A subgroup \(F\le\mathrm{Aut}(B_{d,k})\) satifies the compatibility condition (C) if and only if \(\mathrm{U}_{k}(F)\) is locally action isomorphic to \(F\), see [Tor20, Proposition 3.8]. The term compatibility comes from the following translation of this condition into properties of the \((k-1)\)-local actions of elements of \(F\): The group \(F\) satisfies (C) if and only if
\[\forall \alpha\in F\ \forall\omega\in\Omega\ \exists\beta\in F:\ \sigma_{k-1}(\alpha,b)=\sigma_{k-1}(\beta,b_{\omega}),\ \sigma_{k-1}(\alpha,b_{\omega})=\sigma_{k-1}(\beta,b).\]
This section is concerned with testing compatibility of two given elements (see AreCompatibleBallElements (3.2-1)) and finding an/all elements that is/are compatible with a given one (see CompatibleBallElement (3.2-2), CompatibilitySet (3.2-3)).
‣ AreCompatibleBallElements( d, k, aut1, aut2, dir ) | ( function ) |
Returns: true if aut1 and aut2 are compatible with each other in direction dir, and false otherwise.
The arguments of this method are a degree d \(\in\mathbb{N}_{\ge 3}\), a radius k \(\in\mathbb{N}\), two automorphisms aut1, aut2 \(\in\mathrm{Aut}(B_{d,k})\), and a direction dir \(\in\)[1..d].
gap> AreCompatibleBallElements(3,1,(1,2),(1,2,3),1); true gap> AreCompatibleBallElements(3,1,(1,2),(1,2,3),2); false
gap> a:=(1,3,5)(2,4,6);; a in AutBall(3,2); true gap> LocalAction(1,3,2,a,[]); LocalAction(1,3,2,a,[1]); (1,2,3) (1,2) gap> b:=(1,4)(2,3);; b in AutBall(3,2); true gap> LocalAction(1,3,2,b,[]); LocalAction(1,3,2,b,[1]); (1,2) (1,2,3) gap> AreCompatibleBallElements(3,2,a,b,1); true gap> AreCompatibleBallElements(3,2,a,b,3); false
‣ CompatibleBallElement( F, aut, dir ) | ( function ) |
Returns: an element of F that is compatible with aut in direction dir if one exists, and fail otherwise.
The arguments of this method are a local action F \(\le\mathrm{Aut}(B_{d,k})\), an element aut \(\in\) F, and a direction dir \(\in\)[1..d].
gap> mt:=RandomSource(IsMersenneTwister,1);; gap> a:=Random(mt,AutBall(5,1)); dir:=Random(mt,[1..5]); (1,2,5,4,3) 4 gap> CompatibleBallElement(AutBall(5,1),a,dir); (1,2,5,4,3)
gap> a:=(1,3,5)(2,4,6);; a in AutBall(3,2); true gap> CompatibleBallElement(AutBall(3,2),a,1); (1,4,2,3)
‣ CompatibilitySet( F, aut, dir ) | ( operation ) |
‣ CompatibilitySet( F, aut, dirs ) | ( operation ) |
Returns: the list of elements of F that are compatible with aut in direction dir.
The arguments of this method are a local action F of \(\le\mathrm{Aut}(B_{d,k})\), an automorphism aut \(\in F\), and a direction dir \(\in\)[1..d].
Returns: the list of elements of F that are compatible with aut in all directions of dirs.
The arguments of this method are a local action F of \(\le\mathrm{Aut}(B_{d,k})\), an automorphism aut \(\in F\), and a sublist of directions dirs \(\subseteq\)[1..d].
gap> F:=LocalAction(4,1,TransitiveGroup(4,3)); D(4) gap> G:=LocalAction(4,1,SymmetricGroup(4)); Sym( [ 1 .. 4 ] ) gap> aut:=(1,3);; aut in F; true gap> CompatibilitySet(G,aut,1); RightCoset(Sym( [ 2 .. 4 ] ),(1,3)) gap> CompatibilitySet(F,aut,1); RightCoset(Group([ (2,4) ]),(1,3)) gap> CompatibilitySet(F,aut,[1,3]); RightCoset(Group([ (2,4) ]),(1,3)) gap> CompatibilitySet(F,aut,[1,2]); RightCoset(Group(()),(1,3))
‣ AssembleAutomorphism( d, k, auts ) | ( function ) |
Returns: the automorphism \((\)aut\(,(\)auts\([\)i\(])_{i=1}^{d})\) of \(B_{d,k+1}\), where aut is implicit in \((\)auts\([\)i\(])_{i=1}^{d}\).
The arguments of this method are a degree d \(\in\mathbb{N}_{\ge 3}\), a radius k \(\in\mathbb{N}\), and a list auts of d automorphisms \((\)auts\([\)i\(])_{i=1}^{d}\) of \(B_{d,k}\) which comes from an element \((\)aut\(,(\)auts\([\)i\(])_{i=1}^{d})\) of \(\mathrm{Aut}(B_{d,k+1})\).
gap> mt:=RandomSource(IsMersenneTwister,1);; gap> aut:=Random(mt,AutBall(3,2)); (1,4,5,2,3,6) gap> auts:=[];; gap> for i in [1..3] do auts[i]:=CompatibleBallElement(AutBall(3,2),aut,i); od; gap> auts; [ (1,4,6,2,3,5), (1,3,6,2,4,5), (1,5)(2,6) ] gap> a:=AssembleAutomorphism(3,2,auts); (1,7,9,3,5,11)(2,8,10,4,6,12) gap> a in AutBall(3,3); true gap> LocalAction(2,3,3,a,[]); (1,4,5,2,3,6)
Using the methods of Section 3.2, this section provides methods to test groups for the compatibility condition and search for compatible subgroups inside a given group, e.g. \(\mathrm{Aut}(B_{d,k})\), or with a certain image under some projection.
‣ MaximalCompatibleSubgroup( F ) | ( attribute ) |
Returns: The local action \(C(\)F\()\le\mathrm{Aut}(B_{d,k})\), which is the maximal compatible subgroup of F.
The argument of this attribute is a local action F \(\le\mathrm{Aut}(B_{d,k})\) (see IsLocalAction (2.1-1)).
gap> F:=LocalAction(3,1,Group((1,2))); Group([ (1,2) ]) gap> MaximalCompatibleSubgroup(F); Group([ (1,2) ]) gap> G:=LocalAction(3,2,Group((1,2))); Group([ (1,2) ]) gap> MaximalCompatibleSubgroup(G); Group(())
‣ SatisfiesC( F ) | ( property ) |
Returns: true if F satisfies the compatibility condition (C), and false otherwise.
The argument of this property is a local action F \(\le\mathrm{Aut}(B_{d,k})\) (see IsLocalAction (2.1-1)).
gap> D:=LocalActionDelta(3,SymmetricGroup(3)); Group([ (1,3,6)(2,4,5), (1,3)(2,4), (1,2)(3,4)(5,6) ]) gap> SatisfiesC(D); true
‣ CompatibleSubgroups( F ) | ( function ) |
Returns: the list of all compatible subgroups of F.
The argument of this method is a local action F \(\le\mathrm{Aut}(B_{d,k})\). This method calls AllSubgroups on \(F\) and is therefore slow. Use for instructional purposes on small examples only, and use ConjugacyClassRepsCompatibleSubgroups (3.3-4) or ConjugacyClassRepsCompatibleGroupsWithProjection (3.3-5) for computations.
gap> G:=LocalActionGamma(3,SymmetricGroup(3)); Group([ (1,4,5)(2,3,6), (1,3)(2,4)(5,6) ]) gap> list:=CompatibleSubgroups(G); [ Group(()), Group([ (1,2)(3,5)(4,6) ]), Group([ (1,3)(2,4)(5,6) ]), Group([ (1,6)(2,5)(3,4) ]), Group([ (1,4,5)(2,3,6) ]), Group([ (1,4,5)(2,3,6), (1,3)(2,4)(5,6) ]) ] gap> Size(list); 6 gap> Size(AllSubgroups(SymmetricGroup(3))); 6
‣ ConjugacyClassRepsCompatibleSubgroups( F ) | ( attribute ) |
Returns: a list of compatible representatives of conjugacy classes of F that contain a compatible subgroup.
The argument of this method is a local action F of \(\mathrm{Aut}(B_{d,k})\).
gap> ConjugacyClassRepsCompatibleSubgroups(AutBall(3,2)); [ Group(()), Group([ (1,2)(3,5)(4,6) ]), Group([ (1,4,5)(2,3,6) ]), Group([ (3,5)(4,6), (1,2) ]), Group([ (1,2)(3,5)(4,6), (1,3,6)(2,4,5) ]), Group([ (3,5)(4,6), (1,3,5)(2,4,6), (1,2)(3,4)(5,6) ]), Group([ (1,2)(3,5)(4,6), (1,3,5)(2,4,6), (1,2)(5,6), (1,2)(3,4) ]), Group([ (3,5)(4,6), (1,3,5)(2,4,6), (1,2)(5,6), (1,2)(3,4) ]), Group([ (5,6), (3,4), (1,2), (1,3,5)(2,4,6), (3,5)(4,6) ]) ]
‣ ConjugacyClassRepsCompatibleGroupsWithProjection( l, F ) | ( function ) |
Returns: a list of compatible representatives of conjugacy classes of \(\mathrm{Aut}(B_{d,l})\) that contain a compatible group which projects to F \(\le\mathrm{Aut}(B_{d,r})\).
The arguments of this method are a radius l \(\in\mathbb{N}\), and a local action F \(\le\mathrm{Aut}(B_{d,k})\) for some \(k\le l\).
gap> S3:=LocalAction(3,1,SymmetricGroup(3)); Sym( [ 1 .. 3 ] ) gap> ConjugacyClassRepsCompatibleGroupsWithProjection(2,S3); [ Group([ (1,2)(3,5)(4,6), (1,4,5)(2,3,6) ]), Group([ (1,2)(3,4)(5,6), (1,2)(3,5)(4,6), (1,4,5)(2,3,6) ]), Group([ (3,4)(5,6), (1,2)(3,4), (1,4,5)(2,3,6), (3,5,4,6) ]), Group([ (3,4)(5,6), (1,2)(3,4), (1,4,5)(2,3,6), (3,5)(4,6) ]), Group([ (3,4)(5,6), (1,2)(3,4), (1,4,5)(2,3,6), (5,6), (3,5,4,6) ]) ] gap> A3:=LocalAction(3,1,AlternatingGroup(3)); Alt( [ 1 .. 3 ] ) gap> ConjugacyClassRepsCompatibleGroupsWithProjection(2,A3); [ Group([ (1,4,5)(2,3,6) ]) ]
gap> F:=SymmetricGroup(3);; gap> rho:=SignHomomorphism(F);; gap> H1:=LocalActionPi(2,3,F,rho,[0,1]);; gap> H2:=LocalActionPi(2,3,F,rho,[1]);; gap> Size(ConjugacyClassRepsCompatibleGroupsWithProjection(3,H1)); 2 gap> Size(ConjugacyClassRepsCompatibleGroupsWithProjection(3,H2)); 4
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