217, 1969.
[Eick97]
  Bettina Eick.
  Special presentations for finite soluble groups and computing (pre-)Frattini subgroups.
  In Larry Finkelstein and William M. Kantor, editors, Proceedings of the 2nd DIMACS Workshop held at Rutgers University, New Brunswick, NJ, June 7--10, 1995, volume 28 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, pages 101--112. American Mathematical Society, 1997.
[FJNT95]
  V[olkmar] Felsch, D[avid] L. Johnson, J[oachim] Neubüser, and S[ergey] V. Tsaranov.
  The structure of certain Coxeter groups.
  In C[olin] M. Campbell, T[haddeus] C. Hurley, E[dmund] F. Robertson, S[ean] J. Tobin, and J[ames] J. Ward, editors, Groups '93 Galway / St.Andrews, Galway 1993, Volume 1, volume 211 of London Math. Soc. Lecture Note Series, pages 177--190. Cambridge University Press, 1995.
[FelschNeubueser79]
  Volkmar Felsch and Joachim Neubüser.
  An algorithm for the computation of conjugacy classes and centralizers in p-groups.
  In Edward W. Ng, editor, Symbolic and Algebraic Computation (Proceedings of EUROSAM 79, An International Symposium on Symbolic and Algebraic Manipulation, Marseille, 1979), Lecture Notes in Computer Science, 72, pages 452--465. Springer, Berlin, Heidelberg and New York, 1979.
[Fra82]
  J[ames] S. Frame.
  Recursive computation of tensor power components.
  Bayreuther Math. Schr., 10:153--159, 1982.
[ISSAC98]
  Oliver Gloor, editor.
  Proceedings of the 1998 International Symposium on Symbolic and Algebraic Computation. The Association for Computing Machinery, ACM Press, 1998.
[Hall]
  Marshall Hall, Jr.
  The theory of Groups.
  Macmillan, 1959.
[Hav69]
  George Havas.
  Symbolic and algebraic calculation.
  Basser Computing Dept., Technical Report 89, Basser Department of Computer Science, University of Sydney, Sydney, Australia, 1969.
[Hav74b]
  George Havas.
  A Reidemeister-Schreier program.
  In M[ichael] F. Newman, editor, Proceedings of the Second International Conference on the Theory of Groups, Canberra, 1973, volume 372 of Lecture Notes in Math., pages 347--356. Springer, Berlin, Heidelberg and New York, 1974.
[Hup82]
  B[ertram] Huppert and N[orman] Blackburn.
  Endliche Gruppen II, volume 1242 of Grundlehren Math. Wiss.
  Springer, Berlin, Heidelberg and New York, 1982.
[HIO89]
  Trevor [O.] Hawkes, I. M[artin] Isaacs, and M. Özaydin.
  On the Möbius function of a finite group.
  Rocky Mountain J. Math., 19:1003--1034, 1989.
[HKRR84]
  George Havas, P[eter] E. Kenne, J[ames] S. Richardson, and E[dmund] F. Robertson.
  A Tietze transformation program.
  In Atkinson Durham1982, pages 67--71.
[HM97]
  George Havas and Bohdan Majewski.
  Integer matrix diagonalization.
  J. Symbolic Comput., 24(3/4):399--408, September 1997.
[Howie76]
  J. M. Howie.
  An introduction to semigroup theory.
  Academic Press [Harcourt Brace Jovanovich Publishers], London, 1976.
  L.M.S. Monographs, No. 7.
[HP89]
  Derek F. Holt and W[ilhelm] Plesken.
  Perfect Groups.
  Oxford Math. Monographs. Oxford University Press, 1989.
[HR94]
  Derek [F.] Holt and Sarah Rees.
  Testing modules for irreducibility.
  J. Austral. Math. Soc. Ser. A, 57:1--16, 1994.
[HulpkeTG]
  Alexander Hulpke.
  Constructing transitive permutation groups.
  in preparation.
[Hulpke93]
  Alexander Hulpke.
  Zur Berechnung von Charaktertafeln.
  Diplomarbeit, Lehrstuhl D für Mathematik, Rheinisch Westfälische Technische Hochschule, 1993.
[Hulpke96]
  Alexander Hulpke.
  Konstruktion transitiver Permutationsgruppen.
  Dissertation, Rheinisch Westfälische Technische Hochschule, Aachen, Germany, 1996.
[Hulpke98]
  Alexander Hulpke.
  Computing normal subgroups.
  In Gloor ISSAC98, pages 194--198.
[Hulpke99]
  Alexander Hulpke.
  Computing subgroups invariant under a set of automorphisms.
  J. Symbolic Comput., 27(4):415--427, 1999.
[HulpkeClasses]
  Alexander Hulpke.
  Conjugacy classes in finite permutation groups via homomorphic images.
  Math. Comp., 69(232):1633--1651, 2000.
[HulpkeQuot]
  Alexander Hulpke.
  Representing subgroups of finitely presented groups by quotient subgroups.
  Experimental Mathematics, 10(3):369--381, 2001.
[Hum72]
  James E. Humphreys.
  Introduction to Lie algebras and representation theory.
  Springer-Verlag, New York, 1972.
  Graduate Texts in Mathematics, Vol. 9.
[Hum78]
  James E. Humphreys.
  Introduction to Lie algebras and representation theory.
  Springer-Verlag, New York, 1978.
  Second printing, revised.
[Hup67]
  B[ertram] Huppert.
  Endliche Gruppen I, volume 134 of Grundlehren Math. Wiss.
  Springer, Berlin, Heidelberg and New York, 1967.
[IshibashiEarnest94]
  Hiroyuki Ishibashi and A. G. Earnest.
  Two-element generation of orthogonal groups over finite fields.
  J. Algebra, 165(1):164--171, 1994.
[Isa76]
  I. M. Isaacs.
  Character theory of finite groups, volume 69 of Pure and applied mathematics.
  Academic Press, New York, 1976.
  xii+303 pp., ISBN 0-12-374550-0.
[JLPW95]
  Christoph Jansen, Klaus Lux, Richard [A.] Parker, and Robert [A.] Wilson.
  An Atlas of Brauer Characters, volume 11 of London Math. Soc. Monographs.
  Oxford University Press, 1995.
[Joh97]
  D. L. Johnson.
  Presentations of groups.
  Cambridge University Press, Cambridge, second edition, 1997.
[KleidmanLiebeck90]
  Peter Kleidman and Martin Liebeck.
  The subgroup structure of the finite classical groups.
  Cambridge University Press, 1990.
[Klimyk66]
  A. U. Klimyk.
  Decomposition of the direct product of irreducible representations of semisimple Lie algebras into irreducible representations.
  Ukrain. Mat. \vZ., 18(5):19--27, 1966.
[Klimyk68]
  A. U. Klimyk.
  Decomposition of a direct product of irreducible representations of a semisimple Lie algebra into irreducible representations.
  In American Mathematical Society Translations. Series 2, volume 76, pages 63--73. American Mathematical Society, Providence, R.I., 1968.
[KLM01]
  Gregor Kemper, Frank L"ubeck, and Kay Magaard.
  Matrix generators for the Ree groups \sp 2G\sb 2(q).
  Comm. Algebra, 29(1):407--413, 2001.
[TACP2]
  Donald E. Knuth.
  The Art of Computer Programming, Volume 2: Seminumerical Algorithms.
  Addison-Wesley, third edition, 1998.
[Leon91]
  Jeffrey S. Leon.
  Permutation group algorithms based on partitions, I: theory and algorithms.
  J. Symbolic Comput., 12:533--583, 1991.
[LLL82]
  A. K. Lenstra, H. W. Lenstra, and L. Lovász.
  Factoring polynomials with rational coefficients.
  Math. Ann., 261:513--534, 1982.
[SOGOS]
  R[einhard] Laue, J[oachim] Neubüser, and U[lrich] Schoenwaelder.
  Algorithms for finite soluble groups and the SOGOS system.
  In Atkinson Durham1982, pages 105--135.
[LP91]
  Klaus Lux and Herbert Pahlings.
  Computational aspects of representation theory of finite groups.
  In G. O. Michler and C. R. Ringel, editors, Representation theory of finite groups and finite-dimensional algebras, volume 95 of Progress in Mathematics, pages 37--64. Birkhäuser, Basel, 1991.
[Mac81]
  I. G. Macdonald.
  Numbers of conjugacy classes in some finite classical groups.
  Bull. Austral. Math. Soc., 23(1):23--48, 1981.
[MeckyNeubueser89]
  M[atthias] Mecky and J[oachim] Neubüser.
  Some remarks on the computation of conjugacy classes of soluble groups.
  Bull. Austral. Math. Soc., 40(2):281--292, 1989.
[Mur58]
  F[rancis] D. Murnaghan.
  The orthogonal and symplectic groups.
  Communications Series A 13, Dublin Inst. Adv. Studies, 1958.
[MV97]
  Meena Mahajan and V. Vinay.
  Determinant: combinatorics, algorithms, and complexity.
  Chicago J. Theoret. Comput. Sci., pages Article 5, 26 pp. (electronic), 1997.
[Neb95]
  Gabriele Nebe.
  Endliche rationale Matrixgruppen vom Grad 24.
  Dissertation, Rheinisch Westfälische Technische Hochschule, Aachen, Germany, 1995.
[Neb96]
  Gabriele Nebe.
  Finite subgroups of GLn(Q) for 25 £ n £ 31.
  Comm. Alg., 24 (7):2341--2397, 1996.
[Neu82]
  Joachim Neubüser.
  An elementary introduction to coset table methods in computational group theory.
  In Campbell and Robertson GrpsStAndrews81, pages 1--45.
[neukirch]
  Jürgen Neukirch.
  Algebraische Zahlentheorie.
  Springer, Berlin, Heidelberg and New York, 1992.
[New77]
  M[ichael] F. Newman.
  Determination of groups of prime-power order.
  In R. A. Bryce, J. Cossey, and M[ichael] F. Newman, editors, Group theory, Proc. Miniconf., Austral. Nat. Univ., Canberra, 1975, volume 573 of Lecture Notes in Math., pages 73--84. Springer, Berlin, Heidelberg and New York, 1977.
[New90]
  M[ichael] F. Newman.
  Proving a group infinite.
  Arch. Math. (Basel), 54(3):209--211, 1990.
[NPbook95]
  G[abriele] Nebe and W[ilhelm] Plesken.
  Finite rational matrix groups, volume 556 of AMS Memoirs.
  American Mathematical Society, 1995.
[NP95]
  G[abriele] Nebe and W[ilhelm] Plesken.
  Finite rational matrix groups of degree 16, pages 74--144.
  Volume 556 of AMS Memoirs/ NPbook95, 1995.
[NPP84]
  J[oachim] Neubüser, H[erbert] Pahlings, and W[ilhelm] Plesken.
  CAS; design and use of a system for the handling of characters of finite groups.
  In Atkinson Durham1982, pages 195--247.
[OBr90]
  E[amonn] A. O'Brien.
  The p-group generation algorithm.
  J. Symbolic Comput., 9:677--698, 1990.
[OBr91]
  E[amonn] A. O'Brien.
  The groups of order 256.
  J. Algebra, 143:219--235, 1991.
[Pah93]
  Herbert Pahlings.
  On the Möbius function of a finite group.
  Arch. Math. (Basel), 60:7--14, 1993.
[Par84]
  Richard Parker.
  The Computer Calculation of Modular Characters (the MeatAxe).
  In Atkinson Durham1982, pages 267--274.
[Pfe97]
  G. Pfeiffer.
  The Subgroups of M24, or How to Compute the Table of Marks of a Finite Group.
  Experiment. Math., 6(3):247--270, 1997.
[Ple85]
  W[ilhelm] Plesken.
  Finite unimodular groups of prime degree and circulants.
  J. Algebra, 97:286--312, 1985.
[Ple90]
  W[ilhelm] Plesken.
  Additive decompositions of positive integral quadratic forms.
  The paper is available at Lehrstuhl B für Mathematik, Rheinisch Westfälische Technische Hochschule Aachen, may be it will be published in the near future, 1990.
[PN95]
  W[ilhelm] Plesken and G[abriele] Nebe.
  Finite rational matrix groups, pages 1--73.
  Volume 556 of AMS Memoirs/ NPbook95, 1995.
[Poh87]
  M[ichael] Pohst.
  A modification of the LLL reduction algorithm.
  J. Symbolic Comput., 4:123--127, 1987.
[PP77]
  Wilhelm Plesken and Michael Pohst.
  On maximal finite irreducible subgroups of GL(n,Z). I. the five and seven dimensional cases, II. the six dimensional case.
  Math. Comp., 31:536--576, 1977.
[PP80]
  Wilhelm Plesken and Michael Pohst.
  On maximal finite irreducible subgroups of GL(n,Z). III. the nine dimensional case, IV. remarks on even dimensions with application to n = 8, V. the eight dimensional case and a complete description of dimensions less than ten.
  Math. Comp., 34:245--301, 1980.
[Rin93]
  Michael Ringe.
  The C MeatAxe, Release 1.5.
  Lehrstuhl D für Mathematik, Rheinisch Westfälische Technische Hochschule, Aachen, Germany, 1993.
[Rob88]
  E[dmund] F. Robertson.
  Tietze transformations with weighted substring search.
  J. Symbolic Comput., 6:59--64, 1988.
[Roy87]
  Gordon F. Royle.
  The transitive groups of degree twelve.
  J. Symbolic Comput., pages 255--268, 1987.
[Sch90]
  Gerhard J. A. Schneider.
  Dixon's character table algorithm revisited.
  J. Symbolic Comput., 9:601--606, 1990.
[Sco73]
  L. L. Scott.
  Modular permutation representations.
  Trans. Amer. Math. Soc., 175:101--121, 1973.
[Sho92]
  Mark W. Short.
  The Primitive Soluble Permutation Groups of Degree less than 256, volume 1519 of Lecture Notes in Math.
  Springer, Berlin, Heidelberg and New York, 1992.
[Sim70]
  Charles C. Sims.
  Computational methods in the study of permutation groups.
  In John Leech, editor, Computational Problems in Abstract Algebra, Proc. Conf. Oxford, 1967, pages 169--183. Pergamon Press, Oxford, 1970.
[Sims90b]
  Charles C. Sims.
  Computing the order of a solvable permutation group.
  J. Symbolic Comput., 9:699--705, 1990.
[Sims94]
  C. C. Sims.
  Computation with Finitely Presented Groups.
  Cambridge University Press, 1994.
[Sims97]
  Charles C. Sims.
  Computing with subgroups of automorphism groups of finite groups.
  In Wolfgang Küchlin, editor, Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation, pages 40--403. The Association for Computing Machinery, ACM Press, 1997.
[Sou94]
  Bernd Souvignier.
  Irreducible finite integral matrix groups of degree 8 and 10.
  Math. Comp., 63:335--350, 1994.
[Spa89]
  Lehrstuhl D für Mathematik, Rheinisch Westfälische Technische Hochschule, Aachen, Germany.
  SPAS - Subgroup Presentation Algorithms System, version 2.5, User's reference manual, 1989.
[Tay87]
  D. E. Taylor.
  Pairs of generators for matrix groups. I.
  The Cayley Bulletin, 3, 1987.
[Theissen93]
  Heiko Theißen.
  Methoden zur Bestimmung der rationalen Konjugiertheit in endlichen Gruppen.
  Diplomarbeit, Lehrstuhl D für Mathematik, Rheinisch Westfälische Technische Hochschule, 1993.
[Theissen97]
  Heiko Theißen.
  Eine Methode zur Normalisatorberechnung in Permutationsgruppen mit Anwendungen in der Konstruktion primitiver Gruppen.
  Dissertation, Rheinisch Westfälische Technische Hochschule, Aachen, Germany, 1997.
[vdW76]
  Robert W. van der Waall.
  On symplectic primitive modules and monomial groups.
  Indagationes Math., 38:362--375, 1976.
[Wagon90]
  Stan Wagon.
  Editor's corner: the Euclidean algorithm strikes again.
  Amer. Math. Monthly, 97(2):125--129, 1990.
[AGR]
  Robert. A. Wilson.
  ATLAS of Finite Group Representations.
  http://www.mat.bham.ac.uk/atlas/.
[Wil96]
  R[obert] A. Wilson.
  Standard generators for sporadic simple groups.
  J. Algebra, 184:505--515, 1996.
[Zagier90]
  D. Zagier.
  A one-sentence proof that every prime p º 1 mod 4 is a sum of two squares.
  Amer. Math. Monthly, 97(2):144, 1990.
[Zum89]
  Matthias Zumbroich.
  Grundlagen einer Arithmetik in Kreisteilungskörpern und ihre Implementation in CAS.
  Diplomarbeit, Lehrstuhl D für Mathematik, Rheinisch Westfälische Technische Hochschule, Aachen, Germany, 1989.

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GAP 4 manual
May 2002