A homomorphism \(m\) from a groupoid \(G\) to a groupoid \(H\) consists of a map from the objects of \(G\) to those of \(H\) together with a map from the elements of \(G\) to those of \(H\) which is compatible with tail and head and which preserves multiplication:
\[ m(g1 : o1 \to o2)*m(g2 : o2 \to o3) ~=~ m(g1*g2 : o1 \to o3). \]
Note that when a homomorphism is not injective on objects, the image of the source need not be a subgroupoid of the range. A simple example of this is given by a homomorphism from the two-object, four-element groupoid with trivial group to the free group \(\langle a \rangle\) on one generator, when the image is \([1,a^n,a^{-n}]\) for some \(n>0\).
A variety of homomorphism operations are available.
The basic construction is a homomorphism \(\phi : G \to H\) from a connected groupoid \(G\) to a connected groupoid \(H\), constructed using GroupoidHomomorphismFromSinglePiece, (see 5.1).
Since more than one connected groupoid may be mapped to the same range, we then have the operation GroupoidHomomorphismToSinglePiece, (see 5.4).
The third case arises when the range is a union of connected groupoids, in which case HomomorphismByUnion is called, (see 5.5).
Fourthly, there are is an additional operation for the case where the source is homogeneous and discrete, GroupoidHomomorphismFromHomogeneousDiscrete, (see 5.4-2).
Finally, there are special operations for inclusion mappings, restricted mappings, and groupoid automorphisms (see 5.6).
‣ GroupoidHomomorphismFromSinglePiece( src, rng, gens, images ) | ( operation ) |
‣ GroupoidHomomorphism( args ) | ( function ) |
‣ MappingToSinglePieceData( map ) | ( attribute ) |
The simplest groupoid homomorphism is a mapping \(\phi : G \to H\) from a connected groupoid \(G\) to a connected groupoid \(H\). There are two equivalent sets of input data which may be used. Both require the Source \(G\) and the Range \(H\). The first then requires:
the set of generating arrows GeneratorsOfGroupoid(G);
a list of image arrows in \(H\).
This data is stored in the attribute MappingGeneratorsImages.
The alternative input data consists of:
a homomorphism rhom from the root group of \(G\) to the group at the image object in \(H\);
a list imobs of the images of the objects of \(G\);
a list imrays of the elements in the images of the rays of \(G\), so that the image \(\phi(r_i : o_1 \to o_i)\) of the \(i\)-th ray is (imrays[i]:imobs[1]\(\to\)imobs[i]).
This data is stored in the attribute MappingToSinglePieceData.
So an alternative way to construct a homomorphism of groupoids is to make a call of the form GroupoidHomomorphism(src,rng,rhom,imobs,imrays).
In the following example the same homomorphism is constructed using both methods.
gap> gen1 := GeneratorsOfGroupoid( Gq8 ); [ [x : -28 -> -28], [y : -28 -> -28], [y2 : -28 -> -28], [<identity> of ... : -28 -> -27] ] gap> gen2 := GeneratorsOfGroupoid( Hd8b ); [ [(1,2,3,4) : -14 -> -14], [(1,3) : -14 -> -14], [(1,2,3) : -14 -> -13], [(1,2,4) : -14 -> -12] ] gap> images := [ gen2[1]^2, gen2[1]*gen2[2], IdentityArrow(Hd8b,-14), gen2[4] ]; [ [(1,3)(2,4) : -14 -> -14], [(1,2)(3,4) : -14 -> -14], [() : -14 -> -14], [(1,2,4) : -14 -> -12] ] gap> mor1 := GroupoidHomomorphism( Gq8, Hd8b, gen1, images ); groupoid homomorphism : Gq8 -> Hd8b [ [ [x : -28 -> -28], [y : -28 -> -28], [y2 : -28 -> -28], [<identity> of ... : -28 -> -27] ], [ [(1,3)(2,4) : -14 -> -14], [(1,2)(3,4) : -14 -> -14], [() : -14 -> -14], [(1,2,4) : -14 -> -12] ] ] gap> genq8 := GeneratorsOfGroup( q8 );; gap> imh := [ (1,3)(2,4), (1,2)(3,4), () ];; gap> h := GroupHomomorphismByImages( q8, d8, genq8, imh ); [ x, y, y2 ] -> [ (1,3)(2,4), (1,2)(3,4), () ] gap> mor2 := GroupoidHomomorphism( Gq8, Hd8b, h, [-14,-12], [(),(1,2,4)] );; gap> mor1=mor2; true gap> e := Arrow( Gq8, Product(genq8), -27, -28 ); [x*y*y2 : -27 -> -28] gap> ImageElm( mor2, e ); [(2,4,3) : -12 -> -14]
The properties listed in subsection 3.3 for homomorphisms of magmas with objects also apply to groupoid homomorphisms.
gap> [ IsInjectiveOnObjects( mor2 ), IsSurjectiveOnObjects( mor2 ) ]; [ true, false ] gap> [ IsInjective( mor2 ), IsSurjective( mor2 ) ]; [ false, false ] gap> ad8 := GroupHomomorphismByImages( d8, d8, > [ (1,2,3,4), (1,3) ], [ (1,4,3,2), (2,4) ] );; gap> md8 := GroupoidHomomorphism( Gd8, Gd8, ad8, [-7,-9,-8], [(),(1,3),(2,4)] ); groupoid homomorphism : Gd8 -> Gd8 [ [ [(1,2,3,4) : -9 -> -9], [(1,3) : -9 -> -9], [() : -9 -> -8], [() : -9 -> -7] ], [ [(1,4,3,2) : -7 -> -7], [(2,4) : -7 -> -7], [(1,3) : -7 -> -9], [(2,4) : -7 -> -8] ] ] gap> IsBijectiveOnObjects( md8 ); true gap> [ IsInjective( md8 ), IsSurjective( md8 ) ]; [ true, true ] gap> [ IsEndomorphismWithObjects( md8 ), IsAutomorphismWithObjects( md8 ) ]; [
The attributes of a groupoid homomorphism mor from a single piece groupoid cover both forms of construction defined above.
S = Source(mor) is the source groupoid of the homomorphism;
R = Range(mor) is the range groupoid of the homomorphism;
RootGroupHomomorphism(mor) is the group homomorphism from the root group of S to the group at the image object in R of the root object in S;
ImagesOfObjects(mor) is the list of objects in R which are the images of the objects in S;
ImageElementsOfRays(mor) is the list of group elements in those arrows in R which are the images of the rays in S;
MappingGeneratorsImages(mor) is the two element list containing the list of generators in S and the list of their images in R;
MappingToSinglePieceData(mor) is a list with three elements: the root group homomorphism; the images of the objects; and the images of the rays.
For other types of homomorphism the attributes are very similar.
The operation ObjectGroupHomomorphism, though an operation, is included in this section for convenience.
‣ RootGroupHomomorphism( hom ) | ( attribute ) |
This is the group homomorphism from the root group of the source groupoid to the group at the image object in the range groupoid of the root object in the source.
‣ ImagesOfObjects( hom ) | ( attribute ) |
This is the list of objects in the range groupoid which are the images of the objects in the source.
‣ ImageElementsOfRays( hom ) | ( attribute ) |
This is the list of group elements in those arrows in the range groupoid which are the images of the rays in the source.
gap> RootGroupHomomorphism( mor2 ); [ x, y, y2 ] -> [ (1,3)(2,4), (1,2)(3,4), () ] gap> ImagesOfObjects( mor2 ); [ -14, -12 ] gap> ImageElementsOfRays( mor2 ); [ (), (1,2,4) ]
‣ ObjectGroupHomomorphism( gpdhom, obj ) | ( operation ) |
This operations gives the group homomorphism from an object group of the source to the object group at the image object in the range.
gap> ObjectGroupHomomorphism( mor1, -27 ); [ x, y, y2 ] -> [ (1,4)(2,3), (1,3)(2,4), () ]
In this section we mention inclusion mappings of subgroupoids; and mappings restricted to a source subgroupoid. We also discuss various types of isomorphism: to a different set of objects; to a permutation groupoid; to a pc-groupoid.
‣ InclusionMappingGroupoids( gpd, sgpd ) | ( operation ) |
The operation InclusionMappingGroupoids(gpd,sgpd) returns the inclusion homomorphism from the subgroupoid sgpd to gpd.
gap> inc := InclusionMappingGroupoids( Hs4, Hd8b ); groupoid homomorphism : Hd8b -> Hs4 [ [ [(1,2,3,4) : -14 -> -14], [(1,3) : -14 -> -14], [(1,2,3) : -14 -> -13], [(1,2,4) : -14 -> -12] ], [ [(1,2,3,4) : -14 -> -14], [(1,3) : -14 -> -14], [(1,2,3) : -14 -> -13], [(1,2,4) : -14 -> -12] ] ]
‣ RestrictedMappingGroupoids( mor, sgpd ) | ( operation ) |
‣ ParentMappingGroupoids( mor ) | ( attribute ) |
The operation RestrictedMappingGroupoids(mor,sgpd) returns the restriction of the homomorphism mor to the subgroupoid sgpd of its source. The range is usually set to the ImagesSource of the restriction. The restriction is assigned the attribute ParentMappingGroupoids with value mor (or that of mor is one exists). For another example see section 5.7.
gap> max := MaximalDiscreteSubgroupoid( Hd8b );; gap> res := RestrictedMappingGroupoids( inc, max ); groupoid homomorphism from several pieces : groupoid homomorphism : [ [ [(1,2,3,4) : -14 -> -14], [(1,3) : -14 -> -14] ], [ [(1,2,3,4) : -14 -> -14], [(1,3) : -14 -> -14] ] ] groupoid homomorphism : [ [ [(1,4,2,3) : -13 -> -13], [(1,2) : -13 -> -13] ], [ [(1,4,2,3) : -13 -> -13], [(1,2) : -13 -> -13] ] ] groupoid homomorphism : [ [ [(1,2,4,3) : -12 -> -12], [(2,3) : -12 -> -12] ], [ [(1,2,4,3) : -12 -> -12], [(2,3) : -12 -> -12] ] ]
‣ IsomorphismNewObjects( src, objlist ) | ( operation ) |
The operation IsomorphismNewObjects(gpd,obs) returns the isomorphism from a groupoid gpd to a groupoid with the same object group and ray elements but with a different set obs of objects.
gap> iso1 := IsomorphismNewObjects( Hs4, [-30,-20,-10] ); groupoid homomorphism : [ [ [(1,2,3,4) : -14 -> -14], [(3,4) : -14 -> -14], [() : -14 -> -13], [() : -14 -> -12] ], [ [(1,2,3,4) : -30 -> -30], [(3,4) : -30 -> -30], [() : -30 -> -20], [() : -30 -> -10] ] ] gap> inc2 := mor2*inc*iso1; groupoid homomorphism : [ [ [x : -28 -> -28], [y : -28 -> -28], [y2 : -28 -> -28], [<identity> of ... : -28 -> -27] ], [ [(1,3)(2,4) : -30 -> -30], [(1,2)(3,4) : -30 -> -30], [() : -30 -> -30], [(1,2,4) : -30 -> -10] ] ]
‣ IsomorphismStandardGroupoid( gpd, obs ) | ( operation ) |
The operation IsomorphismStandardGroupoid(gpd,obs) returns the isomorphism from a groupoid with rays to the groupoid of type IsDirectProductWithCompleteDigraphDomain on the given set obs of objects.
gap> iso2 := IsomorphismStandardGroupoid( Hd8b, [-23,-22,-21] ); groupoid homomorphism : [ [ [(1,2,3,4) : -14 -> -14], [(1,3) : -14 -> -14], [(1,2,3) : -14 -> -13], [(1,2,4) : -14 -> -12] ], [ [(1,2,3,4) : -23 -> -23], [(1,3) : -23 -> -23], [() : -23 -> -22], [() : -23 -> -21] ] ] gap> inv2 := InverseGeneralMapping( iso2 ); groupoid homomorphism : [ [ [(1,2,3,4) : -23 -> -23], [(1,3) : -23 -> -23], [() : -23 -> -22], [() : -23 -> -21] ], [ [(1,2,3,4) : -14 -> -14], [(1,3) : -14 -> -14], [(1,2,3) : -14 -> -13], [(1,2,4) : -14 -> -12] ] ]
This operation may also be used to provide a standard form for groupoids of type IsGroupoidByIsomorphisms as described in subsection GroupoidByIsomorphisms (4.6-1).
gap> G2; single piece groupoid with rays: < s3a, [ -7, -6, -5 ], [ [ (), () ], [ (), <identity> of ... ], [ (), <identity ...> ] ] > gap> isoG2 := IsomorphismStandardGroupoid( G2, [-17,-16,-15] ); groupoid homomorphism : [ [ [[ (1,2), (1,2) ] : -7 -> -7], [[ (2,3), (2,3) ] : -7 -> -7], [[ (), <identity> of ... ] : -7 -> -6], [[ (), <identity ...> ] : -7 -> -5] ], [ [(1,2) : -17 -> -17], [(2,3) : -17 -> -17], [() : -17 -> -16], [() : -17 -> -15] ] ]
‣ IsomorphismPermGroupoid( gpd ) | ( attribute ) |
‣ IsomorphismPcGroupoid( gpd ) | ( operation ) |
‣ RegularActionHomomorphismGroupoid( gpd ) | ( attribute ) |
The attribute IsomorphismPermGroupoid(gpd) returns an isomorphism from a groupoid gpd to a groupoid with the same objects but with an isomorphic permutation group. Similarly, IsomorphismPcGroupoid(gpd) changes the group into a pc-group (if appropriate).
The attribute RegularActionHomomorphismGroupoid returns an isomorphism from a groupoid gpd to a groupoid with the same objects but with an isomorphic regular presentation. The example below uses this function because its output is less likely to change than that of IsomorphismPermGroupoid.
gap> N2 := NormalSubgroups( q8 )[2];; gap> Hq8 := SubgroupoidWithRays( Gq8, N2, [ One(q8), q8.1 ] ); single piece groupoid with rays: < N2, [ -28, -27 ], [ <identity> of ..., x ] > gap> regHq8 := RegularActionHomomorphismGroupoid( Hq8 ); groupoid homomorphism : [ [ [y : -28 -> -28], [x : -28 -> -27] ], [ [(1,3,4,7)(2,5,6,8) : -28 -> -28], [(1,2,4,6)(3,8,7,5) : -28 -> -27] ] ]
‣ HomomorphismToSinglePiece( src, rng, piecehoms ) | ( operation ) |
When \(G\) is made up of two or more pieces, all of which get mapped to a connected groupoid, we have a homomorphism to a single piece. The third input parameter in this case is a list of the individual homomorphisms from the single pieces (in the correct order!). See section 3.1 for the corresponding operation on homomorphisms of magmas with objects.
In the following example the source V3 of homV3 has three pieces, and one of the component homomorphisms is an IdentityMapping .
gap> gend12 := [ (15,16,17,18,19,20), (15,20)(16,19)(17,18) ];; gap> d12 := Group( gend12 );; gap> Gd12 := Groupoid( d12, [-37,-36,-35,-34] );; gap> SetName( d12, "d12" ); gap> SetName( Gd12, "Gd12" ); gap> s3 := Subgroup( d12, [ (15,17,19)(16,18,20), (15,20)(16,19)(17,18) ] );; gap> Gs3 := SubgroupoidByPieces( Gd12, [ [ s3, [-36,-35,-34] ] ] );; gap> SetName( s3, "s3" ); gap> SetName( Gs3, "Gs3" ); gap> gend8 := GeneratorsOfGroup( d8 );; gap> imhd8 := [ ( ), (15,20)(16,19)(17,18) ];; gap> hd8 := GroupHomomorphismByImages( d8, s3, gend8, imhd8 );; gap> homd8 := GroupoidHomomorphism( Gd8, Gs3, hd8 ); groupoid homomorphism : Gd8 -> Gs3 [ [ [(1,2,3,4) : -9 -> -9], [(1,3) : -9 -> -9], [() : -9 -> -8], [() : -9 -> -7] ], [ [() : -36 -> -36], [(15,20)(16,19)(17,18) : -36 -> -36], [() : -36 -> -35], [() : -36 -> -34] ] ] gap> hc6 := GroupHomomorphismByImages( c6, s3, > [(5,6,7)(8,9)], [(15,16)(17,20)(18,19)] );; gap> Fs3 := SubgroupoidByObjects( Gs3, [ -35 ] );; gap> SetName( Fs3, "Fs3" ); gap> homc6 := GroupoidHomomorphism( Gc6, Fs3, hc6 );; gap> incFs3 := InclusionMappingGroupoids( Gs3, Fs3 );; gap> ihomc6 := homc6 * incFs3; groupoid homomorphism : Gc6 -> Gs3 [ [ [(5,6,7)(8,9) : -6 -> -6] ], [ [(15,16)(17,20)(18,19) : -35 -> -35] ] ] gap> idGs3 := IdentityMapping( Gs3 );; gap> V3 := ReplaceOnePieceInUnion( U3, 1, Gs3 ); groupoid with 3 pieces: [ Gs3, Gd8, Gc6 ] gap> homs3 := [ idGs3, homd8, ihomc6 ];; gap> homV3 := HomomorphismToSinglePiece( V3, Gs3, homs3 );; gap> Display( homV3 ); homomorphism to single piece groupoid with pieces: (1) : groupoid mapping: [ Gs3 ] -> [ Gs3 ] root homomorphism: [ [ (15,17,19)(16,18,20), (15,20)(16,19)(17,18) ], [ (15,17,19)(16,18,20), (15,20)(16,19)(17,18) ] ] images of objects: [ -36, -35, -34 ] images of rays: [ [() : -36 -> -36], [() : -36 -> -35], [() : -36 -> -34] ] (2) : groupoid mapping: [ Gd8 ] -> [ Gs3 ] root homomorphism: [ [ (1,2,3,4), (1,3) ], [ (), (15,20)(16,19)(17,18) ] ] images of objects: [ -36, -35, -34 ] images of rays: [ [() : -36 -> -36], [() : -36 -> -35], [() : -36 -> -34] ] (3) : groupoid mapping: [ Gc6 ] -> [ Gs3 ] root homomorphism: [ [ (5,6,7)(8,9) ], [ (15,16)(17,20)(18,19) ] ] images of objects: [ -35 ] images of rays: [ [() : -35 -> -35] ]
‣ GroupoidHomomorphismFromHomogeneousDiscrete( src, rng, homs, oims ) | ( operation ) |
This operation requires the source and range; a list of homomorphisms from object group to object group; and a list of the image objects.
gap> Dd8 := MaximalDiscreteSubgroupoid( Gd8 ); homogeneous, discrete groupoid: < d8, [ -9, -8, -7 ] > gap> id8 := IdentityMapping( d8 );; gap> GroupoidHomomorphismFromHomogeneousDiscrete( Dd8, Gd8, [id8,id8,id8], > [-8,-7,-9] ); groupoid homomorphism : morphism from a homogeneous discrete groupoid: [ -9, -8, -7 ] -> [ -8, -7, -9 ] object homomorphisms: IdentityMapping( d8 ) IdentityMapping( d8 ) IdentityMapping( d8 )
‣ HomomorphismByUnion( src, rng, homs ) | ( operation ) |
As in section 3.3, when the range \(H\) has more than one connected component, a homomorphism is a union of homomorphisms, one for each piece in the range.
gap> isoq8 := IsomorphismNewObjects( Gq8, [-38,-37] ); groupoid homomorphism : [ [ [x : -28 -> -28], [y : -28 -> -28], [y2 : -28 -> -28], [<identity> of ... : -28 -> -27] ], [ [x : -38 -> -38], [y : -38 -> -38], [y2 : -38 -> -38], [<identity> of ... : -38 -> -37] ] ] gap> Gq8b := Range( isoq8 );; gap> SetName( Gq8b, "Gq8b" ); gap> V4 := UnionOfPieces( [ V3, Gq8 ] ); groupoid with 4 pieces: [ Gs3, Gq8, Gd8, Gc6 ] gap> SetName( V4, "V4" ); gap> Vs3q8b := UnionOfPieces( [ Gs3, Gq8b ] );; gap> SetName( Vs3q8b, "Vs3q8b" ); gap> hom4 := HomomorphismByUnion( V4, Vs3q8b, [ homV3, isoq8 ] );; gap> Display( hom4 ); magma homomorphism: V4 -> Vs3q8b with pieces : [ Pcgs([ x, y, y2 ]) -> [ x, y, y2 ], [ -38, -37 ], [ <identity> of ..., <identity> of ... ] ] (1) : groupoid mapping: [ Gs3 ] -> [ Gs3 ] root homomorphism: [ [ (15,17,19)(16,18,20), (15,20)(16,19)(17,18) ], [ (15,17,19)(16,18,20), (15,20)(16,19)(17,18) ] ] images of objects: [ -36, -35, -34 ] images of rays: [ [() : -36 -> -36], [() : -36 -> -35], [() : -36 -> -34] ] (2) : groupoid mapping: [ Gd8 ] -> [ Gs3 ] root homomorphism: [ [ (1,2,3,4), (1,3) ], [ (), (15,20)(16,19)(17,18) ] ] images of objects: [ -36, -35, -34 ] images of rays: [ [() : -36 -> -36], [() : -36 -> -35], [() : -36 -> -34] ] (3) : groupoid mapping: [ Gc6 ] -> [ Gs3 ] root homomorphism: [ [ (5,6,7)(8,9) ], [ (15,16)(17,20)(18,19) ] ] images of objects: [ -35 ] images of rays: [ [() : -35 -> -35] ]
‣ IsomorphismGroupoids( A, B ) | ( operation ) |
When \(A,B\) are two single piece groupoids, they are isomorphic provided they have the same number of objects and the root groups are isomorphic.
When \(A=[A_1,\ldots,A_n],~ B=[B_1,\ldots,B_n]\) are both unions of connected groupoids, they are isomorphic if there is a permutation \(\pi\) of \([1,\ldots,n]\) such that \(A_i\) is isomorphic to \(B_{\pi(i)}\) for all \(i\).
gap> s3a := Group( (1,2,3), (2,3) );; gap> s3b := Group( (4,6,8)(5,7,9), (4,9)(5,8)(6,7) );; gap> s3c := Group( (4,6,8)(5,7,9), (5,9)(6,8) );; gap> Ga := SinglePieceGroupoid( s3a, [-23,-22,-21] );; gap> SetName( Ga, "Ga" ); gap> Gb := SinglePieceGroupoid( s3b, [-6,-5,-4] );; gap> SetName( Gb, "Gb" ); gap> Gc := SinglePieceGroupoid( s3c, [-9,-8,-7] );; gap> SetName( Gc, "Gc" ); gap> c6a := Group( (1,2,3,4,5,6) );; gap> c6b := Group( (7,8)(9,10,11) );; gap> c6c := Group( (12,13)(14,15)(16,17,18)(19,20,21)(22,23,24,25,26,27) );; gap> Ha := SinglePieceGroupoid( c6a, [-3,-2,-1] );; gap> SetName( Ha, "Ha" ); gap> Hb := SinglePieceGroupoid( c6b, [-16,-15,-14] );; gap> SetName( Hb, "Hb" ); gap> Hc := SinglePieceGroupoid( c6c, [-19,-18,-17] );; gap> SetName( Hc, "Hc" ); gap> IsomorphismGroupoids( Ga, Gb ); groupoid homomorphism : Ga -> Gb [ [ [(1,2,3) : -23 -> -23], [(2,3) : -23 -> -23], [() : -23 -> -22], [() : -23 -> -21] ], [ [(4,6,8)(5,7,9) : -6 -> -6], [(4,9)(5,8)(6,7) : -6 -> -6], [() : -6 -> -5], [() : -6 -> -4] ] ] gap> IsomorphismGroupoids( Ga, Ha ); fail gap> A := UnionOfPieces( [ Ha, Gb, Gc, Hb ] ); groupoid with 4 pieces: [ Hb, Gc, Gb, Ha ] gap> B := UnionOfPieces( [ Gc, Hb, Hc, Ga ] ); groupoid with 4 pieces: [ Ga, Hc, Hb, Gc ] gap> iso := IsomorphismGroupoids( A, B );; gap> Print( List( PiecesOfMapping(iso), p -> [Source(p),Range(p)] ) ); [ [ Gc, Ga ], [ Hb, Hc ], [ Ha, Hb ], [ Gb, Gc ] ]
In this sections we consider automorphisms of single piece groupoids, then homogeneous discrete groupoids, and then homogeneous groupoids.
‣ GroupoidAutomorphismByObjectPerm( gpd, imobs ) | ( operation ) |
‣ GroupoidAutomorphismByGroupAuto( gpd, gpiso ) | ( operation ) |
‣ GroupoidAutomorphismByRayShifts( gpd, imrays ) | ( operation ) |
‣ GroupoidInnerAutomorphism( gpd, arrow ) | ( operation ) |
We first describe automorphisms of a groupoid \(G\) where \(G\) is the direct product of a group \(g\) and a complete digraph. The automorphism group is generated by three types of automorphism:
given a permutation \(\pi\) of the \(n\) objects, we define \(\pi : G \to G,~ (o_i,g,o_j) \mapsto (o_{\pi i},g,o_{\pi j})\);
given an automorphism \(\alpha\) of the root group \(g\), we define \(\alpha : G \to G,~ (o_i,g,o_j) \mapsto (o_i,\alpha g,o_j)\);
given \(L = [1,g_2,g_3,\ldots,g_n] \in g^n\) we define \(\theta_L : G \to G,~ (o_i,g,o_j) \mapsto (o_i,g_i^{-1}gg_j,o_j)\) so that, in particular, for all \(j\) the rays \((r_j : o_1 \to o_j)\) are shifted by \(g_j\;\): they map to \((r_jg_j : o_1 \to o_j)\);
given \(g \in G\), the inner automorphism of \(G\) by \(g\) is the mapping \(h \mapsto h^g\) wehere conjugation of arrows is defined in section 4.5.
gap> a4 := Subgroup( s4, [(1,2,3),(2,3,4)] );; gap> SetName( a4, "a4" ); gap> gensa4 := GeneratorsOfGroup( a4 );; gap> Ga4 := SubgroupoidByPieces( Gs4, [ [a4, [-15,-13,-11]] ] ); single piece groupoid: < a4, [ -15, -13, -11 ] > gap> SetName( Ga4, "Ga4" ); gap> d := Arrow( Ga4, (1,3,4), -11, -13 ); [(1,3,4) : -11 -> -13] gap> aut1 := GroupoidAutomorphismByObjectPerm( Ga4, [-13,-11,-15] );; gap> Display( aut1 ); groupoid mapping: [ Ga4 ] -> [ Ga4 ] root homomorphism: [ [ (1,2,3), (2,3,4) ], [ (1,2,3), (2,3,4) ] ] images of objects: [ -13, -11, -15 ] images of rays: [ [() : -13 -> -13], [() : -13 -> -11], [() : -13 -> -15] ] gap> d1 := ImageElm( aut1, d ); [(1,3,4) : -15 -> -11] gap> h2 := GroupHomomorphismByImages( a4, a4, gensa4, [(2,3,4), (1,3,4)] );; gap> aut2 := GroupoidAutomorphismByGroupAuto( Ga4, h2 );; gap> Display( aut2 ); groupoid mapping: [ Ga4 ] -> [ Ga4 ] root homomorphism: [ [ (1,2,3), (2,3,4) ], [ (2,3,4), (1,3,4) ] ] images of objects: [ -15, -13, -11 ] images of rays: [ [() : -15 -> -15], [() : -15 -> -13], [() : -15 -> -11] ] gap> d2 := ImageElm( aut2, d1 ); [(1,2,4) : -15 -> -11] gap> im3 := [(), (1,3,2), (2,4,3)];; gap> aut3 := GroupoidAutomorphismByRayShifts( Ga4, im3 );; gap> Display( aut3 ); groupoid mapping: [ Ga4 ] -> [ Ga4 ] root homomorphism: [ [ (1,2,3), (2,3,4) ], [ (1,2,3), (2,3,4) ] ] images of objects: [ -15, -13, -11 ] images of rays: [ [() : -15 -> -15], [(1,3,2) : -15 -> -13], [(2,4,3) : -15 -> -11] ] gap> d3 := ImageElm( aut3, d2 ); [(1,4)(2,3) : -15 -> -11] gap> d0 := Arrow( Ga4, (2,3,4), -11, -13 );; gap> aut4 := GroupoidInnerAutomorphism( Ga4, d0 );; gap> Display( aut4 ); groupoid mapping: [ Ga4 ] -> [ Ga4 ] root homomorphism: [ [ (1,2,3), (2,3,4) ], [ (1,2,3), (2,3,4) ] ] images of objects: [ -15, -11, -13 ] images of rays: [ [() : -15 -> -15], [(2,4,3) : -15 -> -11], [(2,3,4) : -15 -> -13] ] gap> d4 := ImageElm( aut4, d3 ); [(1,2,4) : -15 -> -13]
Let \(S\) be a wide subgroupoid with rays of a standard groupoid \(G\).
An automorphism of the root group extends to the whole of \(S\) with the rays fixed by the automorphism.
An automorphism of \(G\) by permuting the objects may map \(S\) to a different subgroupoid. So we construct an isomorphism \(\iota\) from \(S\) to a standard groupoid \(T\), constgruct \(\alpha\) permuting the objects of \(T\), and return \(\iota*\alpha*\iota^{-1}\).
For an automorphism by ray shifts we require that the shifts are elements of the root group of \(S\).
gap> s4c := Group( (1,2,3,4), (3,4) );; gap> SetName( s4c, "s4c" ); gap> s3c := Subgroup( s4c, [ (1,2), (2,3) ] );; gap> SetName( s3c, "s3c" ); gap> Gs4c := SinglePieceGroupoid( s4c, [-9,-8,-7,-6] );; gap> SetName( Gs4c, "Gs4c" ); gap> Hs3c := SubgroupoidWithRays( Gs4c, s3c, [ (), (1,4), (2,4), (3,4) ] );; gap> SetName( Hs3c, "Hs3c" ); gap> ## (1) automorphism by group auto gap> a1 := GroupHomomorphismByImages( s3c, s3c, [(1,2),(2,3)], [(1,3),(2,3)] );; gap> aut1 := GroupoidAutomorphismByGroupAuto( Hs3c, a1 ); groupoid homomorphism : Hs3c -> Hs3c [ [ [(1,2) : -9 -> -9], [(2,3) : -9 -> -9], [(1,4) : -9 -> -8], [(2,4) : -9 -> -7], [(3,4) : -9 -> -6] ], [ [(1,3) : -9 -> -9], [(2,3) : -9 -> -9], [(1,4) : -9 -> -8], [(2,4) : -9 -> -7], [(3,4) : -9 -> -6] ] ] gap> a := Arrow( Hs3c, (2,3,4), -8, -8 ); [(2,3,4) : -8 -> -8] gap> ImageElm( aut1, a ); [(2,4,3) : -8 -> -8] gap> b := Arrow( Hs3c, (1,2,3,4), -7, -6 ); [(1,2,3,4) : -7 -> -6] gap> ## b = (2,4)(1,2)(3,4) -> (2,4)(1,3)(3,4) gap> ImageElm( aut1, b ); [(1,4,2,3) : -7 -> -6] gap> ## (2) automorphism by object perm gap> aut2 := GroupoidAutomorphismByObjectPerm( Hs3c, [-8,-7,-6,-9] ); groupoid homomorphism : Hs3c -> Hs3c [ [ [(1,2) : -9 -> -9], [(2,3) : -9 -> -9], [(1,4) : -9 -> -8], [(2,4) : -9 -> -7], [(3,4) : -9 -> -6] ], [ [(2,4) : -8 -> -8], [(2,3) : -8 -> -8], [(1,2,4) : -8 -> -7], [(1,3,4) : -8 -> -6], [(1,4) : -8 -> -9] ] ] gap> ImageElm( aut2, a ); [(1,4,3) : -7 -> -7] gap> ImageElm( aut2, b ); [(1,2)(3,4) : -6 -> -9] gap> ## (3) automorphism by ray shifts gap> aut3 := GroupoidAutomorphismByRayShifts( Hs3c, [(),(2,3),(1,3),(1,2)] ); groupoid homomorphism : Hs3c -> Hs3c [ [ [(1,2) : -9 -> -9], [(2,3) : -9 -> -9], [(1,4) : -9 -> -8], [(2,4) : -9 -> -7], [(3,4) : -9 -> -6] ], [ [(1,2) : -9 -> -9], [(2,3) : -9 -> -9], [(1,4)(2,3) : -9 -> -8], [(1,3)(2,4) : -9 -> -7], [(1,2)(3,4) : -9 -> -6] ] ] gap> ImageElm( aut3, a ); [(2,4,3) : -8 -> -8] gap> ImageElm( aut3, b ); [(1,4,2,3) : -7 -> -6] gap> e86 := Arrow( Hs3c, (1,3,2,4), -8, -6 );; gap> aut86 := GroupoidInnerAutomorphism( Hs3c, e86 ); groupoid homomorphism : Hs3c -> Hs3c [ [ [(1,2) : -9 -> -9], [(2,3) : -9 -> -9], [(1,4) : -9 -> -8], [(2,4) : -9 -> -7], [(3,4) : -9 -> -6] ], [ [(1,2) : -9 -> -9], [(2,3) : -9 -> -9], [(2,4,3) : -9 -> -6], [(2,4) : -9 -> -7], [(1,4)(2,3) : -9 -> -8] ] ]
‣ AutomorphismGroupOfGroupoid( gpd ) | ( operation ) |
‣ NiceObjectAutoGroupGroupoid( gpd, aut ) | ( operation ) |
As above, let \(G\) be the direct product of a group \(g\) and a complete digraph with \(n\) objects. The AutomorphismGroup \(\Aut(G)\) of \(G\) is isomorphic to the quotient of \(S_n \times A \times g^n\) by a subgroup isomorphic to \(g\), where \(A\) is the automorphism group of \(g\) and \(S_n\) is the symmetric group on the \(n\) objects. This is one of the main topics in [AW10].
If \(H\) is the union of \(k\) groupoids, all isomorphic to \(G\), then \(\Aut(H)\) is isomorphic to \(S_k \ltimes \Aut(G)\).
The function NiceObjectAutoGroupGroupoid takes a groupoid and a subgroup of its automorphism group and retuns a nice monomorphism from this automorphism group to a pc-group, if one is available. The current implementation is experimental. Note that ImageElm at present only works on generating elements.
gap> AGa4 := AutomorphismGroupOfGroupoid( Ga4 ); Aut(Ga4) gap> Length( GeneratorsOfGroup( AGa4 ) ); 8 gap> AGgens := GeneratorsOfGroup( AGa4);; gap> NGa4 := NiceObject( AGa4 );; gap> MGa4 := NiceMonomorphism( AGa4 );; gap> Size( AGa4 ); 20736 gap> SetName( AGa4, "AGa4" ); gap> SetName( NGa4, "NGa4" ); gap> ## cannot test images of AGgens because of random variations gap> ## Now do some tests! gap> mgi := MappingGeneratorsImages( MGa4 );; gap> autgen := mgi[1];; gap> pcgen := mgi[2];; gap> ngen := Length( autgen );; gap> ForAll( [1..ngen], i -> Order(autgen[i]) = Order(pcgen[i]) ); true
The inner automorphism subgroup \(\mathrm{Inn}(G)\) of the automorphism group of \(G\) is the group of inner automorphisms \(\wedge a : b \mapsto b^a\) for \(a \in G\). It is not the case that the map \(G \to \mathrm{Inn}(G), a \mapsto \wedge a\) preserves multiplication. Indeed, when \(a=(o,g,p), b=(p,h,r) \in G\) with objects \(p,q,r\) all distict, then
\[ \wedge(ab) ~=~ (\wedge a)(\wedge b)(\wedge a) ~=~ (\wedge b)(\wedge a)(\wedge b). \]
(Compare this with the permutation identity \((pq)(qr)(pq) = (pr) = (qr)(pq)(qr)\).) So the map \(G \to \mathrm{Inn}(G)\) is of type IsMappingWithObjectsByFunction.
In the example we convert the automorphism group AGa4 into a single object groupoid, and then define the inner automorphism map.
gap> AGa40 := Groupoid( AGa4, [0] ); single piece groupoid: < Aut(Ga4), [ 0 ] > gap> conj := function(a) > return ArrowNC( true, GroupoidInnerAutomorphism(Ga4,a), 0, 0 ); > end;; gap> inner := MappingWithObjectsByFunction( Ga4, AGa40, conj, [0,0,0] );; gap> a1 := Arrow( Ga4, (1,2,3), -15, -13 );; gap> inn1 := ImageElm( inner, a1 );; gap> a2 := Arrow( Ga4, (2,3,4), -13, -11 );; gap> inn2 := ImageElm( inner, a2 );; gap> a3 := a1*a2; [(1,3)(2,4) : -15 -> -11] gap> inn3 := ImageElm( inner, a3 ); [groupoid homomorphism : Ga4 -> Ga4 [ [ [(1,2,3) : -15 -> -15], [(2,3,4) : -15 -> -15], [() : -15 -> -13], [() : -15 -> -11] ], [ [(1,3,4) : -11 -> -11], [(1,2,4) : -11 -> -11], [(1,3)(2,4) : -11 -> -13], [() : -11 -> -15] ] ] : 0 -> 0] gap> (inn3 = inn1*inn2*inn1) and (inn3 = inn2*inn1*inn2); true
‣ GroupoidAutomorphismByGroupAutos( gpd, auts ) | ( operation ) |
Homogeneous, discrete groupoids are the second type of groupoid for which a method is provided for AutomorphismGroupOfGroupoid. This is used in the XMod package for constructing crossed modules of groupoids. The two types of generating automorphism are GroupoidAutomorphismByGroupAutos, which requires a list of group automorphisms, one for each object group, and GroupoidAutomorphismByObjectPerm, which permutes the objects.
gap> Hs3 := HomogeneousDiscreteGroupoid( s3, [ -13..-10] ); homogeneous, discrete groupoid: < s3, [ -13 .. -10 ] > gap> aut4 := GroupoidAutomorphismByObjectPerm( Hs3, [-12,-10,-11,-13] ); groupoid homomorphism : morphism from a homogeneous discrete groupoid: [ -13, -12, -11, -10 ] -> [ -12, -10, -11, -13 ] object homomorphisms: IdentityMapping( s3 ) IdentityMapping( s3 ) IdentityMapping( s3 ) IdentityMapping( s3 ) gap> gens3 := GeneratorsOfGroup( s3 );; gap> g1 := gens3[1];; gap> g2 := gens3[2];; gap> b1 := GroupHomomorphismByImages( s3, s3, gens3, [g1, g2^g1 ] );; gap> b2 := GroupHomomorphismByImages( s3, s3, gens3, [g1^g2, g2 ] );; gap> b3 := GroupHomomorphismByImages( s3, s3, gens3, [g1^g2, g2^(g1*g2) ] );; gap> b4 := GroupHomomorphismByImages( s3, s3, gens3, [g1^(g2*g1), g2^g1 ] );; gap> aut5 := GroupoidAutomorphismByGroupAutos( Hs3, [b1,b2,b3,b4] ); groupoid homomorphism : morphism from a homogeneous discrete groupoid: [ -13, -12, -11, -10 ] -> [ -13, -12, -11, -10 ] object homomorphisms: GroupHomomorphismByImages( s3, s3, [ (15,17,19)(16,18,20), (15,20)(16,19)(17,18) ], [ (15,17,19)(16,18,20), (15,18)(16,17)(19,20) ] ) GroupHomomorphismByImages( s3, s3, [ (15,17,19)(16,18,20), (15,20)(16,19)(17,18) ], [ (15,19,17)(16,20,18), (15,20)(16,19)(17,18) ] ) GroupHomomorphismByImages( s3, s3, [ (15,17,19)(16,18,20), (15,20)(16,19)(17,18) ], [ (15,19,17)(16,20,18), (15,16)(17,20)(18,19) ] ) GroupHomomorphismByImages( s3, s3, [ (15,17,19)(16,18,20), (15,20)(16,19)(17,18) ], [ (15,19,17)(16,20,18), (15,18)(16,17)(19,20) ] ) gap> genAHs3 := GeneratorsOfGroup( AHs3 );; gap> Length( genAHs3 ); 4 gap> ids3 := IdentityMapping( s3 );; gap> aut5 := GroupoidAutomorphismByGroupAutos( Hs3, [b1^2,ids3,ids3,ids3] );; gap> aut6 := GroupoidAutomorphismByGroupAutos( Hs3, [b2,ids3,ids3,ids3] );; gap> aut7 := GroupoidAutomorphismByObjectPerm( Hs3, [ -12, -11, -10, -13 ] );; gap> aut8 := GroupoidAutomorphismByObjectPerm( Hs3, [ -12, -13, -11, -10 ] );; gap> ok := ForAll( genAHs3, a -> a in[ aut5, aut6, aut7, aut8 ] ); true gap> nobAHs3 := NiceObject( AHs3 );; gap> nmonAHs3 := NiceMonomorphism( AHs3 );; gap> w := genAHs3[1];; gap> w1 := ImageElm( nmonAHs3, w );; gap> x := genAHs3[2];; gap> x1 := ImageElm( nmonAHs3, x );; gap> y := genAHs3[3];; gap> y1 := ImageElm( nmonAHs3, y );; gap> z := genAHs3[4];; gap> z1 := ImageElm( nmonAHs3, z );; gap> u := z*w*y*x*z; groupoid homomorphism : morphism from a homogeneous discrete groupoid: [ -13, -12, -11, -10 ] -> [ -11, -13, -10, -12 ] object homomorphisms: IdentityMapping( s3 ) ConjugatorAutomorphism( s3, (15,19,17)(16,20,18) ) IdentityMapping( s3 ) ConjugatorAutomorphism( s3, (15,20)(16,19)(17,18) ) gap> u1 := z1*w1*y1*x1*z1; (1,2,4,3)(5,17,23,16,8,20,26,13)(6,18,24,15,7,19,25,14)(9,21,27,12,10,22,28, 11) gap> imu := ImageElm( nmonAHs3, u );; gap> u1 = imu; true
‣ AutomorphismGroupoidOfGroupoid( gpd ) | ( attribute ) |
If \(G\) is a single piece groupoid with automorphism group \(\Aut(G)\), and if \(H\) is the union of \(k\) pieces, all isomorphic to \(G\), then the automorphism group of \(H\) is the wreath product \(S_k \ltimes \Aut(G)\). However, we find it more convenient to construct the automorphism groupoid of \(H\). This is a single piece groupoid \(\Aut(H)\) with \(k\) objects -- the object lists of the pieces of \(H\) -- and root group \(\Aut(G)\). Isomorphisms between the root groups of the \(k\) pieces may be applied to the generators of \(\Aut(G)\) to construct automorphism groups of these pieces, and then isomorphisms between these automorphism groups. We then construct \(\AUT(H)\) using GroupoidByIsomorphisms.
In the special case that \(H\) is homogeneous, there is no need to construct a collection of automorphism groups. Rather, the rays of \(\AUT(H)\) are given by IsomorphismNewObjects. For the example we use Hd8 constructed in subsection HomogeneousGroupoid (4.1-5).
gap> Hd8 := HomogeneousGroupoid( Gd8, > [ [-20,-19,-18], [-12,-11,-10], [-16,-15,-14] ] );; gap> AHd8 := AutomorphismGroupoidOfGroupoid( Hd8 ); gap> ObjectList( AHd8 ); [ [ -20, -19, -18 ], [ -16, -15, -14 ], [ -12, -11, -10 ] ] gap> RaysOfGroupoid( AHd8 ){[2..3]}; [ groupoid homomorphism : [ [ [(1,2,3,4) : -20 -> -20], [(1,3) : -20 -> -20], [() : -20 -> -19], [() : -20 -> -18] ], [ [(1,2,3,4) : -16 -> -16], [(1,3) : -16 -> -16], [() : -16 -> -15], [() : -16 -> -14] ] ], groupoid homomorphism : [ [ [(1,2,3,4) : -20 -> -20], [(1,3) : -20 -> -20], [() : -20 -> -19], [() : -20 -> -18] ], [ [(1,2,3,4) : -12 -> -12], [(1,3) : -12 -> -12], [() : -12 -> -11], [() : -12 -> -10] ] ] ] gap> ObjectGroup( AHd8, [ -12, -11, -10 ] ); <group with 8 generators>
Suppose that gpd is the direct product of a group \(G\) and a complete digraph, and that \(\rho : G \to M\) is an isomorphism to a matrix group \(M\). Then if rep is the isomorphic groupoid with the same objects and root group \(M\) there is an isomorphism \(\mu\) from gpd to rep mapping \((g : i \to j)\) to \((\rho g : i \to j)\).
When gpd is a groupoid with rays, a representation can be obtained by restricting a representation of its parent.
gap> reps := IrreducibleRepresentations( s4 );; gap> rep4 := reps[4];; gap> Rs4 := Groupoid( Image( rep4 ), ObjectList( Gs4 ) ); single piece groupoid: < Group([ [ [ 0, 1, 0 ], [ 1, 0, 0 ], [ 0, 0, 1 ] ], [ [ 0, 0, 1 ], [ 1, 0, 0 ], [ 0, 1, 0 ] ], [ [ -1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, -1 ] ], [ [ 1, 0, 0 ], [ 0, -1, 0 ], [ 0, 0, -1 ] ] ]), [ -15, -14, -13, -12, -11 ] > gap> IsMatrixGroupoid( Rs4 ); true gap> gens := GeneratorsOfGroupoid( Gs4 ); [ [(1,2,3,4) : -15 -> -15], [(3,4) : -15 -> -15], [() : -15 -> -14], [() : -15 -> -13], [() : -15 -> -12], [() : -15 -> -11] ] gap> images := List( gens, > g -> Arrow( Rs4, ImageElm(rep4,g![1]), g![2], g![3] ) ); [ [[ [ -1, 0, 0 ], [ 0, 0, 1 ], [ 0, -1, 0 ] ] : -15 -> -15], [[ [ 0, 1, 0 ], [ 1, 0, 0 ], [ 0, 0, 1 ] ] : -15 -> -15], [[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] : -15 -> -14], [[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] : -15 -> -13], [[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] : -15 -> -12], [[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] : -15 -> -11] ] gap> mor := GroupoidHomomorphismFromSinglePiece( Gs4, Rs4, gens, images ); groupoid homomorphism : [ [ [(1,2,3,4) : -15 -> -15], [(3,4) : -15 -> -15], [() : -15 -> -14], [() : -15 -> -13], [() : -15 -> -12], [() : -15 -> -11] ], [ [[ [ -1, 0, 0 ], [ 0, 0, 1 ], [ 0, -1, 0 ] ] : -15 -> -15], [[ [ 0, 1, 0 ], [ 1, 0, 0 ], [ 0, 0, 1 ] ] : -15 -> -15], [[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] : -15 -> -14], [[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] : -15 -> -13], [[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] : -15 -> -12], [[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] : -15 -> -11] ] ] gap> a := Arrow( Hs4, (1,4,2), -12, -13 ); [(1,4,2) : -12 -> -13] gap> ImageElm( mor, a ); [[ [ 0, 0, -1 ], [ -1, 0, 0 ], [ 0, 1, 0 ] ] : -12 -> -13] gap> rmor := RestrictedMappingGroupoids( mor, Hd8b ); groupoid homomorphism : [ [ [(1,2,3,4) : -14 -> -14], [(1,3) : -14 -> -14], [(1,2,3) : -14 -> -13], [(1,2,4) : -14 -> -12] ], [ [[ [ -1, 0, 0 ], [ 0, 0, 1 ], [ 0, -1, 0 ] ] : -14 -> -14], [[ [ 1, 0, 0 ], [ 0, 0, -1 ], [ 0, -1, 0 ] ] : -14 -> -14], [[ [ 0, 0, 1 ], [ -1, 0, 0 ], [ 0, -1, 0 ] ] : -14 -> -13], [[ [ 0, -1, 0 ], [ 0, 0, 1 ], [ -1, 0, 0 ] ] : -14 -> -12] ] ]
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