This page is devoted to answering some basic questions along the line “How do I construct … in GAP?” You may view the html source code for the GAP commands without the output or GAP prompt.
Please send suggestions, additions, corrections to David Joyner.
This page itself is under construction…
Questions
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How
do I … a polynomial? |
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How do I find the … of a group
representation? |
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How do I compute an mod m, where A is …?
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Given a group G, how do I compute … ?
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Answers
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- permutation:
To construct a permutation group, write down generators in disjoint cycle notation, put them in a list (i.e., surround them by square brackets), and the permutation group G generated by the cycles (1,2)(3,4) and (1,2,3):
- permutation:
gap> G:=Group((1,2)(3,4),(1,2,3)); Group([ (1,2)(3,4), (1,2,3) ])
This is of course a subgroup of the symmetric group S4 on 4 letters. Indeed, this G is in fact the alternating group on four letters, A4.
By virtue of the fact that the permutations generating G employ integers less than or equal to 4, this group G is a subgroup of the symmetric group S4 on 4 letters. Some permutation groups have special constructions:
gap> S4:=SymmetricGroup(4); Sym( [ 1 .. 4 ] ) gap> A4:=AlternatingGroup(4); Alt( [ 1 .. 4 ] ) gap> IsSubgroup(S4,G); true gap> IsSubgroup(A4,G); true gap> S3:=SymmetricGroup(3); Sym( [ 1 .. 3 ] ) gap> IsSubgroup(S3,G); false
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- dihedral
To construct a dihedral group, use the special “DihedralGroup” command:
- dihedral
gap> G:=DihedralGroup(6); gap> Size(G); 6 gap> f:=GeneratorsOfGroup( G ); [ f1, f2 ] gap> f[1]^2; f[2]^3; identity of ... identity of ... gap> f[1]^2= f[2]^3; true
- cyclic group
To construct a cyclic group, you may construct integers mod n:gap> R:=ZmodnZ( 12); (Integers mod 12) gap> a:=Random(R); ZmodnZObj( 11, 12 ) gap> 4*a; ZmodnZObj( 8, 12 ) gap> b:=Random(R); ZmodnZObj( 9, 12 ) gap> a+b; ZmodnZObj( 8, 12 )
or use the special “CyclicGroup” command
gap> G:=CyclicGroup(12); pc group of size 12 with 3 generators gap> a:=Random(G); f3^2 gap> f:=GeneratorsOfGroup( G ); [ f1, f2, f3 ] gap> f[1]^4; f3 gap> f[1]^12; identity of ...
- conjugacy:
The conjugacy classes of a group G are computed using the “ConjugacyClasses” command. This is a list of classes {x^-1*g*x | x in G}.gap> G:=SL(2,7); SL(2,7) gap> CG:=ConjugacyClasses(G); [ [ [ Z(7)^0, 0*Z(7) ], [ 0*Z(7), Z(7)^0 ] ]^G, [ [ 0*Z(7), Z(7)^3 ], [ Z(7)^0, Z(7)^5 ] ]^G, [ [ 0*Z(7), Z(7)^4 ], [ Z(7)^5, Z(7)^5 ] ]^G, [ [ Z(7)^3, 0*Z(7) ], [ 0*Z(7), Z(7)^3 ] ]^G, [ [ 0*Z(7), Z(7)^3 ], [ Z(7)^0, Z(7)^2 ] ]^G, [ [ 0*Z(7), Z(7)^4 ], [ Z(7)^5, Z(7)^2 ] ]^G, [ [ 0*Z(7), Z(7)^3 ], [ Z(7)^0, 0*Z(7) ] ]^G, [ [ 0*Z(7), Z(7)^3 ], [ Z(7)^0, Z(7)^4 ] ]^G, [ [ 0*Z(7), Z(7)^3 ], [ Z(7)^0, Z(7) ] ]^G, [ [ Z(7)^4, 0*Z(7) ], [ 0*Z(7), Z(7)^2 ] ]^G, [ [ Z(7)^5, 0*Z(7) ], [ 0*Z(7), Z(7) ] ]^G ] gap> g:=Representative(CG[3]); Order(g); [ [ 0*Z(7), Z(7)^4 ], [ Z(7)^5, Z(7)^5 ] ] 14 gap> g:=Representative(CG[4]); Order(g); [ [ Z(7)^3, 0*Z(7) ], [ 0*Z(7), Z(7)^3 ] ] 2 gap> g:=Representative(CG[5]); Order(g); [ [ 0*Z(7), Z(7)^3 ], [ Z(7)^0, Z(7)^2 ] ] 7 gap> g:=Representative(CG[6]); Order(g); [ [ 0*Z(7), Z(7)^4 ], [ Z(7)^5, Z(7)^2 ] ] 7 gap>
- presented
To construct a finitely presented group in GAP, use the “FreeGroup” and “FpGroupPresentation” commands. Here is one example.gap> M12 := MathieuGroup( 12 ); Group([ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6), (1,12)(2,11)(3,6)(4,8)(5,9)(7,10) ]) gap> F := FreeGroup( "a", "b", "c" ); free group on the generators [ a, b, c ] gap> words := [ F.1, F.2 ]; [ a, b ] gap> P := PresentationViaCosetTable( M12, F, words ); presentation with 3 gens and 10 rels of total length 97 gap> TzPrintRelators( P ); #I 1. c^2 #I 2. b^4 #I 3. a*c*a*c*a*c #I 4. a*b^2*a*b^-2*a*b^-2 #I 5. a^11 #I 6. a^2*b*a^-2*b^2*a*b^-1*a^2*b^-1 #I 7. a*b*a^-1*b*a^-1*b^-1*a*b*a^-1*b*a^-1*b^-1 #I 8. a^2*b*a^2*b^2*a^-1*b*a^-1*b^-1*a^-1*b^-1 #I 9. a*b*a*b*a^2*b^-1*a^-1*b^-1*a*c*b*c #I 10. a^4*b*a^2*b*a^-2*c*a*b*a^-1*c gap> G := FpGroupPresentation( P ); fp group on the generators [ a, b, c ] gap> RelatorsOfFpGroup( G ); [ c^2, b^4, a*c*a*c*a*c, a*b^-2*a*b^-2*a*b^-2, a^11, a^2*b*a^-2*b^-2*a*b^-1*a^2*b^-1, a*b*a^-1*b*a^-1*b^-1*a*b*a^-1*b*a^-1*b^-1, a^2*b*a^2*b^-2*a^-1*b*a^-1*b^-1*a^-1*b^-1, a*b*a*b*a^2*b^-1*a^-1*b^-1*a*c*b*c, a^4*b*a^2*b*a^-2*c*a*b*a^-1*c ] gap> Size(M12); 95040 gap> Size(G); 95040 gap> IsomorphismGroups(G,M12); ????????
The last command is computationally intensive and requires more than the default memory allocation of 256M of RAM.
Here is another example.
gap> F := FreeGroup( "a", "b"); free group on the generators [ a, b ] gap> G:=F/[F.1^2,F.2^3,F.1*F.2*F.1^(-1)*F.2^(-1)]; fp group on the generators [ a, b ] gap> Size(G); 6
- rref
The key command for row reduction is “TriangulizeMat”. The following example illustrates the syntax.gap> M:=[[1,2,3,4,5],[1,2,1,2,1],[1,1,0,0,0]]; [ [ 1, 2, 3, 4, 5 ], [ 1, 2, 1, 2, 1 ], [ 1, 1, 0, 0, 0 ] ] gap> TriangulizeMat(M); gap> M; [ [ 1, 0, 0, -1, 1 ], [ 0, 1, 0, 1, -1 ], [ 0, 0, 1, 1, 2 ] ] gap> Display(M); [ [ 1, 0, 0, -1, 1 ], [ 0, 1, 0, 1, -1 ], [ 0, 0, 1, 1, 2 ] ] gap> M:=Z(3)^0*[[1,2,3,4,5],[1,2,1,2,1],[1,1,0,0,0]]; [ [ Z(3)^0, Z(3), 0*Z(3), Z(3)^0, Z(3) ], [ Z(3)^0, Z(3), Z(3)^0, Z(3), Z(3)^0 ], [ Z(3)^0, Z(3)^0, 0*Z(3), 0*Z(3), 0*Z(3) ] ] gap> TriangulizeMat(M); gap> Display(M); 1 . . 2 1 . 1 . 1 2 . . 1 1 2 gap>
- kernel:
There are different methods for matrices over the integers and matrices over a field. For integer entries, related commands include “NullspaceIntMat” and “SolutionNullspaceIntMat” in section 25.1 “Linear equations over the integers and Integral Matrices” of the reference manual.gap> M:=[[1,2,3],[4,5,6],[7,8,9]]; [ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 7, 8, 9 ] ] gap> NullspaceIntMat(M); [ [ 1, -2, 1 ] ] gap> SolutionNullspaceIntMat(M,[0,0,1]); [ fail, [ [ 1, -2, 1 ] ] ] gap> SolutionNullspaceIntMat(M,[0,0,0]); [ [ 0, 0, 0 ], [ [ 1, -2, 1 ] ] ] gap> SolutionNullspaceIntMat(M,[1,2,3]); [ [ 1, 0, 0 ], [ [ 1, -2, 1 ] ] ]
Here (0,0,1) is not in the image of M
(under v-> v*M) but (0,0,0) and (1,2,3) are.For field entries, related commands include “NullspaceMat” and “TriangulizedNullspaceMat” in section 24.6 “Matrices Representing Linear Equations and the Gaussian Algorithm”
of the reference manual.gap> M:=[[1,2,3],[4,5,6],[7,8,9]]; [ [ 1, 2, 3 ], [ 4, 5, 6 ], [ 7, 8, 9 ] ] gap> NullspaceMat(M); [ [ 1, -2, 1 ] ] gap> TriangulizedNullspaceMat(M); [ [ 1, -2, 1 ] ] gap> M:=[[1,2,3,1,1],[4,5,6,1,1],[7,8,9,1,1],[1,2,3,1,1]]; [ [ 1, 2, 3, 1, 1 ], [ 4, 5, 6, 1, 1 ], [ 7, 8, 9, 1, 1 ], [ 1, 2, 3, 1, 1 ] ] gap> NullspaceMat(M); [ [ 1, -2, 1, 0 ], [ -1, 0, 0, 1 ] ] gap> TriangulizedNullspaceMat(M); [ [ 1, 0, 0, -1 ], [ 0, 1, -1/2, -1/2 ] ]
- characteristic polynomial:
Please see section 24.12.1 of the GAP reference manual for examples of characteristic polynomial of a square matrix (“CharacteristicPolynomial”) and section 56.3 for examples of the “characteristic polynomial” (called a “TracePolynomial”) of an element of a field extension.
- character:
GAP contains very extensive character theoretic functions and data libraries (including an interface the character table in the Atlas). Here is just one simple example.gap> G:=Group((1,2)(3,4),(1,2,3)); Group([ (1,2)(3,4), (1,2,3) ]) gap> T:=CharacterTable(G); CharacterTable( Alt( [ 1 .. 4 ] ) ) gap> Display(T); CT1 2 2 2 . . 3 1 . 1 1 1a 2a 3a 3b 2P 1a 1a 3b 3a 3P 1a 2a 1a 1a X.1 1 1 1 1 X.2 1 1 A /A X.3 1 1 /A A X.4 3 -1 . . A = E(3)^2 = (-1-ER(-3))/2 = -1-b3 gap> irr:=Irr(G); [ Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 1, 1, 1, 1 ] ), Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 1, 1, E(3)^2, E(3) ] ), Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 1, 1, E(3), E(3)^2 ] ), Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 3, -1, 0, 0 ] ) ] gap> Display(irr); [ [ 1, 1, 1, 1 ], [ 1, 1, E(3)^2, E(3) ], [ 1, 1, E(3), E(3)^2 ], [ 3, -1, 0, 0 ] ] gap> chi:=irr[2]; gamma:=CG[3]; g:=Representative(gamma); g^chi; Character( CharacterTable( Alt( [ 1 .. 4 ] ) ), [ 1, 1, E(3)^2, E(3) ] ) (1,2,3)^G (1,2,3) E(3)^2For further details and examples, see chapters 69– 72 of the GAP reference manual.
- brauer:
Just a simple example of what GAP can do here. To construct a Brauer character table:gap> G:=Group((1,2)(3,4),(1,2,3)); Group([ (1,2)(3,4), (1,2,3) ]) gap> irr:=IrreducibleRepresentations(G,GF(7)); [ [ (1,2)(3,4), (1,2,3) ] -> [ [ [ Z(7)^0 ] ], [ [ Z(7)^0 ] ] ], [ (1,2)(3,4), (1,2,3) ] -> [ [ [ Z(7)^0 ] ], [ [ Z(7)^4 ] ] ], [ (1,2)(3,4), (1,2,3) ] -> [ [ [ Z(7)^0 ] ], [ [ Z(7)^2 ] ] ], [ (1,2)(3,4), (1,2,3) ] -> [ [ [ 0*Z(7), Z(7)^3, Z(7)^0 ], [ 0*Z(7), Z(7)^3, 0*Z(7) ], [ Z(7)^0, Z(7)^3, 0*Z(7) ] ], [ [ 0*Z(7), Z(7)^0, 0*Z(7) ], [ 0*Z(7), 0*Z(7), Z(7)^0 ], [ Z(7)^0, 0*Z(7), 0*Z(7) ] ] ] ] gap> brvals := List(irr,chi-> List(ConjugacyClasses(G),c-> BrauerCharacterValue(Image(chi, Representative(c))))); [ [ 1, 1, 1, 1 ], [ 1, 1, E(3)^2, E(3) ], [ 1, 1, E(3), E(3)^2 ], [ 3, -1, 0, 0 ] ] gap> Display(brvals); [ [ 1, 1, 1, 1 ], [ 1, 1, E(3)^2, E(3) ], [ 1, 1, E(3), E(3)^2 ], [ 3, -1, 0, 0 ] ] gap>List(ConjugacyClasses(G),c->BrauerCharacterValue(Image(chi, Representative(c)))));
#Display(brvals);
T:=CharacterTable(G);
Display(T);
–>
- polynomial
There are various ways to construct a polynomial in GAP.gap> Pts:=Z(7)^0*[1,2,3]; [ Z(7)^0, Z(7)^2, Z(7) ] gap> Vals:=Z(7)^0*[1,2,6]; [ Z(7)^0, Z(7)^2, Z(7)^3 ] gap> g:=InterpolatedPolynomial(GF(7),Pts,Vals); Z(7)^5*x_1^2+Z(7)
Or:
gap> p:=3;; F:=GF(p);; gap> R:=PolynomialRing(F,["x1","x2"]); PolynomialRing(..., [ x1, x2 ]) gap> vars:=IndeterminatesOfPolynomialRing(R);; gap> x1:=vars[1]; x2:=vars[2]; x1 x2 gap> p:=x1^5-x2^5; x1^5-x2^5 gap> DivisorsMultivariatePolynomial(p,R); [ x1^4+x1^3*x2+x1^2*x2^2+x1*x2^3+x2^4, x1-x2 ]
Or:
gap> x:=X(Rationals); x_1 gap> f:=x+x^2+1; x_1^2+x_1+1 gap> Value(f,[x],[1]); 3
- factor
To factor a polynomial in GAP, there is one command for univariate polynomials (“Factors”) and another command for multivariate polynomials (“DivisorsMultivariatePolynomial”). For a factoring a univariate polynomial, GAP provides only methods over finite fields and over subfields of cyclotomic fields. Please see the examples given in section 64.10 “Polynomial Factorization” for more details. For multivariate polynomials, a very slow algorithm has been implemented in GAP and an interface to a very fast algorithm in Singular has been implemented for those who have both Singular and the GAP Singular package installed. The former of these was illustrated above in “polynomial” above. (Again, the ground field must be a finite field or a subfields of cyclotomic fields.) For the latter, please see the example in the (GAP-)Singular manual FactorsUsingSingularNC.
- roots
There are some situations where GAP can find the roots of a polynomial but GAP does not do this generally. (The roots must generate either a finite field or a subfield of a cyclotomic field.) However, there is a package called RadiRoot which must be installed which does help to do this for polynomials with rational coefficients (radiroot itself requires other packages to be installed; please see the webpage for more details). The “Factors” command actually has an option which allows you to increase the groundfield so that a factorization actually returns the roots. Please see the examples given in section 64.10 “Polynomial Factorization” for more details. Here is a second approach.gap> p:=3; n:=4; F:=GF(p^n); c:=Random(F); r:=2; 3 4 GF(3^4) Z(3^4)^79 2 gap> x:=X(F,1); f:=x^r-c*x+c-1; x_1 x_1^2+Z(3^4)^39*x_1+Z(3^4)^36 gap> F_f:=FieldExtension( F, f ); AsField( GF(3^4), GF(3^8) ) gap> alpha:=RootOfDefiningPolynomial(F_f); Z(3^4)^36 gap> Value(f,[x],[alpha]); 0*Z(3)
Here is a third. First, enter the following program
RootOfPolynomial:=function(f,R) local F0,Ff,a; F0:=CoefficientsRing(R); Ff:=FieldExtension(F0,f); a:=RootOfDefiningPolynomial(Ff); return a; end;
Here’s how this can be used to find a root:
gap> F:=Rationals; Rationals gap> x:=X(F,1); f:=x^2+x+1; x_1 x_1^2+x_1+1 gap> R:=PolynomialRing( F, [ x ]); PolynomialRing(..., [ x_1 ]) gap> a:=RootOfPolynomial(f,R); E(3) gap> # check: gap> Value(f,[x],[a]); 0
Related links:
- In the GAP Forum: Hensel lifting discussion.
- In the manual, Galois groups.
- evaluate:
The relevant command is “Value”. There are several examples already on this page. For others, please see the examples given in section 64.7 Multivariate polynomials of the manual. For sparse uivariate polynomials, there is also the command “ValuePol” in section 23.6 of the manual.
- integer power
This is easy and intuitive:gap> a:=1000; n:=100000; m:=123; 1000 100000 123 gap> a^n mod m; 1
- matrix power:
This too is easy and intuitive:gap> A:=[[1,2],[3,4]]; n:=100000; m:=123; [ [ 1, 2 ], [ 3, 4 ] ] 100000 123 gap> A^n mod m; [ [ 1, 41 ], [ 0, 1 ] ]
- polynomial power
GAP allows you to do arithmetic over the polynomial ring R[x], where R = Z/nZ (where n is a positive integer). Here’s an example.gap> Z4:=ZmodnZ(4); (Integers mod 4) gap> R:=UnivariatePolynomialRing(Z4,1); PolynomialRing(..., [ x ]) gap> x:=IndeterminatesOfPolynomialRing(R)[1]; x gap> I:=TwoSidedIdealByGenerators( R,[x^8-x^0]); two-sided ideal in PolynomialRing(..., [ x ]), (1 generators) gap> gen:=x^8-x^0; x8-ZmodnZObj(1,4) gap> QuotientRemainder(R,x^8,gen); [ ZmodnZObj(1,4), ZmodnZObj(1,4) ] gap> QuotientRemainder(R,x^15,gen); [ x^7, x^7 ] gap> QuotientRemainder(R,x^15+x^8,gen); [ x^7+ZmodnZObj(1,4), x^7+ZmodnZObj(1,4) ] gap> PowerMod( R, x+x^0, 15, gen ); ZmodnZObj(0,4) gap> PowerMod( R, x, 15, gen ); x^7
- Groebner basis
GAP’s Groebner bases algorithms are relatively slow and are included mostly for simple examples and for teaching purposes. However, a GAP interface to a very fast algorithm in Singular has been implemented for those who have both Singular and the GAP Singular package installed. The former of these is illustrated in section 64.17 Groebner bases of the GAP manual. For the latter, please see the example in the (GAP-)Singular manual GroebnerBasis.
- normal subgroup:
Here is an example:gap> G := AlternatingGroup( 5 ); Group( (1,2,5), (2,3,5), (3,4,5) ) gap> normal := NormalSubgroups( G ); [ Subgroup( Group( (1,2,5), (2,3,5), (3,4,5) ), [ ] ), Subgroup( Group( (1,2,5), (2,3,5), (3,4,5) ), [ (1,2)(3,4), (1,3)(4,5), (1,4)(2,3) ] ) ]Related links:
- Please see Volkmar Felsch’s GAP Forum response to a related question.
- The xgap package (or, on a mac, Gap.app) displays subgroup lattices graphically.
- abelian subgroup
One idea to compute all the abelian subgroups is to compute all the subgroups then “filter” out the abelian ones. Here is an illustration, taked from a GAP Forum response Volkmar Felsch.gap> G := AlternatingGroup( 5 ); Group( (1,2,5), (2,3,5), (3,4,5) ) gap> classes := ConjugacyClassesSubgroups( G ); [ ConjugacyClassSubgroups( Group( (1,2,5), (2,3,5), (3,4,5) ), Subgroup( Group( (1,2,5), (2,3,5), (3,4,5) ), [ ] ) ), ConjugacyClassSubgroups( Group( (1,2,5), (2,3,5), (3,4,5) ), Subgroup( Group( (1,2,5), (2,3,5), (3,4,5) ), [ (2,3)(4,5) ] ) ), ConjugacyClassSubgroups( Group( (1,2,5), (2,3,5), (3,4,5) ), Subgroup( Group( (1,2,5), (2,3,5), (3,4,5) ), [ (3,4,5) ] ) ), ConjugacyClassSubgroups( Group( (1,2,5), (2,3,5), (3,4,5) ), Subgroup( Group( (1,2,5), (2,3,5), (3,4,5) ), [ (2,3)(4,5), (2,4)(3,5) ] ) ), ConjugacyClassSubgroups( Group( (1,2,5), (2,3,5), (3,4,5) ), Subgroup( Group( (1,2,5), (2,3,5), (3,4,5) ), [ (1,2,3,4,5) ] ) ), ConjugacyClassSubgroups( Group( (1,2,5), (2,3,5), (3,4,5) ), Subgroup( Group( (1,2,5), (2,3,5), (3,4,5) ), [ (3,4,5), (1,2)(4,5) ] ) ), ConjugacyClassSubgroups( Group( (1,2,5), (2,3,5), (3,4,5) ), Subgroup( Group( (1,2,5), (2,3,5), (3,4,5) ), [ (1,2,3,4,5), (2,5)(3,4) ] ) ), ConjugacyClassSubgroups( Group( (1,2,5), (2,3,5), (3,4,5) ), Subgroup( Group( (1,2,5), (2,3,5), (3,4,5) ), [ (2,3)(4,5), (2,4)(3,5), (3,4,5) ] ) ), ConjugacyClassSubgroups( Group( (1,2,5), (2,3,5), (3,4,5) ), Group( (1,2,5), (2,3,5), (3,4,5) ) ) ] gap> cl := classes[4]; ConjugacyClassSubgroups( Group( (1,2,5), (2,3,5), (3,4,5) ), Subgroup( Group( (1,2,5), (2,3,5), (3,4,5) ), [ (2,3)(4,5), (2,4)(3,5) ] ) ) gap> length := Size( cl ); 5 gap> rep := Representative( cl ); Subgroup( Group( (1,2,5), (2,3,5), (3,4,5) ), [ (2,3)(4,5), (2,4)(3,5) ] ) gap> order := Size( rep ); 4 gap> IsAbelian( rep ); true gap> abel := Filtered( classes, cl -> IsAbelian( Representative( cl ) ) ); [ ConjugacyClassSubgroups( Group( (1,2,5), (2,3,5), (3,4,5) ), Subgroup( Group( (1,2,5), (2,3,5), (3,4,5) ), [ ] ) ), ConjugacyClassSubgroups( Group( (1,2,5), (2,3,5), (3,4,5) ), Subgroup( Group( (1,2,5), (2,3,5), (3,4,5) ), [ (2,3)(4,5) ] ) ), ConjugacyClassSubgroups( Group( (1,2,5), (2,3,5), (3,4,5) ), Subgroup( Group( (1,2,5), (2,3,5), (3,4,5) ), [ (3,4,5) ] ) ), ConjugacyClassSubgroups( Group( (1,2,5), (2,3,5), (3,4,5) ), Subgroup( Group( (1,2,5), (2,3,5), (3,4,5) ), [ (2,3)(4,5), (2,4)(3,5) ] ) ), ConjugacyClassSubgroups( Group( (1,2,5), (2,3,5), (3,4,5) ), Subgroup( Group( (1,2,5), (2,3,5), (3,4,5) ), [ (1,2,3,4,5) ] ) ) ]
- homology
This depends on how the group is given. For example, suppose that G is a permutation group with generators genG and H is a permutation group with generators genH. To find a homomorphism from G to H, one may use the “GroupHomomorphismByImages” or “GroupHomomorphismByImagesNC” commands. For examples of the syntax, please see section 38.1 Creating Group Homomorphisms. Here’s an illustration of how to convert a finitely presented group into a permutation group.gap> p:=7; 7 gap> G:=PSL(2,p); Group([ (3,7,5)(4,8,6), (1,2,6)(3,4,8) ]) gap> H:=SchurCover(G); fp group of size 336 on the generators [ f1, f2, f3 ] gap> iso:=IsomorphismPermGroup(H); [ f1, f2, f3 ] -> [ (1,2,4,3)(5,9,7,10)(6,11,8,12)(13,14,15,16), (2,5,6)(3,7,8)(11,13,14)(12,15,16), (1,4)(2,3)(5,7)(6,8)(9,10)(11,12)(13, 15)(14,16) ] gap> H0:=Image(iso); # 2-cover of PSL2 Group([ (1,2,4,3)(5,9,7,10)(6,11,8,12)(13,14,15,16), (2,5,6)(3,7,8)(11,13,14)(12,15,16), (1,4)(2,3)(5,7)(6,8)(9,10)(11,12)(13, 15)(14,16) ]) gap> IdGroup(H0); [ 336, 114 ] gap> IdGroup(SL(2,7)); [ 336, 114 ] gap>
- semi-direct product(Contributed by Nilo de Roock):
As you can easily verify, D8 is isomorphic to C2:C4. Or in GAP…N:=CyclicGroup(IsPermGroup,4); G:=CyclicGroup(IsPermGroup,2); AutN:=AutomorphismGroup(N); f:=GroupHomomorphismByImages(G,AutN,GeneratorsOfGroup(G),[Elements(AutN)[2]]); NG:=SemidirectProduct(G,f,N);
Verify with
StructureDescription(NG);
- semi-direct products(Contributed by Nilo de Roock):
The following shows how to construct all non-abelian groups of order 12 as semi-direct products. These products are not trivial yet small enough to verify by hand.#D12 = (C2 x C2) : C3 G1:=CyclicGroup(IsPermGroup,2); G2:=CyclicGroup(IsPermGroup,2); G:=DirectProduct(G1,G2); N:=CyclicGroup(IsPermGroup,3); AutN:=AutomorphismGroup(N); f:=GroupHomomorphismByImages(G,AutN,[Elements(G)[1],Elements(G)[2],Elements(G)[3],Elements(G)[4]],[Elements(AutN)[1],Elements(AutN)[2],Elements(AutN)[1],Elements(AutN)[2]]); NG:=SemidirectProduct(G,f,N); Print(str(NG)); Print("\n");#T = C4 : C3 G:=CyclicGroup(IsPermGroup,4); N:=CyclicGroup(IsPermGroup,3); AutN:=AutomorphismGroup(N); f:=GroupHomomorphismByImages(G,AutN,[Elements(G)[1],Elements(G)[2],Elements(G)[3],Elements(G)[4]],[Elements(AutN)[1],Elements(AutN)[2],Elements(AutN)[1],Elements(AutN)[2]]); NG:=SemidirectProduct(G,f,N); Print(str(NG)); Print("\n");#A4 = C3 : (C2 x C2) G:=CyclicGroup(IsPermGroup,3); N1:=CyclicGroup(IsPermGroup,2); N2:=CyclicGroup(IsPermGroup,2); N:=DirectProduct(G1,G2); AutN:=AutomorphismGroup(N); f:=GroupHomomorphismByImages(G,AutN,[Elements(G)[1],Elements(G)[2],Elements(G)[3]],[Elements(AutN)[1],Elements(AutN)[4],Elements(AutN)[5]]); NG:=SemidirectProduct(G,f,N); Print(str(NG)); Print("\n");
- cohomology
GAP will compute the Schur multiplier H2(G,C) using the
“AbelianInvariantsMultiplier” command. Here is an example showing how to find H2(A5,C), where A5 is the alternating group on 5 letters.gap> A5:=AlternatingGroup(5); Alt( [ 1 .. 5 ] ) gap> AbelianInvariantsMultiplier(A5); [ 2 ]
So, H2(A5,C) is Z/2Z.
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