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<a id="X803E01577A2B37D2" name="X803E01577A2B37D2"></a>
<div class="ChapSects"><a href="chap6.html#X803E01577A2B37D2">6 Automorphisms of Groupoids</a>
<div class="ContSect"> <a href="chap6.html#X7935832881320667">6.1 Automorphisms of single piece groupoids</a>

<div class="ContSSBlock">
 <a href="chap6.html#X79DA704A8051CB72">6.1-1 GroupoidAutomorphismByObjectPerm</a>
 <a href="chap6.html#X86BD61567B41B139">6.1-2 GroupoidInnerAutomorphism</a>
 <a href="chap6.html#X8278B93E7F78038C">6.1-3 Automorphisms of a groupoid with rays</a>

 <a href="chap6.html#X7DB9D5737EB8C260">6.1-4 AutomorphismGroupOfGroupoid</a>
 <a href="chap6.html#X7EC237ED7E1978B0">6.1-5 Inner automorphisms</a>

 <a href="chap6.html#X80F9594481CA9FDE">6.1-6 GroupoidAutomorphismByGroupAutos</a>
 <a href="chap6.html#X7A93C6507BC697CD">6.1-7 AutomorphismGroupoidOfGroupoid</a>
</div></div>
<div class="ContSect"> <a href="chap6.html#X8003AF117B956D16">6.2 Matrix representations of groupoids</a>

</div>
<div class="ContSect"> <a href="chap6.html#X7DB8E7EF7A51F1BE">6.3 Groupoid actions</a>

<div class="ContSSBlock">
 <a href="chap6.html#X7C4DCB287B3DD0CD">6.3-1 GroupoidActionByConjugation</a>
</div></div>
</div>

<h3>6 Automorphisms of Groupoids</h3>

In this chapter we consider automorphisms of single piece groupoids; then homogeneous discrete groupoids; and finally homogeneous groupoids. We also consider matrix representations and groupooid actions.

<a id="X7935832881320667" name="X7935832881320667"></a>

<h4>6.1 Automorphisms of single piece groupoids</h4>

<a id="X79DA704A8051CB72" name="X79DA704A8051CB72"></a>

<h5>6.1-1 GroupoidAutomorphismByObjectPerm</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GroupoidAutomorphismByObjectPerm</code>( <var class="Arg">gpd</var>, <var class="Arg">imobs</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GroupoidAutomorphismByGroupAuto</code>( <var class="Arg">gpd</var>, <var class="Arg">gpiso</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GroupoidAutomorphismByNtuple</code>( <var class="Arg">gpd</var>, <var class="Arg">imrays</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GroupoidAutomorphismByRayShifts</code>( <var class="Arg">gpd</var>, <var class="Arg">imrays</var> )</td><td class="tdright">( operation )</td></tr></table></div>
We first describe automorphisms of a groupoid G where G is the direct product of a group g and a complete digraph with n objects.. The automorphism group is generated by three types of automorphism:

<ul>
<li>given a permutation π of the n objects, we define


\alpha_{\pi} : G \to G,~
(g : o_i \to o_j) \mapsto (g : o_{\pi i} \to o_{\pi j});


</li>
<li>given an automorphism θ of the root group g, we define


\alpha_{\theta} : G \to G,~ (g : o_i \to o_j) \mapsto (\theta g : o_i \to o_j);


</li>
<li>given L = [g_1,g_2,g_3,...,g_n] ∈ g^n we define


\alpha_L : G \to G,~ (g : o_i \to o_j) \mapsto (g_i^{-1}gg_j : o_i \to o_j).


If g_1 = 1_g, then for all j the rays (r_j : o_1 -> o_j) are shifted by g_j: so they map to (r_jg_j : o_1 -> o_j). So the operation <code class="code">GroupoidAutomorphismByRayShifts</code> is the special case of <code class="code">GroupoidAutomorphismByNtuple</code> when g_1=1.

</li>
</ul>

<div class="example"><pre>

gap> perm1 := [-13,-12,-14];;
gap> aut1 := GroupoidAutomorphismByObjectPerm( Ha4, perm1 );; 
gap> Display( aut1 ); 
homomorphism to single piece groupoid: Ha4 -> Ha4
root group homomorphism:
(1,2,3) -> (1,2,3)
(2,3,4) -> (2,3,4)
object map: [ -14, -13, -12 ] -> [ -13, -12, -14 ]
ray images: [ (), (), () ]
gap> d := Arrow( Ha4, (1,3,4), -12, -13 ); 
[(1,3,4) : -12 -> -13]
gap> d1 := ImageElm( aut1, d ); 
[(1,3,4) : -14 -> -12]
gap> gensa4 := GeneratorsOfGroup( a4 );; 
gap> alpha2 := GroupHomomorphismByImages( a4, a4, gensa4, [(2,3,4), (1,3,4)] );; 
gap> aut2 := GroupoidAutomorphismByGroupAuto( Ha4, alpha2 );; 
gap> Display( aut2 ); 
homomorphism to single piece groupoid: Ha4 -> Ha4
root group homomorphism:
(1,2,3) -> (2,3,4)
(2,3,4) -> (1,3,4)
object map: [ -14, -13, -12 ] -> [ -14, -13, -12 ]
ray images: [ (), (), () ]
gap> d2 := ImageElm( aut2, d1 );
[(1,2,4) : -14 -> -12]
gap> L3 := [(1,2)(3,4), (1,3)(2,4), (1,4)(2,3)];; 
gap> aut3 := GroupoidAutomorphismByNtuple( Ha4, L3 );; 
gap> Display( aut3 ); 
homomorphism to single piece groupoid: Ha4 -> Ha4
root group homomorphism:
(1,2,3) -> (1,4,2)
(2,3,4) -> (1,4,3)
object map: [ -14, -13, -12 ] -> [ -14, -13, -12 ]
ray images: [ (), (1,4)(2,3), (1,3)(2,4) ]
gap> d3 := ImageElm( aut3, d2 );
[(2,3,4) : -14 -> -12]
gap> L4 := [(), (1,3,2), (2,4,3)];; 
gap> aut4 := GroupoidAutomorphismByRayShifts( Ha4, L4 );; 
gap> Display( aut4 ); 
homomorphism to single piece groupoid: Ha4 -> Ha4
root group homomorphism:
(1,2,3) -> (1,2,3)
(2,3,4) -> (2,3,4)
object map: [ -14, -13, -12 ] -> [ -14, -13, -12 ]
ray images: [ (), (1,3,2), (2,4,3) ]
gap> d4 := ImageElm( aut4, d3 );
[() : -14 -> -12]
gap> h4 := Arrow( Ha4, (2,3,4), -12, -13 );; 
gap> aut1234 := aut1*aut2*aut3*aut4;; 
gap> Display( aut1234 ); 
homomorphism to single piece groupoid: Ha4 -> Ha4
root group homomorphism:
(1,2,3) -> (1,4,3)
(2,3,4) -> (1,2,3)
object map: [ -14, -13, -12 ] -> [ -13, -12, -14 ]
ray images: [ (), (2,3,4), (1,3,4) ]
gap> d4 = ImageElm( aut1234, d );
true
gap> inv1234 := InverseGeneralMapping( aut1234 );; 
gap> Display( inv1234 ); 
homomorphism to single piece groupoid: Ha4 -> Ha4
root group homomorphism:
(1,2,3) -> (1,4,3)
(2,3,4) -> (2,4,3)
object map: [ -14, -13, -12 ] -> [ -12, -14, -13 ]
ray images: [ (), (1,3,2), (1,3,4) ]

</pre></div>

<a id="X86BD61567B41B139" name="X86BD61567B41B139"></a>

<h5>6.1-2 GroupoidInnerAutomorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GroupoidInnerAutomorphism</code>( <var class="Arg">gpd</var>, <var class="Arg">arrow</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GroupoidInnerAutomorphismNormalSubgroupoid</code>( <var class="Arg">gpd</var>, <var class="Arg">subgpd</var>, <var class="Arg">arrow</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Given an arrow a = (c : p -> q) ∈ G with p ≠ q, the inner automorphism α_a of G by a is the mapping g ↦ g^a where conjugation of arrows is defined in section <a href="chap4.html#X8653FC9786E3209A">4.5</a>. It is easily checked that if L_a = [1,...,1,c^-1,1,...,1,c,1,...,1], with c^-1 in position p and c in position q, then


\alpha_a ~=~ \alpha_{(p,q)} * \alpha_{L_a}.


Similarly, when p=q, then α_a = α_L_a where now L_a = [1,...,1,c,1,...,1], with c in position p.

<div class="example"><pre>

gap> inn1 := GroupoidInnerAutomorphism( Ha4, h4 );;
gap> Display( inn1 );
homomorphism to single piece groupoid: Ha4 -> Ha4
root group homomorphism:
(1,2,3) -> (1,2,3)
(2,3,4) -> (2,3,4)
object map: [ -14, -13, -12 ] -> [ -14, -12, -13 ]
ray images: [ (), (2,4,3), (2,3,4) ]
gap> d5 := ImageElm( inn1, d4 );
[(2,3,4) : -14 -> -13]

</pre></div>

Conjugation may also be applied to certain normal subgroupoids of G. Firstly, let N be the wide subgroupoid of G determined by a normal subgroup n of the root group. Then, provided the group element of a is in n, the inner automorphism by a may be applied to N.

<div class="example"><pre>

gap> Nk4 := SubgroupoidBySubgroup( Ha4, k4 );;
gap> SetName( Nk4, "Nk4" );
gap> e4 := Arrow( Ha4, (1,2)(3,4), -14, -13 );;
gap> inn2 := GroupoidInnerAutomorphismNormalSubgroupoid( Ha4, Nk4, e4 );;
gap> Display( inn2 );
homomorphism to single piece groupoid: Nk4 -> Nk4
root group homomorphism:
(1,2)(3,4) -> (1,2)(3,4)
(1,3)(2,4) -> (1,3)(2,4)
object map: [ -14, -13, -12 ] -> [ -13, -14, -12 ]
ray images: [ (), (), (1,2)(3,4) ]

</pre></div>

Secondly, if H is a homogeneous, discrete subgroupoid of G and if the group element of a is in the common vertex groups, then the inner automorphism may be applied to H.

<div class="example"><pre>

gap> Ma4 := MaximalDiscreteSubgroupoid( Ha4 );;
gap> SetName( Ma4, "Ma4" );
gap> inn3 := GroupoidInnerAutomorphism( Ha4, Ma4, e4 );;
gap> Display( inn3 );
homogeneous discrete groupoid mapping: [ Ma4 ] -> [ Ma4 ]
images of objects: [ -13, -14, -12 ]
object homomorphisms:
GroupHomomorphismByImages( a4, a4, [ (1,2,3), (2,3,4) ], [ (1,4,2), (1,4,3) ] )
GroupHomomorphismByImages( a4, a4, [ (1,2,3), (2,3,4) ], [ (1,4,2), (1,4,3) ] )
GroupHomomorphismByImages( a4, a4, [ (1,2,3), (2,3,4) ], [ (1,2,3), (2,3,4) ] )

</pre></div>

<a id="X8278B93E7F78038C" name="X8278B93E7F78038C"></a>

<h5>6.1-3 Automorphisms of a groupoid with rays</h5>

Let S be a wide subgroupoid with rays of a standard groupoid G.

An automorphism α of the root group H extends to the whole of S with the rays fixed by the automorphism: (r^-1_ihr_j : o_i -> o_j) ↦ (r^-1_i (α h)r_j : o_i -> o_j).

An automorphism of G obtained by permuting the objects may map S to a different subgroupoid. So we construct an isomorphism ι from S to a standard groupoid T, construct α permuting the objects of T, and return ι*α*ι^-1.

For an automorphism by ray shifts we require that the shifts are elements of the root group of S.

<div class="example"><pre>

gap> ## (1) automorphism by group auto 
gap> a6 := GroupHomomorphismByImages( k4, k4,
> [ (1,2)(3,4), (1,3)(2,4) ], [ (1,3)(2,4), (1,4)(2,3) ] );;
gap> aut6 := GroupoidAutomorphismByGroupAuto( Kk4, a6 );
groupoid homomorphism : Kk4 -> Kk4
[ [ [(1,2)(3,4) : -14 -> -14], [(1,3)(2,4) : -14 -> -14],
 [(1,3,4) : -14 -> -13], [(1,4)(2,3) : -14 -> -12] ],
 [ [(1,3)(2,4) : -14 -> -14], [(1,4)(2,3) : -14 -> -14],
 [(1,3,4) : -14 -> -13], [(1,4)(2,3) : -14 -> -12] ] ]
gap> a := Arrow( Kk4, (1,3)(2,4), -12, -12 );;
gap> ImageElm( aut6, a );
[(1,4)(2,3) : -12 -> -12]
gap> b := Arrow( Kk4, (1,4,2), -12, -13 );;
gap> ImageElm( aut6, b );
[(1,2,3) : -12 -> -13]
gap> ## (2) automorphism by object perm 
gap> aut7 := GroupoidAutomorphismByObjectPerm( Kk4, [-13,-12,-14] );
groupoid homomorphism : Kk4 -> Kk4
[ [ [(1,2)(3,4) : -14 -> -14], [(1,3)(2,4) : -14 -> -14],
 [(1,3,4) : -14 -> -13], [(1,4)(2,3) : -14 -> -12] ],
 [ [(1,4)(2,3) : -13 -> -13], [(1,2)(3,4) : -13 -> -13],
 [(2,3,4) : -13 -> -12], [(1,4,3) : -13 -> -14] ] ]
gap> ImageElm( aut7, a ); 
[(1,3)(2,4) : -14 -> -14]
gap> ImageElm( aut7, b ); 
[(1,3)(2,4) : -14 -> -12]
gap> ## (3) automorphism by ray shifts 
gap> aut8 := GroupoidAutomorphismByRayShifts( Kk4,
> [ (), (1,4)(2,3), (1,3)(2,4) ] );
groupoid homomorphism : Kk4 -> Kk4
[ [ [(1,2)(3,4) : -14 -> -14], [(1,3)(2,4) : -14 -> -14],
 [(1,3,4) : -14 -> -13], [(1,4)(2,3) : -14 -> -12] ],
 [ [(1,2)(3,4) : -14 -> -14], [(1,3)(2,4) : -14 -> -14],
 [(1,2,3) : -14 -> -13], [(1,2)(3,4) : -14 -> -12] ] ]
gap> ImageElm( aut8, a );
[(1,3)(2,4) : -12 -> -12]
gap> ImageElm( aut8, b );
[(1,2,3) : -12 -> -13]
gap> ## (4) combine these three automorphisms 
gap> aut678 := aut6 * aut7 * aut8;
groupoid homomorphism : Kk4 -> Kk4
[ [ [(1,2)(3,4) : -14 -> -14], [(1,3)(2,4) : -14 -> -14],
 [(1,3,4) : -14 -> -13], [(1,4)(2,3) : -14 -> -12] ],
 [ [(1,2)(3,4) : -13 -> -13], [(1,3)(2,4) : -13 -> -13],
 [(1,4,3) : -13 -> -12], [(1,3,2) : -13 -> -14] ] ]
gap> ImageElm( aut678, a ); 
[(1,4)(2,3) : -14 -> -14]
gap> ImageElm( aut678, b );
[(1,4)(2,3) : -14 -> -12]
gap> ## (5) conjgation by an arrow
gap> e8 := Arrow( Kk4, (1,3)(2,4), -14, -12 );;
gap> aut9 := GroupoidInnerAutomorphism( Kk4, e8 );
groupoid homomorphism : Kk4 -> Kk4
[ [ [(1,2)(3,4) : -14 -> -14], [(1,3)(2,4) : -14 -> -14],
 [(1,3,4) : -14 -> -13], [(1,4)(2,3) : -14 -> -12] ],
 [ [(1,2)(3,4) : -12 -> -12], [(1,3)(2,4) : -12 -> -12],
 [(1,4,2) : -12 -> -13], [(1,4)(2,3) : -12 -> -14] ] ]

</pre></div>

<a id="X7DB9D5737EB8C260" name="X7DB9D5737EB8C260"></a>

<h5>6.1-4 AutomorphismGroupOfGroupoid</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AutomorphismGroupOfGroupoid</code>( <var class="Arg">gpd</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NiceObjectAutoGroupGroupoid</code>( <var class="Arg">gpd</var>, <var class="Arg">aut</var> )</td><td class="tdright">( operation )</td></tr></table></div>
As above, let G be the direct product of a group g and a complete digraph with n objects. The <code class="code">AutomorphismGroup</code> Aut(G) of G is isomorphic to the quotient of S_n × A × 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 <a href="chapBib.html#biBAlWe">[AW10]</a>.

If H is the union of k groupoids, all isomorphic to G, then Aut(H) is isomorphic to S_k ⋉ Aut(G).

The function <code class="code">NiceObjectAutoGroupGroupoid</code> 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 <code class="code">ImageElm</code> at present only works on generating elements.

<div class="example"><pre>

gap> AHa4 := AutomorphismGroupOfGroupoid( Ha4 ); 
Aut(Ha4)
gap> Agens := GeneratorsOfGroup( AHa4);; 
gap> Length( Agens );
8
gap> NHa4 := NiceObject( AHa4 );; 
gap> MHa4 := NiceMonomorphism( AHa4 );; 
gap> Size( AHa4 ); ## (3!)x24x(12^2)
20736
gap> SetName( AHa4, "AHa4" ); 
gap> SetName( NHa4, "NHa4" );
gap> ## either of these names may be returned
gap> names := [ "(((A4 x A4 x A4) : C2) : C3) : C2",
> "(C2 x C2 x C2 x C2 x C2 x C2) : (((C3 x C3 x C3) : C3) : (C2 x C2))" ];;
gap> StructureDescription( NHa4 ) in names;
true
gap> ## cannot test images of Agens because of random variations 
gap> ## Now do some tests!
gap> mgi := MappingGeneratorsImages( MHa4 );; 
gap> autgen := mgi[1];;
gap> pcgen := mgi[2];;
gap> ngen := Length( autgen );; 
gap> ForAll( [1..ngen], i -> Order(autgen[i]) = Order(pcgen[i]) ); 
true

</pre></div>

<a id="X7EC237ED7E1978B0" name="X7EC237ED7E1978B0"></a>

<h5>6.1-5 Inner automorphisms</h5>

The inner automorphism subgroup mathrmInn(G) of the automorphism group of G is the group of inner automorphisms ∧ a : b ↦ b^a for a ∈ G. It is not the case that the map G -> mathrmInn(G), a ↦ ∧ a preserves multiplication. Indeed, when a=(o,g,p), b=(p,h,r) ∈ 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 -> mathrmInn(G) is of type <code class="code">IsMappingWithObjectsByFunction</code>.

In the example we convert the automorphism group <code class="code">AGa4</code> into a single object groupoid, and then define the inner automorphism map.

<div class="example"><pre>

gap> AHa40 := Groupoid( AHa4, [0] );
single piece groupoid: < Aut(Ha4), [ 0 ] >
gap> conj := function(a) 
> return ArrowNC( Ha4, true, GroupoidInnerAutomorphism(Ha4,a), 0, 0 ); 
> end;; 
gap> inner := MappingWithObjectsByFunction( Ha4, AHa40, conj, [0,0,0] );;
gap> a1 := Arrow( Ha4, (1,2,3), -14, -13 );;
gap> inner1 := ImageElm( inner, a1 );; 
gap> a2 := Arrow( Ha4, (2,3,4), -13, -12 );;
gap> inner2 := ImageElm( inner, a2 );; 
gap> a3 := a1*a2; 
[(1,3)(2,4) : -14 -> -12]
gap> inner3 := ImageElm( inner, a3 ); 
[groupoid homomorphism : Ha4 -> Ha4
[ [ [(1,2,3) : -14 -> -14], [(2,3,4) : -14 -> -14], [() : -14 -> -13],
 [() : -14 -> -12] ],
 [ [(1,3,4) : -12 -> -12], [(1,2,4) : -12 -> -12], [(1,3)(2,4) : -12 -> -13],
 [() : -12 -> -14] ] ] : 0 -> 0]
gap> (inner3 = inner1*inner2*inner1) and (inner3 = inner2*inner1*inner2);
true
true

</pre></div>

<a id="X80F9594481CA9FDE" name="X80F9594481CA9FDE"></a>

<h5>6.1-6 GroupoidAutomorphismByGroupAutos</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GroupoidAutomorphismByGroupAutos</code>( <var class="Arg">gpd</var>, <var class="Arg">auts</var> )</td><td class="tdright">( operation )</td></tr></table></div>
Homogeneous, discrete groupoids are the second type of groupoid for which a method is provided for <code class="code">AutomorphismGroupOfGroupoid</code>. This is used in the XMod package for constructing crossed modules of groupoids. The two types of generating automorphism are <code class="code">GroupoidAutomorphismByGroupAutos</code>, which requires a list of group automorphisms, one for each object group, and <code class="code">GroupoidAutomorphismByObjectPerm</code>, which permutes the objects. So, if the object groups g have automorphism group Aut(g) and there are n objects, the autmorphism group of the groupoid has size n!|Aut(g)|^n.

<div class="example"><pre>

gap> Dd8 := HomogeneousDiscreteGroupoid( d8, [ -13..-10] ); 
homogeneous, discrete groupoid: < d8, [ -13 .. -10 ] >
gap> aut10 := GroupoidAutomorphismByObjectPerm( Dd8, [-12,-10,-11,-13] ); 
groupoid homomorphism : morphism from a homogeneous discrete groupoid:
[ -13, -12, -11, -10 ] -> [ -12, -10, -11, -13 ]
object homomorphisms:
IdentityMapping( d8 )
IdentityMapping( d8 )
IdentityMapping( d8 )
IdentityMapping( d8 )
gap> gend8 := GeneratorsOfGroup( d8 );; 
gap> g1 := gend8[1];; 
gap> g2 := gend8[2];;
gap> b1 := IdentityMapping( d8 );;
gap> b2 := GroupHomomorphismByImages( d8, d8, gend8, [g1, g2*g1 ] );; 
gap> b3 := GroupHomomorphismByImages( d8, d8, gend8, [g1^g2, g2 ] );; 
gap> b4 := GroupHomomorphismByImages( d8, d8, gend8, [g1^g2, g2^(g1*g2) ] );; 
gap> aut11 := GroupoidAutomorphismByGroupAutos( Dd8, [b1,b2,b3,b4] ); 
groupoid homomorphism : morphism from a homogeneous discrete groupoid:
[ -13, -12, -11, -10 ] -> [ -13, -12, -11, -10 ]
object homomorphisms:
IdentityMapping( d8 )
GroupHomomorphismByImages( d8, d8, [ (5,6,7,8), (5,7) ],
[ (5,6,7,8), (5,8)(6,7) ] )
GroupHomomorphismByImages( d8, d8, [ (5,6,7,8), (5,7) ], [ (5,8,7,6), (5,7) ] )
GroupHomomorphismByImages( d8, d8, [ (5,6,7,8), (5,7) ], [ (5,8,7,6), (6,8) ] )
gap> ADd8 := AutomorphismGroupOfGroupoid( Dd8 ); 
<group with 4 generators>
gap> Size( ADd8 ); ## 4!*8^4
98304
gap> genADd8 := GeneratorsOfGroup( ADd8 );;
gap> Length( genADd8 ); 
4
gap> w := GroupoidAutomorphismByGroupAutos( Dd8, [b2,b1,b1,b1] );; 
gap> x := GroupoidAutomorphismByGroupAutos( Dd8, [b3,b1,b1,b1] );; 
gap> y := GroupoidAutomorphismByObjectPerm( Dd8, [ -12, -11, -10, -13 ] );; 
gap> z := GroupoidAutomorphismByObjectPerm( Dd8, [ -12, -13, -11, -10 ] );; 
gap> ok := ForAll( genADd8, a -> a in[ w, x, y, z ] ); 
true
gap> NADd8 := NiceObject( ADd8 );; 
gap> MADd8 := NiceMonomorphism( ADd8 );;
gap> w1 := ImageElm( MADd8, w );; 
gap> x1 := ImageElm( MADd8, x );; 
gap> y1 := ImageElm( MADd8, y );; 
gap> z1 := ImageElm( MADd8, 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( d8 )
GroupHomomorphismByImages( d8, d8, [ (5,6,7,8), (5,7) ],
[ (5,6,7,8), (5,8)(6,7) ] )
IdentityMapping( d8 )
GroupHomomorphismByImages( d8, d8, [ (5,6,7,8), (5,7) ], [ (5,8,7,6), (5,7) ] )
gap> u1 := z1*w1*y1*x1*z1; 
(1,2,4,3)(5,17,23,11,6,18,24,16)(7,19,25,15,9,21,27,13)(8,20,26,14,10,22,28,12)
gap> imu := ImageElm( MADd8, u );; 
gap> u1 = imu;
true

</pre></div>

<a id="X7A93C6507BC697CD" name="X7A93C6507BC697CD"></a>

<h5>6.1-7 AutomorphismGroupoidOfGroupoid</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AutomorphismGroupoidOfGroupoid</code>( <var class="Arg">gpd</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
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 ⋉ 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 <code class="code">GroupoidByIsomorphisms</code>.

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 <code class="code">IsomorphismNewObjects</code>. For the example we use <code class="code">HGd8</code> constructed in subsection <code class="func">HomogeneousGroupoid</code> (<a href="chap4.html#X855F318181808814">4.1-5</a>).

<div class="example"><pre>

gap> HGd8 := HomogeneousGroupoid( Gd8, 
> [ [-39,-38,-37], [-36,-35,-34], [-33,-32,-31] ] );;
gap> SetName( HGd8, "HGd8" );
gap> AHGd8 := AutomorphismGroupoidOfGroupoid( HGd8 ); 
Aut(HGd8)
gap> ObjectList( AHGd8 );
[ [ -39, -38, -37 ], [ -36, -35, -34 ], [ -33, -32, -31 ] ]
gap> RaysOfGroupoid( AHGd8 ){[2..3]};
[ groupoid homomorphism :
 [ [ [(5,6,7,8) : -39 -> -39], [(5,7) : -39 -> -39], [() : -39 -> -38],
 [() : -39 -> -37] ],
 [ [(5,6,7,8) : -36 -> -36], [(5,7) : -36 -> -36], [() : -36 -> -35],
 [() : -36 -> -34] ] ], groupoid homomorphism :
 [ [ [(5,6,7,8) : -39 -> -39], [(5,7) : -39 -> -39], [() : -39 -> -38],
 [() : -39 -> -37] ],
 [ [(5,6,7,8) : -33 -> -33], [(5,7) : -33 -> -33], [() : -33 -> -32],
 [() : -33 -> -31] ] ] ]
gap> obgp := ObjectGroup( AHGd8, [ -36, -35, -34 ] );; 
gap> Size( obgp ); ## 3!*8^3
3072

</pre></div>

<a id="X8003AF117B956D16" name="X8003AF117B956D16"></a>

<h4>6.2 Matrix representations of groupoids</h4>

Suppose that <code class="code">gpd</code> is the direct product of a group G and a complete digraph, and that ρ : G -> M is an isomorphism to a matrix group M. Then, if <code class="code">rep</code> is the isomorphic groupoid with the same objects and root group M, there is an isomorphism μ from <code class="code">gpd</code> to <code class="code">rep</code> mapping (g : i -> j) to (ρ g : i -> j).

When <code class="code">gpd</code> is a groupoid with rays, a representation can be obtained by restricting a representation of its parent.

<div class="example"><pre>

gap> reps := IrreducibleRepresentations( a4 );; 
gap> rep4 := reps[4]; 
Pcgs([ (2,4,3), (1,3)(2,4), (1,2)(3,4) ]) ->
[ [ [ 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 ] ] ]
gap> Ra4 := Groupoid( Image( rep4 ), Ga4!.objects );; 
gap> ObjectList( Ra4 ) = [ -15 .. -11 ];
true
gap> gens := GeneratorsOfGroupoid( Ga4 );
[ [(1,2,3) : -15 -> -15], [(2,3,4) : -15 -> -15], [() : -15 -> -14],
 [() : -15 -> -13], [() : -15 -> -12], [() : -15 -> -11] ]
gap> images := List( gens, 
> g -> Arrow( Ra4, ImageElm(rep4,g![2]), g![3], g![4] ) ); 
[ [[ [ 0, 0, -1 ], [ 1, 0, 0 ], [ 0, -1, 0 ] ] : -15 -> -15],
 [[ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ] : -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( Ga4, Ra4, gens, images );

groupoid homomorphism :
[ [ [(1,2,3) : -15 -> -15], [(2,3,4) : -15 -> -15], [() : -15 -> -14],
 [() : -15 -> -13], [() : -15 -> -12], [() : -15 -> -11] ],
 [ [[ [ 0, 0, -1 ], [ 1, 0, 0 ], [ 0, -1, 0 ] ] : -15 -> -15],
 [[ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ] : -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> IsMatrixGroupoid( Ra4 ); 
true
gap> a := Arrow( Ha4, (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, Ha4 );
groupoid homomorphism :
[ [ [(1,2,3) : -14 -> -14], [(2,3,4) : -14 -> -14], [() : -14 -> -13],
 [() : -14 -> -12] ],
 [ [[ [ 0, 0, -1 ], [ 1, 0, 0 ], [ 0, -1, 0 ] ] : -14 -> -14],
 [[ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ] : -14 -> -14],
 [[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] : -14 -> -13],
 [[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] : -14 -> -12] ] ]
gap> ParentMappingGroupoids( rmor ) = mor;
true

</pre></div>

<a id="X7DB8E7EF7A51F1BE" name="X7DB8E7EF7A51F1BE"></a>

<h4>6.3 Groupoid actions</h4>

Recall from sections <a href="chap4.html#X8653FC9786E3209A">4.5</a> and <code class="func">GroupoidInnerAutomorphism</code> (<a href="chap6.html#X86BD61567B41B139">6.1-2</a>) the notion of conjugation in a groupoid, and the associated inner automorphisms.

It was mentioned there that the map ∧ : G -> Aut(G),~ a -> ∧ a, is not a groupoid homomorphism. It is in fact a groupoid action which we now define. Let {p,q,r,u,v} be distinct objects in G and let:

a_1 = (c_1 : p -> q),~~ a_2 = (c_2 : q -> r),~~ a_3 = (c_3 : q -> p),~~ a_4 = (c_4 : u -> v),

b_1 = (d_1 : p -> p),~~ b_2 = (d_2 : p -> p),~~ b_3 = (d_3 : q -> q),~~ b_4 = (c_3c_1 : q -> q) be arrows in G. Then the following conjugation identities must be satisfied:

<ul>
<li>∧(a_1a_2) ~=~ (∧ a_1)*(∧ a_2)*(∧ a_1) ~=~ (∧ a_2)*(∧ a_1)*(∧ a_2),

</li>
<li>∧(b_1a_1) ~=~ (∧ b_1)*(∧ a_1)*(∧ b_1)^-1,

</li>
<li>∧(a_1b_3) ~=~ (∧ b_3)^-1*(∧ a_1)*(∧ b_3),

</li>
<li>∧(b_1b_2) ~=~ (∧ b_1)*(∧ b_2),

</li>
<li>∧(a_1a_3) ~=~ (∧ a_1)*(∧ a_3)*(∧ b_4),

</li>
<li>(∧ a_1)*(∧ a_4) ~=~ (∧ a_4)*(∧ a_1).

</li>
</ul>
In the following example we check the first of these identities in one particular case.

<div class="example"><pre>

gap> c1 := Arrow( Ha4, (1,2)(3,4), -14, -13);;
gap> innc1 := GroupoidInnerAutomorphism( Ha4, c1 );
groupoid homomorphism : Ha4 -> Ha4
[ [ [(1,2,3) : -14 -> -14], [(2,3,4) : -14 -> -14], [() : -14 -> -13],
 [() : -14 -> -12] ],
 [ [(1,4,2) : -13 -> -13], [(1,4,3) : -13 -> -13], [() : -13 -> -14],
 [(1,2)(3,4) : -13 -> -12] ] ]
gap> c2 := Arrow( Ha4, (1,4,2), -13, -12);;
gap> innc2 := GroupoidInnerAutomorphism( Ha4, c2 );
groupoid homomorphism : Ha4 -> Ha4
[ [ [(1,2,3) : -14 -> -14], [(2,3,4) : -14 -> -14], [() : -14 -> -13],
 [() : -14 -> -12] ],
 [ [(1,2,3) : -14 -> -14], [(2,3,4) : -14 -> -14], [(1,4,2) : -14 -> -12],
 [(1,2,4) : -14 -> -13] ] ]
gap> c12 := c1 * c2;
[(2,4,3) : -14 -> -12]
gap> innc12 := GroupoidInnerAutomorphism( Ha4, c12 );
groupoid homomorphism : Ha4 -> Ha4
[ [ [(1,2,3) : -14 -> -14], [(2,3,4) : -14 -> -14], [() : -14 -> -13],
 [() : -14 -> -12] ],
 [ [(1,4,2) : -12 -> -12], [(2,3,4) : -12 -> -12], [(2,3,4) : -12 -> -13],
 [(2,4,3) : -12 -> -14] ] ]
gap> [ innc1 * innc2 * innc1 = innc12, innc2 * innc1 * innc2 = innc12 ];
[ true, true ]

</pre></div>

<a id="X7C4DCB287B3DD0CD" name="X7C4DCB287B3DD0CD"></a>

<h5>6.3-1 GroupoidActionByConjugation</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GroupoidActionByConjugation</code>( <var class="Arg">gpd</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGroupoidAction</code>( <var class="Arg">map</var> )</td><td class="tdright">( category )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ActionMap</code>( <var class="Arg">act</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
The operation <code class="code">GroupoidInnerAutomorphism</code>, which produces the conjugation action of G on itself, does satisfy the conjugation identities and so provides a standard example of an action.

An action is a record with fields <code class="code">Source</code>, <code class="code">Range</code> and <code class="code">ActionMap</code>.

The examples repeat those in section <code class="func">GroupoidInnerAutomorphism</code> (<a href="chap6.html#X86BD61567B41B139">6.1-2</a>): firstly with a groupoid acting on itself.

<div class="example"><pre>

gap> act1 := GroupoidActionByConjugation( Ha4 );
<general mapping: Ha4 -> Aut(Ha4) >
gap> IsGroupoidAction( act1 );
true
gap> amap1 := ActionMap( act1 );;
gap> amap1( h4 ) = inn1;
true

</pre></div>

Secondly with an action on a single piece, normal subgroupoid.

<div class="example"><pre>

gap> act2 := GroupoidActionByConjugation( Ha4, Nk4 );
<general mapping: Ha4 -> Aut(Nk4) >
gap> IsGroupoidAction( act2 );
true
gap> amap2 := ActionMap( act2 );;
gap> amap2( e4 ) = inn2;
true

</pre></div>

Thirly with an action on a homogeneous, discrete subgroupoid.

<div class="example"><pre>

gap> act3 := GroupoidActionByConjugation( Ha4, Ma4 );
<general mapping: Ha4 -> Aut(Ma4) >
gap> IsGroupoidAction( act3 );
true
gap> amap3 := ActionMap( act3 );;
gap> amap3( e4 ) = inn3;
true

</pre></div>

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