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This is a page on GAP 3, which is still available, but no longer supported. The present version is GAP 4  (See  Status of GAP 3).

GAP 3 Share Package "anupq"

ANU Prime Quotient algorithm for finitely presented groups

Share package since release 3.2, about December 1992.
Corresponding GAP 4 package: GAP 4 package "anupq".


Eamonn O'Brien.


Language: C
Operating system: UNIX
Current version: 1.4 (in the 3.4.4 distribution)


The program provides access to implementations of the following algorithms:

  • A p-quotient algorithm to compute a power-commutator presentation for a p-group. The algorithm implemented here is based on that described in [NB96], [HN80], and papers referred to there. The current implementation incorporates the following features:
    • collection from the left, see [Vau90]; Vaughan-Lee's implementation of this collection algorithm is used in the program;
    • an improved consistency algorithm, see [Vau84];
    • new exponent law enforcement and power routines;
    • closing of relations under the action of automorphisms;
    • some formula evaluation.
  • A p-group generation algorithm to generate descriptions of p-groups. The algorithm implemented here is based on the algorithms described in [New77] and [OBr90].
  • A standard presentation algorithm used to compute a canonical power-commutator presentation of a p-group. The algorithm implemented here is described in [OBr94].
  • An algorithm which can be used to compute the automorphism group of a p-group. The algorithm implemented here is described in [OBr95].


An ANU Pq manual is given in chapter 57 of the GAP 3 manual.


[HN80]  G. Havas and M. F. Newman, Application of computers to questions like those of Burnside. In: Burnside groups (Bielefeld, 1977), Lecture Notes in Math. 806, Springer-Verlag 1980, pp. 211-230.

[NB96]  M. F. Newman and E. A. O'Brien, Application of computers to questions like those of Burnside II". Internat. J. Algebra Comput. 6 (1996), 593-605.

[New77]  M. F. Newman, Determination of groups of prime-power order. In: Group theory (Canberra, 1975), Lecture Notes in Math. 573, Springer-Verlag 1977, pp. 73-84.

[OBr90]  E. A. O'Brien, The p-group generation algorithm. J. Symbolic Comput. 9 (1990), 677-698.

[OBr94]  E. A. O'Brien, Isomorphism testing for p-groups. J. Symbolic Comput. 17 (1994), 133-147.

[OBr95]  E. A. O'Brien, Computing automorphism groups of p-groups. In: Computational Algebra and Number Theory (Sydney, 1992), Kluwer Academic Publishers 1995, pp. 83-90.

[Vau84]  M. R. Vaughan-Lee, An Aspect of the Nilpotent Quotient Algorithm. In: Computational Group Theory (Durham, 1982), Academic Press 1984, pp. 75-84.

[Vau90]  M. R. Vaughan-Lee, Collection from the left. J. Symbolic Comput. 9 (1990), 725-733.

Contact addresses

Eamonn O'Brien
Department of Mathematics
University of Auckland
Auckland, Private Bag 92019
New Zealand

Werner Nickel
Fachbereich Mathematik
TU Darmstadt
Schlossgartenstr. 7
D-64289 Darmstadt