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{{About|a software package|other uses|Gap (disambiguation)}}
{{use dmy dates|date=December 2022}}
{{infobox software
| name = GAP
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* [[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]]
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| license = [[GNU General Public License]]
| website = {{URL|https://www.gap-system.org/}}
| logo = GAP computer algebra system icon.png
}}
'''GAP''' ([[Group (mathematics)|Groups]], [[Algorithm]]s and [[Computer programming|Programming]]) is
==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=September 27, 2019}}</ref> In the summer of 2005 coordination was transferred to an equal partnership of four 'GAP Centres', located at the
==Features==
▲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=September 27, 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=April 18, 2020}}</ref>
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|Finite groups]] 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|finite fields]] (which are represented using [[Conway polynomial (finite fields)|Conway polynomials]]). [[Ring (mathematics)|Rings]], [[Module (mathematics)|modules]] and [[Lie algebra|Lie algebras]] 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 [[
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|
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]] ===
<pc group of size 8 with 3 generators>▼
{{sxhl|lang=gap-console|1=
<span style="color: darkblue;">gap></span> <span style="color: #B60000;">i:=IsomorphismPermGroup(G);</span> <span style="color: darkgray;"># Find an isomorphism from G to a group of permutations.</span>▼
gap> G:=SmallGroup(8,1); # Set G to be the 1st group (in GAP catalogue) of order 8.
<action isomorphism>▼
▲
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> Image(i,G); # Generators for the image of G under i - written as products of disjoint cyclic permutations.
<nowiki>[ (), (1,2)(3,4)(5,6)(7,8), (1,3,2,4)(5,7,6,8), (1,4,2,3)(5,8,6,7), ▼
gap> Elements(Image(i,G)); # All the elements of im G.
▲
}}
=== [[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==
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==External links==
* {{Official website|
* {{GitHub|gap-system}}
{{Computer algebra systems}}
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{{DEFAULTSORT:Gap Computer Algebra System}}
[[Category:Computer algebra system software for Linux]]
[[Category:Computer algebra system software for
[[Category:Computer algebra system software for Windows]]
[[Category:Free computer algebra systems]]
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