View source for GAP (computer algebra system)

{{Short description|Computer algebra system}}
{{About|a software package|other uses|Gap (disambiguation)}}
{{Use dmy dates|date=October 2025}}
{{infobox software
| name                   = GAP
<!--
| logo                   = [[image:|128px]]
| screenshot             = [[image:|248px]]
| caption                = 
-->
| developer              = {{plainlist|
* [[University of St Andrews]]
* [[RWTH Aachen University|RWTH Aachen]]
* [[Braunschweig University of Technology|TU Braunschweig]]
* [[Colorado State University]]
* [[University of Kaiserslautern|TU Kaiserslautern]]
}}
| released = 1988
| latest release version = {{wikidata|property|reference|P348}}
| latest release date    = {{start date and age|{{wikidata|qualifier|P348|P577}}}}
| programming language   = [[C (programming language)|C]]
| operating_system       = [[Cross-platform]]
| genre                  = [[Computer algebra system]]
| license                = [[GNU General Public License]]
| website                = {{URL|https://www.gap-system.org/}}
| logo                   = Logo of the GAP computer algebra system.svg
}}

'''GAP''' ([[Group (mathematics)|groups]], [[algorithm]]s and [[Computer programming|programming]]) is an [[open-source]] [[computer algebra system]] for computational [[Discrete mathematics#Algebraic structures|discrete algebra]] with particular emphasis on [[computational group theory]].

==History==
GAP was developed at Lehrstuhl D für Mathematik (LDFM), [[RWTH Aachen University|Rheinisch-Westfälische Technische Hochschule Aachen]], Germany from 1986 to 1997. After the retirement of [[Joachim 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.<ref>{{cite web|url=https://www.gap-system.org/Doc/History/history.html|title=Some history of GAP|website=Official GAP website|access-date=27 September 2019}}</ref> In the summer of 2005 coordination was transferred to an equal partnership of four 'GAP Centres', located at the University of St Andrews, RWTH Aachen, [[Braunschweig University of Technology|Technische Universität Braunschweig]], and [[Colorado State University]] at [[Fort Collins, Colorado|Fort Collins]]; in April 2020, a fifth GAP Centre located at the [[University of Kaiserslautern|TU Kaiserslautern]] was added.<ref>{{cite web|url=https://www.gap-system.org/Contacts/centres.html|title=GAP Centres|website=Official GAP website|access-date=18 April 2020}}</ref>

==Features==
GAP contains a [[Procedural programming|procedural programming language]] and a large collection of functions to create and manipulate various mathematical objects. It supports integers and rational numbers of arbitrary size, memory permitting. [[Finite group]]s can be defined as [[Permutation group|groups of permutations]] and it is also possible to define [[Finitely-presented group|finitely presented groups]] by specifying generators and relations.  Several databases of important finite groups are included. GAP also allows to work with [[Matrix (mathematics)|matrices]] and with [[finite field]]s (which are represented using [[Conway polynomial (finite fields)|Conway polynomials]]). [[Ring (mathematics)|Rings]], [[Module (mathematics)|modules]] and [[Lie algebra]]s are also supported.

==Distribution==
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 [[Microsoft Windows|Windows]], and on [[Mac_(computer)|Mac]] systems. The standard distribution requires about 300 MB (about 400 MB if all the packages are loaded).

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|2021|03}}, there are 151 packages distributed with GAP, of which approximately 71 have been through this process.

An interface is available for using the [[Singular (software)|SINGULAR]] computer algebra system from within GAP.  GAP is also included in the mathematical software system [[SageMath]].

==Sample session==

=== Permutation group ===
{{sxhl|lang=gap-console|1=
gap> G:=SmallGroup(8,1); # Set G to be the 1st group (in GAP catalogue) 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); # Generators for the image of G under i – written as products of disjoint cyclic permutations.
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) ]
}}

=== Euclidean ring ===
{{sxhl|lang=gap-console|1=
gap> # test consistency of EuclideanDegree, EuclideanQuotient, EuclideanRemainder,
gap> # and QuotientRemainder for some ring and elements of it
gap> checkEuclideanRing :=
> function(R, colls...)
>   local coll1, coll2, a, b, deg_b, deg_r, q, r, qr;
>   if Length(colls) >= 1 then coll1:=colls[1];
>   elif Size(R) <= 100 then coll1 := R;
>   else coll1 := List([1..100],i->Random(R));
>   fi;
>   if Length(colls) >= 2 then coll2:=colls[2];
>   elif Size(R) <= 100 then coll2 := R;
>   else coll2 := List([1..100],i->Random(R));
>   fi;
>   for b in coll1 do
>     if IsZero(b) then continue; fi;
>     deg_b := EuclideanDegree(R, b);
>     for a in coll2 do
>       q := EuclideanQuotient(R, a, b); Assert(0, q in R);
>       r := EuclideanRemainder(R, a, b); Assert(0, r in R);
>       if a <> q*b + r then Error("a <> q*b + r for ", [R,a,b]); fi;
>       deg_r := EuclideanDegree(R, r);
>       if not IsZero(r) and deg_r >= deg_b then Error("Euclidean degree did not decrease for ",[R,a,b]); fi;
>       qr := QuotientRemainder(R, a, b);
>       if qr <> [q, r] then Error("QuotientRemainder inconsistent for ", [R,a,b]); fi;
>     od;
>   od;
>   return true;
> end;;

gap> # rings in characteristic 0
gap> checkEuclideanRing(Integers,[-100..100],[-100..100]);
true
gap> checkEuclideanRing(Rationals);
true
gap> checkEuclideanRing(GaussianIntegers);
true
gap> checkEuclideanRing(GaussianRationals);
true

gap> # finite fields
gap> ForAll(Filtered([2..50], IsPrimePowerInt), q->checkEuclideanRing(GF(q)));
true

gap> # ZmodnZ
gap> ForAll([1..50], m -> checkEuclideanRing(Integers mod m));
true
gap> checkEuclideanRing(Integers mod ((2*3*5)^2));
true
gap> checkEuclideanRing(Integers mod ((2*3*5)^3));
true
gap> checkEuclideanRing(Integers mod ((2*3*5*7)^2));
true
gap> checkEuclideanRing(Integers mod ((2*3*5*7)^3));
true
}}<ref>https://pygments.org/docs/lexers/#pygments.lexers.algebra.GAPConsoleLexer {{Bare URL inline|date=August 2025}}</ref>

==See also==
{{Portal|Free and open-source software}}
* [[Comparison of computer algebra systems]]

==References==
{{Reflist}}

==External links==
* {{Official website|https://www.gap-system.org}}
* {{GitHub|gap-system}} 

{{Computer algebra systems}}

{{DEFAULTSORT:Gap Computer Algebra System}}
[[Category:Computer algebra system software for Linux]]
[[Category:Computer algebra system software for macOS]]
[[Category:Computer algebra system software for Windows]]
[[Category:Free computer algebra systems]]