References for Methods of Computational Group Theory
This page provides some guide to theoretical background material for the
methods implemented in GAP and its packages.
The most comprehensive (and extremely wellwritten) textbook on
computational group theory (excluding
computational representation theory) is the

Handbook of Computational Group Theory [Ho05]
by Derek Holt.
The Computer Algebra Handbook [GKW03] aims
to provide an overview of the full field of computer algebra by individual
articles written by different authors in the style of an encyclopedia.
There are surveys on different areas and aspects as well as descriptions
of program systems, a bibliography with over 2100 entries, and pointers
to many conferences since 1979 and their proceedings. (It should be noted,
however, that most articles of the handbook were already written about
four years before its publication.) The two most relevant articles in our
context are also available via the internet:

Computational Group Theory by Charles C. Sims (pp. 6483 in
[GKW03]), available as Postscript or PDF file from
Selected Publications of Charles C. Sims
(note that for reading the PDF file you will probably need acrobat6),
and

Algorithms of Representation Theory by Gerhard Hiss (pp. 8488
in [GKW03]), available as
PDF file.
There are some monographs covering special areas:

Permutation Groups
Greg Butler [Bu91] gives an elementary introduction.
Ákos Seress [Se03]
gives an uptodate survey on permutattion group algorithms and analyses
their complexity.

Finitely Presented Groups
David Johnson's book [Jo97] is a very readable
introduction to the general subject of fp groups touching computational
aspects. The authoritative text on the subject of computing methods
for fp groups is the book [Si94] by Charles C. Sims.

Polycyclic Groups
There is no published textbook yet, the Habilitationsschrift
[Ei01] of Bettina Eick is presently the most
comprehensive source.

Representation Theory
The book Representations of Groups, A Computational Approach by Klaus Lux and Herbert Pahlings
[LP10]
provides a joint development of both ordinary and modular representation theory together
with the wealth of algorithms and their implementations that are playing such an important
role in the recent development of the field. GAP programs are used and
documented throughout the book. An important feature is that the book is accompanied by an own website
http://www.math.rwthaachen.de/~RepresentationsOfGroups/
which i.a. provides errata, solutions for the numerous exercises in the book and additional material.
This feature will in particular allow further updating of this important text.
Still, another good source are the two survey papers [LP91] and [LP99].

Lie Algebras
The book [deG00] by Willem A. de Graaf covers the
standard topics of Lie Algebra theory with strong emphasis on algorithmic
aspects.

Algebraic Number Theory and Commutative Algebra
These areas, some methods of which are used in GAP and
its packages, are e. g. presented by the book [Co00]
on computational number theory by Henri Cohen and the book
[GP02] on commutative algebra by GerdMartin Greuel
and Gerhard Pfister.
We refrain from listing any of the several hundred papers having contributed
to the development of algorithms in computational group theory. Rather
we refer to the bibliographies of the quoted books and the
Algebra Database in BibTeX, containing many such titles,
which has been compiled by Eamonn O'Brien.
Bibliography
 Bu91

Gregory Butler,
Fundamental Algorithms for Permutation Groups.
Lecture Notes in Computer Science, vol. 559,
Springer Verlag 1991, xii + 238 p.
 Co00

Henri Cohen,
A Course in Computational Number Theory.
Graduate texts in mathematics, vol. 138,
Springer Verlag, 4th ed. 2000, xx + 545 p.
 deG00

Willem A. de Graaf,
Lie Algebras: Theory and Algorithms.
NorthHolland mathematical Library, vol. 56,
Elsevier 2000, xii + 393 p.
 Ei01

Bettina Eick,
Algorithms for Polycyclic Groups.
Habilitationsschrift, Universität Kassel, 2000, 113 p.
 GKW03

Johannes Grabmeier, Erich Kaltofen, Volker Weispfenning, eds.,
Computer Algebra Handbook.
Springer Verlag 2003, xx + 637 p.
 GP02

GerdMartin Greuel, Gerhard Pfister
A SINGULAR Introduction to Commutative Algebra.
Springer Verlag 2003, xvii + 588 p.
 Ho05

Derek F. Holt,
Handbook of Computational Group Theory.
In the series 'Discrete Mathematics and its Applications',
Chapman & Hall/CRC 2005, xvi + 514 p.
 Jo97

David L. Johnson,
Presentations of Groups.
LMS Student Texts, vol. 15,
Cambridge University Press, 2nd ed. 1997, x + 216 p.
 LP91

Klaus Lux, Herbert Pahlings,
Computational Aspects of Representation Theory of Finite Groups.
pp. 3764 in: Representation Theory of Finite Groups and
FiniteDimensional Algebras, G. O. Michler, C. M. Ringel, eds., 1991.
 LP99

Klaus Lux, Herbert Pahlings,
Computational Aspects of Representation Theory of Finite Groups II.
pp. 381397 in: Algorithmic Algebra and Number Theory,
B. H. Matzat, G.M. Greuel, G. Hiss, eds., 1999.
 LP10

Klaus Lux, Herbert Pahlings,
Representations of Groups, A Computational Approach.
Cambridge Studies in Advanced Mathematics 124, Cambridge University Press 2010, x+460 p.
 Se03

Ákos Seress
Permutation Group Algorithms.
Cambridge Tracts in Mathematics, vol 152,
Cambridge University Press 2003, ix + 264 p.
A
sample of the book, including contents and introduction, can be
looked at in the web.
 Si94

Charles C. Sims,
Computation with finitely presented groups.
Encyclopedia of mathematics and its applications, vol. 48,
Cambridge University Press 1994, xiii + 604 p.
