GAP (Groups, Algorithms and Programming) is a computer algebra system for computational discrete algebra similar to Mathematica with particular emphasis on, but not restricted to, computational group theory. GAP was developed at Lehrstuhl D für Mathematik (LDFM), RWTH Aachen, Germany from 1986 to 1997. After the retirement of J. Neubüser from the chair of LDFM, the development and maintenance of GAP was coordinated by the School of Mathematical and Computational Sciences at the University of St. Andrews, Scotland. In the summer of 2005 coordination was transferred to an equal partnership of 4 `GAP Centres', located at St Andrews; LDFM; the Technical University of Braunschweig and Colorado State University at Fort Collins. In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... Flowcharts are often used to represent algorithms. ... Wikibooks has more about this subject: Computer programming Computer programming (often simply programming) is the craft of implementing one or more interrelated abstract algorithms using a particular programming language to produce a concrete computer program. ... A computer algebra system (CAS) is a software program that facilitates symbolic mathematics. ... Mathematica is a widely-used computer algebra system originally developed by Stephen Wolfram and sold by his company Wolfram Research. ... Group theory is that branch of mathematics concerned with the study of groups. ... The RWTH Aachen University is a large university located in Aachen (Germany). ... University of St Andrews The University of St Andrews was founded between 1410-1413 and is the oldest university in Scotland and the third oldest in the United Kingdom. ... Royal motto: Nemo me impune lacessit (Latin: No one provokes me with impunity) Scotlands location within the UK Main languages English Scots Scottish Gaelic Doric Capital Edinburgh Largest city Glasgow First Minister Jack McConnell Area  - Total  - % water Ranked 2nd UK 78,782 km² 1. ... Map of Germany showing Braunschweig Braunschweig [ˈbraunʃvaik] (English Brunswick) is a city of 245,500 people (as of December 31, 2004), located in Lower Saxony, Germany. ... Colorado State University is a public land grant institution of higher learning located in Fort Collins, Colorado in the United States. ... Horsetooth Rock, atop Horsetooth Mountain, is often used as a symbol of Fort Collins Fort Collins, situated on the Cache la Poudre River, is the largest city and county seat of Larimer County, Colorado. ...

GAP and its sources, including packages (sets of user contributed programs), data library (including a list of small groups) and the manual, are distributed freely, subject to "copyleft" conditions. GAP runs on any Unix system, under Windows, and on Macintosh systems. It requires a minimum of 32 MB disk space; the full distribution takes about 300 MB. To run GAP one needs a minimum of 20 MB of main memory - for most purposes 128 MB are sufficient. The following list in mathematics contains the finite groups of small order up to group isomorphism. ... The reversed c is the copyleft symbol. ... Unix or UNIX is a computer operating system originally developed in the 1960s and 1970s by a group of AT&T Bell Labs employees including Ken Thompson, Dennis Ritchie, and Douglas McIlroy. ... Microsoft Windows is a range of operating environments for personal computers and servers. ... The box for Mac OS X v10. ...

The user contributed packages are an important feature of the system, adding a great deal of functionality. GAP offers package authors the opportunity to submit these packages for a process of peer review, hopefully improving the quality of the final packages, and providing recognition akin to an academic publication for their authors. As of August 2005 there are 50 packages distributed with GAP, of which approximately 30 have been through this process. Peer review (known as refereeing in some academic fields) is a scholarly process used in the publication of manuscripts and in the awarding of funding for research. ...

The current version is 4.4.5, as of May 2005. GAP 3 (last release: 3.4.4) is still available but no longer supported.

An interface is available for using the SINGULAR computer algebra system from within GAP. SINGULAR is a computer algebra system for polynomial computations with special emphasis on the needs of commutative algebra, algebraic geometry, and singularity theory. ...

Sample session

 gap> G:=SmallGroup(8,1); # Set G to be a group of order 8. <pc group of size 8 with 3 generators> gap> i:=IsomorphismPermGroup(G); # Find an isomorphism from G to a group of permutations <action isomorphism> gap> Image(i,G); # The image of G under I - these are the generators of im G. Group([ (1,5,3,7,2,6,4,8), (1,3,2,4)(5,7,6,8), (1,2)(3,4)(5,6)(7,8) ]) gap> Elements(Image(i,G)); # All the elements of im G. [ (), (1,2)(3,4)(5,6)(7,8), (1,3,2,4)(5,7,6,8), (1,4,2,3)(5,8,6,7), (1,5,3,7,2,6,4,8), (1,6,3,8,2,5,4,7), (1,7,4,5,2,8,3,6), (1,8,4,6,2,7,3,5) ] 

External links

• GAP official site: Details about the system, the software itself and instructions how to obtain and to install it.
• The interface from GAP to Singular