#############################################################################
##
#W  verify.tst
#Y  Copyright (C) 2011                                   James D. Mitchell
##
##  Licensing information can be found in the README file of this package.
##
#############################################################################
##

# this file contains everything required to verify the computations in the
# paper:

# ‘Groups that together with any transformation generate regular semigroups or
# idempotent generated semigroups’ by J. Araujo, J. D. Mitchell, and Csaba
# Schneider. 

# by doing:
#
#   ReadTest(Filename(DirectoriesPackageLibrary("citrus","dev/companion"),
#    "verify.tst"));
#
# all the computations on the webpage:
#
#   http://tinyurl.com/jdmitchell/companion.html
#
# can be verified in around 2 minutes.

gap> START_TEST("verify.tst 0.1");
gap> LoadPackage("citrus", false);;
gap> if not IsBound(OnCTSK) then 
> ReadPackage("citrus/dev/companion/companion.gd");
> ReadPackage("citrus/dev/companion/companion.gi");
> fi;
gap> Print("The following should take approx. 2 minutes.\n\n");

# Groups with the universal transversal property that do not always lead to
# idempotent generated semigroups [Theorem 1.1 (ii)=>(iii)]

# C(5)

gap> g:=PrimitiveGroup(5,1);;
gap> f:=Transformation( [ 1, 3, 2, 2, 2 ] );;
gap> gens:=List(GeneratorsOfGroup(g), x-> AsTransformation(x, Degree(f)));;
gap> s:=Semigroup(Concatenation(gens, [f]));;
gap> e:=Idempotents(s);;
gap> t:=Semigroup(e);;
gap> Length(e); Size(t); Size(s)-Size(g);
66
281
355

# D(2*5)

gap> g:=PrimitiveGroup(5,2);;
gap> f:=Transformation( [ 1, 2, 3, 3, 3 ] );;
gap> gens:=List(GeneratorsOfGroup(g), x-> AsTransformation(x, Degree(f)));;
gap> s:=Semigroup(Concatenation(gens, [f]));;
gap> e:=Idempotents(s);;
gap> t:=Semigroup(e);;
gap> Length(e); Size(t); Size(s)-Size(g);
66
281
355

# AGL(1,7)

gap> g:=PrimitiveGroup(7,4);;
gap> f:=Transformation( [ 1, 2, 3, 3, 3, 3, 3 ] );;
gap> gens:=List(GeneratorsOfGroup(g), x-> AsTransformation(x, Degree(f)));;
gap> s:=Semigroup(Concatenation(gens, [f]));;
gap> e:=Idempotents(s);;
gap> t:=Semigroup(e);;
gap> Length(e); Size(t); Size(s)-Size(g);
302
2066
2947

# PGL(2,7)

gap> g:=PrimitiveGroup(8,5);;
gap> f:=Transformation( [ 6, 2, 3, 4, 6, 6, 6, 6 ] );;
gap> gens:=List(GeneratorsOfGroup(g), x-> AsTransformation(x, Degree(f)));;
gap> s:=Semigroup(Concatenation(gens, [f]));;
gap> e:=Idempotents(s);;
gap> t:=Semigroup(e);;
gap> Length(e); Size(t); Size(s)-Size(g);
3201
89833
108648

# PSL(2,8)

gap> g:=PrimitiveGroup(9,8);;
gap> f:=Transformation( [ 1, 2, 3, 5, 4, 5, 4, 4, 5 ] );;
gap> gens:=List(GeneratorsOfGroup(g), x-> AsTransformation(x, Degree(f)));;
gap> s:=Semigroup(Concatenation(gens, [f]));;
gap> e:=Idempotents(s, Rank(f));;
gap> t:=Semigroup(e);;
gap> ForAll(Idempotents(s), x-> x=One(s) or x in t);
true 
#22s
gap> Length(e); Size(t); Size(s)-Size(g);
756
4019625
4654665

# PGammaL(2,8)

gap> g:=PrimitiveGroup(9,9);;
gap> f:=Transformation( [ 1, 2, 3, 5, 4, 5, 4, 4, 5 ] );;
gap> gens:=List(GeneratorsOfGroup(g), x-> AsTransformation(x, Degree(f)));;
gap> s:=Semigroup(Concatenation(gens, [f]));;
gap> e:=Idempotents(s, Rank(f));;
gap> t:=Semigroup(e);;
gap> ForAll(Idempotents(s), x-> x=One(s) or x in t);
true 
#26s
gap> Length(e); Size(t); Size(s)-Size(g);
756
4019625
4654665

# Groups with universal transversal property that always lead to idempotent
# generated semigroups [Theorem 1.1 (iii)=>(i)]

# AGL(1,5)

gap> SetInfoLevel(InfoCompan, 3);;
gap> g:=PrimitiveGroup(5,3);;
gap> IsAlwaysIdempotentGenerated(g);
#I  ##################################################
#I  TESTING TRANSFORMATIONS OF RANK 1
#I  ##################################################
#I  kernel : [ 1, 1, 1, 1, 1 ]
#I  image  : [ 1 ]
#I  transversal of stabilizer of image has length 1
#I  all 1 semigroups are idempotent generated
#I  ##################################################
#I  ##################################################
#I  TESTING TRANSFORMATIONS OF RANK 2
#I  ##################################################
#I  kernel : [ 1, 1, 2, 2, 1 ]
#I  image  : [ 1, 3 ]
#I  transversal of stabilizer of image has length 1
#I  all 1 semigroups are idempotent generated
#I  ##################################################
#I  kernel : [ 1, 2, 1, 1, 1 ]
#I  image  : [ 1, 2 ]
#I  transversal of stabilizer of image has length 1
#I  all 1 semigroups are idempotent generated
#I  ##################################################
#I  ##################################################
#I  TESTING TRANSFORMATIONS OF RANK 3
#I  ##################################################
#I  kernel : [ 1, 2, 3, 3, 1 ]
#I  image  : [ 1, 2, 3 ]
#I  transversal of stabilizer of image has length 3
#I  all 3 semigroups are idempotent generated
#I  ##################################################
#I  kernel : [ 1, 2, 3, 1, 1 ]
#I  image  : [ 1, 2, 3 ]
#I  transversal of stabilizer of image has length 3
#I  all 3 semigroups are idempotent generated
#I  ##################################################
#I  kernel : [ 1, 2, 2, 3, 1 ]
#I  image  : [ 1, 3, 4 ]
#I  transversal of stabilizer of image has length 3
#I  all 3 semigroups are idempotent generated
#I  ##################################################
#I  ##################################################
#I  TESTING TRANSFORMATIONS OF RANK 4
#I  ##################################################
#I  kernel : [ 1, 2, 3, 4, 1 ]
#I  image  : [ 1, 2, 3, 4 ]
#I  transversal of stabilizer of image has length 6
#I  all 6 semigroups are idempotent generated
#I  ##################################################
[ true, 18 ]

# PSL(2,5)

gap> g:=PrimitiveGroup(6,1);;
gap> IsAlwaysIdempotentGenerated(g);
#I  ##################################################
#I  TESTING TRANSFORMATIONS OF RANK 1
#I  ##################################################
#I  kernel : [ 1, 1, 1, 1, 1, 1 ]
#I  image  : [ 1 ]
#I  transversal of stabilizer of image has length 1
#I  all 1 semigroups are idempotent generated
#I  ##################################################
#I  ##################################################
#I  TESTING TRANSFORMATIONS OF RANK 2
#I  ##################################################
#I  kernel : [ 1, 1, 2, 2, 2, 1 ]
#I  image  : [ 2, 5 ]
#I  transversal of stabilizer of image has length 1
#I  all 1 semigroups are idempotent generated
#I  ##################################################
#I  kernel : [ 1, 2, 2, 1, 1, 1 ]
#I  image  : [ 1, 2 ]
#I  transversal of stabilizer of image has length 1
#I  all 1 semigroups are idempotent generated
#I  ##################################################
#I  kernel : [ 1, 2, 1, 1, 1, 1 ]
#I  image  : [ 1, 2 ]
#I  transversal of stabilizer of image has length 1
#I  all 1 semigroups are idempotent generated
#I  ##################################################
#I  ##################################################
#I  TESTING TRANSFORMATIONS OF RANK 3
#I  ##################################################
#I  kernel : [ 1, 2, 2, 3, 3, 1 ]
#I  image  : [ 1, 2, 5 ]
#I  transversal of stabilizer of image has length 1
#I  all 1 semigroups are idempotent generated
#I  ##################################################
#I  kernel : [ 1, 2, 2, 3, 3, 1 ]
#I  image  : [ 1, 2, 4 ]
#I  transversal of stabilizer of image has length 1
#I  all 1 semigroups are idempotent generated
#I  ##################################################
#I  kernel : [ 1, 2, 3, 3, 1, 1 ]
#I  image  : [ 1, 2, 3 ]
#I  transversal of stabilizer of image has length 1
#I  all 1 semigroups are idempotent generated
#I  ##################################################
#I  kernel : [ 1, 2, 3, 3, 1, 1 ]
#I  image  : [ 1, 2, 4 ]
#I  transversal of stabilizer of image has length 1
#I  all 1 semigroups are idempotent generated
#I  ##################################################
#I  kernel : [ 1, 2, 3, 1, 1, 1 ]
#I  image  : [ 1, 2, 3 ]
#I  transversal of stabilizer of image has length 1
#I  all 1 semigroups are idempotent generated
#I  ##################################################
#I  kernel : [ 1, 2, 3, 1, 1, 1 ]
#I  image  : [ 2, 3, 5 ]
#I  transversal of stabilizer of image has length 1
#I  all 1 semigroups are idempotent generated
#I  ##################################################
#I  kernel : [ 1, 1, 2, 3, 3, 1 ]
#I  image  : [ 1, 3, 4 ]
#I  transversal of stabilizer of image has length 1
#I  all 1 semigroups are idempotent generated
#I  ##################################################
#I  kernel : [ 1, 1, 2, 3, 3, 1 ]
#I  image  : [ 2, 3, 5 ]
#I  transversal of stabilizer of image has length 1
#I  all 1 semigroups are idempotent generated
#I  ##################################################
#I  kernel : [ 1, 1, 2, 2, 3, 3 ]
#I  image  : [ 2, 4, 5 ]
#I  transversal of stabilizer of image has length 1
#I  all 1 semigroups are idempotent generated
#I  ##################################################
#I  kernel : [ 1, 1, 2, 2, 3, 3 ]
#I  image  : [ 2, 3, 5 ]
#I  transversal of stabilizer of image has length 1
#I  all 1 semigroups are idempotent generated
#I  ##################################################
#I  ##################################################
#I  TESTING TRANSFORMATIONS OF RANK 4
#I  ##################################################
#I  kernel : [ 1, 2, 3, 4, 4, 1 ]
#I  image  : [ 1, 2, 3, 4 ]
#I  transversal of stabilizer of image has length 6
#I  all 6 semigroups are idempotent generated
#I  ##################################################
#I  kernel : [ 1, 2, 3, 4, 1, 1 ]
#I  image  : [ 1, 2, 3, 4 ]
#I  transversal of stabilizer of image has length 6
#I  all 6 semigroups are idempotent generated
#I  ##################################################
#I  kernel : [ 1, 2, 3, 3, 4, 1 ]
#I  image  : [ 2, 4, 5, 6 ]
#I  transversal of stabilizer of image has length 6
#I  all 6 semigroups are idempotent generated
#I  ##################################################
#I  kernel : [ 1, 2, 3, 4, 4, 4 ]
#I  image  : [ 1, 2, 3, 4 ]
#I  transversal of stabilizer of image has length 6
#I  all 6 semigroups are idempotent generated
#I  ##################################################
#I  ##################################################
#I  TESTING TRANSFORMATIONS OF RANK 5
#I  ##################################################
#I  kernel : [ 1, 2, 3, 4, 5, 1 ]
#I  image  : [ 1, 2, 3, 4, 5 ]
#I  transversal of stabilizer of image has length 12
#I  all 12 semigroups are idempotent generated
#I  ##################################################
[ true, 50 ]

# PGL(2,5)

gap> g:=PrimitiveGroup(6,2);;
gap> IsAlwaysIdempotentGenerated(g);
#I  ##################################################
#I  TESTING TRANSFORMATIONS OF RANK 1
#I  ##################################################
#I  kernel : [ 1, 1, 1, 1, 1, 1 ]
#I  image  : [ 1 ]
#I  transversal of stabilizer of image has length 1
#I  all 1 semigroups are idempotent generated
#I  ##################################################
#I  ##################################################
#I  TESTING TRANSFORMATIONS OF RANK 2
#I  ##################################################
#I  kernel : [ 1, 1, 2, 2, 2, 1 ]
#I  image  : [ 2, 5 ]
#I  transversal of stabilizer of image has length 1
#I  all 1 semigroups are idempotent generated
#I  ##################################################
#I  kernel : [ 1, 2, 2, 1, 1, 1 ]
#I  image  : [ 1, 2 ]
#I  transversal of stabilizer of image has length 1
#I  all 1 semigroups are idempotent generated
#I  ##################################################
#I  kernel : [ 1, 2, 1, 1, 1, 1 ]
#I  image  : [ 1, 2 ]
#I  transversal of stabilizer of image has length 1
#I  all 1 semigroups are idempotent generated
#I  ##################################################
#I  ##################################################
#I  TESTING TRANSFORMATIONS OF RANK 3
#I  ##################################################
#I  kernel : [ 1, 2, 2, 3, 3, 1 ]
#I  image  : [ 1, 2, 5 ]
#I  transversal of stabilizer of image has length 1
#I  all 1 semigroups are idempotent generated
#I  ##################################################
#I  kernel : [ 1, 2, 3, 3, 1, 1 ]
#I  image  : [ 1, 2, 3 ]
#I  transversal of stabilizer of image has length 1
#I  all 1 semigroups are idempotent generated
#I  ##################################################
#I  kernel : [ 1, 2, 3, 1, 1, 1 ]
#I  image  : [ 1, 2, 3 ]
#I  transversal of stabilizer of image has length 1
#I  all 1 semigroups are idempotent generated
#I  ##################################################
#I  kernel : [ 1, 1, 2, 2, 3, 3 ]
#I  image  : [ 2, 4, 5 ]
#I  transversal of stabilizer of image has length 1
#I  all 1 semigroups are idempotent generated
#I  ##################################################
#I  ##################################################
#I  TESTING TRANSFORMATIONS OF RANK 4
#I  ##################################################
#I  kernel : [ 1, 2, 3, 4, 4, 1 ]
#I  image  : [ 1, 2, 3, 4 ]
#I  transversal of stabilizer of image has length 3
#I  all 3 semigroups are idempotent generated
#I  ##################################################
#I  kernel : [ 1, 2, 3, 4, 1, 1 ]
#I  image  : [ 1, 2, 3, 4 ]
#I  transversal of stabilizer of image has length 3
#I  all 3 semigroups are idempotent generated
#I  ##################################################
#I  kernel : [ 1, 2, 3, 3, 4, 1 ]
#I  image  : [ 2, 4, 5, 6 ]
#I  transversal of stabilizer of image has length 3
#I  all 3 semigroups are idempotent generated
#I  ##################################################
#I  ##################################################
#I  TESTING TRANSFORMATIONS OF RANK 5
#I  ##################################################
#I  kernel : [ 1, 2, 3, 4, 5, 1 ]
#I  image  : [ 1, 2, 3, 4, 5 ]
#I  transversal of stabilizer of image has length 6
#I  all 6 semigroups are idempotent generated
#I  ##################################################
[ true, 23 ]

# Groups that satisfy the universal transversal property [Theorem 2.7]

gap> g:=[PrimitiveGroup(5,1), PrimitiveGroup(5,2), PrimitiveGroup(5,3), 
> PrimitiveGroup(6,1), PrimitiveGroup(6,2), PrimitiveGroup(7,4), 
> PrimitiveGroup(8,5), PrimitiveGroup(9,8), PrimitiveGroup(9,9)];;
gap> ForAll(g, IsUniversalTransversalGroup);
true 
#9s 

# Groups that do not satisfy the universal transversal property [Table 1]

# Groups of degree 7

gap> g:=[PrimitiveGroup(7,5), PrimitiveGroup(7,3), PrimitiveGroup(7,2)];;

gap> o:=Orb(g[1], [ 1, 2, 4 ], OnSets);; Enumerate(o);;
gap> ForAny(o, x-> IsInjectiveTransOnList([ 1, 2, 3, 2, 2, 2, 1 ], x));
false

gap> o:=Orb(g[2], [ 1, 2, 3 ], OnSets);; Enumerate(o);;
gap> ForAny(o, x-> IsInjectiveTransOnList([ 1, 2, 2, 3, 3, 3, 3 ], x));
false

gap> o:=Orb(g[3], [ 1, 2, 4 ], OnSets);; Enumerate(o);;
gap> ForAny(o, x-> IsInjectiveTransOnList([ 1, 1, 2, 2, 3, 3, 3 ], x));
false

# Group of degree 8

gap> g:=PrimitiveGroup(8,4);;

gap> o:=Orb(g, [ 1, 2, 3, 5 ], OnSets);; Enumerate(o);;
gap> ForAny(o, x-> IsInjectiveTransOnList([ 1, 2, 3, 4, 2, 2, 2, 1 ], x));
false

# Groups of degree 10

gap> g:=[PrimitiveGroup(10, 1), PrimitiveGroup(10,2), PrimitiveGroup(10, 3), 
> PrimitiveGroup(10, 4), PrimitiveGroup(10, 5), PrimitiveGroup(10, 6), 
> PrimitiveGroup(10, 7)];;

gap> o:=Orb(g[1], [ 1, 2, 3 ], OnSets);; Enumerate(o);;
gap> ForAny(o, x-> IsInjectiveTransOnList([ 1, 1, 2, 2, 3, 3, 3, 3, 3, 3 ], x));
false

gap> o:=Orb(g[2], [ 1, 2, 3 ], OnSets);; Enumerate(o);;
gap> ForAny(o, x-> IsInjectiveTransOnList([ 1, 1, 2, 2, 3, 3, 3, 3, 3, 3 ], x));
false

gap> o:=Orb(g[3], [ 1, 2, 3, 10 ], OnSets);; Enumerate(o);;
gap> ForAny(o, x-> IsInjectiveTransOnList([ 1, 2, 3, 3, 4, 4, 4, 4, 1, 1 ], x));
false

gap> o:=Orb(g[4], [ 1, 2, 3, 10 ], OnSets);; Enumerate(o);;
gap> ForAny(o, x-> IsInjectiveTransOnList([ 1, 2, 3, 4, 4, 4, 4, 1, 1, 1 ], x));
false

gap> o:=Orb(g[5], [ 1, 2, 3, 10 ], OnSets);; Enumerate(o);;
gap> ForAny(o, x-> IsInjectiveTransOnList([ 1, 2, 3, 3, 4, 4, 4, 4, 1, 1 ], x));
false

gap> o:=Orb(g[6], [ 1, 2, 3, 10 ], OnSets);; Enumerate(o);;
gap> ForAny(o, x-> IsInjectiveTransOnList([ 1, 2, 3, 4, 4, 4, 4, 1, 1, 1 ], x));
false

gap> o:=Orb(g[7], [ 1, 2, 3, 10 ], OnSets);; Enumerate(o);;
gap> ForAny(o, x-> IsInjectiveTransOnList([ 1, 2, 3, 4, 4, 4, 4, 1, 1, 1 ], x));
false

# Groups of degree 11

gap> g:=[PrimitiveGroup(11,5), PrimitiveGroup(11,6)];;

gap> o:=Orb(g[1], [ 1, 2, 3, 5 ], OnSets);; Enumerate(o);;
gap> ForAny(o, x-> 
> IsInjectiveTransOnList([ 1, 2, 3, 3, 3, 3, 3, 3, 4, 3, 3 ], x));
false

gap> o:=Orb(g[2], [ 1, 2, 3, 4, 6 ], OnSets);; Enumerate(o);;
gap> ForAny(o, x-> 
> IsInjectiveTransOnList([ 1, 2, 3, 4, 4, 4, 4, 5, 5, 4, 4 ], x));
false

# Groups of degree 12

gap> g:=[PrimitiveGroup(12,2), PrimitiveGroup(12,1), PrimitiveGroup(12,4), 
> PrimitiveGroup(12,3)];;

gap> o:=Orb(g[1], [ 1 .. 6 ], OnSets);; Enumerate(o);;
gap> ForAny(o, x-> 
> IsInjectiveTransOnList( [ 1, 1, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6 ], x));
false

gap> o:=Orb(g[2], [ 1, 2, 3, 4, 11, 12 ], OnSets);; Enumerate(o);;
gap> ForAny(o, x->
> IsInjectiveTransOnList([ 1, 2, 3, 4, 5, 6, 6, 6, 6, 6, 6, 6 ], x));
false

gap> o:=Orb(g[3], [ 1, 2, 3, 4, 6, 7 ], OnSets);; Enumerate(o);;
gap> ForAny(o, x->
> IsInjectiveTransOnList([ 1, 2, 3, 4, 5, 6, 6, 6, 6, 6, 6, 6 ], x));
false

gap> o:=Orb(g[4], [ 1, 2, 3, 4, 6, 7 ], OnSets);; Enumerate(o);;
gap> ForAny(o, x-> 
> IsInjectiveTransOnList([ 1, 2, 3, 4, 5, 6, 6, 6, 6, 6, 6, 6 ], x));
false

# Group of degree 13

gap> g:=PrimitiveGroup(13,7);;

gap> o:=Orb(g, [ 1, 2, 3, 4, 5, 7, 8 ], OnSets);; Enumerate(o);;
gap> ForAny(o, x-> 
> IsInjectiveTransOnList([ 1, 2, 3, 4, 5, 6, 7, 7, 7, 7, 7, 7, 7 ], x));
false

# Groups of degree 14

gap> g:=[PrimitiveGroup(14,2), PrimitiveGroup(14,1)];;

gap> o:=Orb(g[1], [ 1, 2, 3, 4, 5, 6, 9, 12 ], OnSets);; Enumerate(o);;
gap> ForAny(o, x->
> IsInjectiveTransOnList([ 1, 2, 3, 4, 5, 6, 7, 8, 8, 8, 8, 8, 8, 8 ], x));
false

gap> o:=Orb(g[2], [ 1, 2, 3, 4, 5, 6, 9, 12 ], OnSets);; Enumerate(o);;
gap> ForAny(o, x->
> IsInjectiveTransOnList([ 1, 2, 3, 4, 5, 6, 7, 8, 8, 8, 8, 8, 8, 8 ], x));
false

# Groups of degree 15

gap> g:=[PrimitiveGroup(15,4), PrimitiveGroup(15,1)];;

gap> o:=Orb(g[1], [ 1, 2, 3, 4, 5, 6, 8, 12 ], OnSets);; Enumerate(o);;
gap> ForAny(o, x-> IsInjectiveTransOnList([ 1, 2, 3, 4, 5, 6, 7, 8, 8, 8, 8, 8,
> 8, 8, 8 ], x));
false

gap> o:=Orb(g[2], [ 1, 2, 3, 4, 5, 6, 8, 12 ], OnSets);;
gap> ForAny(o, x-> IsInjectiveTransOnList([ 1, 2, 3, 4, 5, 6, 7, 8, 8, 8, 8, 8,
> 8, 8, 8 ], x));
false

# Groups of degree 17

gap> g:=[PrimitiveGroup(17, 6), PrimitiveGroup(17, 7), PrimitiveGroup(17, 8)];;

gap> o:=Orb(g[1], [ 1, 2, 3, 4, 5, 6, 7, 11, 14 ], OnSets);; Enumerate(o);;
gap> ForAny(o, x-> IsInjectiveTransOnList([ 1, 2, 3, 4, 5, 6, 7, 8, 9, 9, 9, 9,
> 9, 9, 9, 9, 9 ], x));
false

gap> o:=Orb(g[2], [ 8, 9, 10, 11, 12, 13, 14, 16, 17 ], OnSets);; Enumerate(o);;
gap> ForAny(o, x-> IsInjectiveTransOnList([ 1, 2, 3, 4, 5, 6, 7, 8, 9, 9, 9, 9,
> 9, 9, 9, 9, 9 ], x));
false

gap> o:=Orb(g[3], [ 1, 2, 3, 4, 5, 6, 7, 11, 14 ], OnSets);; Enumerate(o);;
gap> ForAny(o, x-> IsInjectiveTransOnList([ 1, 2, 3, 4, 5, 6, 7, 8, 9, 9, 9, 9,
> 9, 9, 9, 9, 9 ], x));
false

# Group of degree 18

gap> g:=PrimitiveGroup(18,2);;

gap> o:=Orb(g, [ 1, 2, 3, 4, 5, 6, 7, 8, 10, 11 ], OnSets);; Enumerate(o);;
gap> ForAny(o, x-> IsInjectiveTransOnList([ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 10,
> 10, 10, 10, 10, 10, 10, 10 ], x));
false

# Groups of degree 21

gap> g:=[PrimitiveGroup(21,5), PrimitiveGroup(21,6), PrimitiveGroup(21, 7)];;

gap> o:=Orb(g[1], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13 ], OnSets);;
gap> Enumerate(o);;
gap> ForAny(o, x-> 
> IsInjectiveTransOnList([ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 11, 11, 11, 11,
> 11, 11, 11, 11, 11, 11 ], x));
false

gap> o:=Orb(g[2], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13 ], OnSets);;
gap> Enumerate(o);;
gap> ForAny(o, x-> IsInjectiveTransOnList([ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
> 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11 ], x));
false

gap> o:=Orb(g[3], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13 ], OnSets);; 
gap> Enumerate(o);;
gap> ForAny(o, x-> IsInjectiveTransOnList([ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
> 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11 ], x));
false

# Groups of degree 22

gap> g:=[PrimitiveGroup(22,1), PrimitiveGroup(22,2)];;

gap> o:=Orb(g[1], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15 ], OnSets);;
gap> Enumerate(o);;
gap> ForAny(o, x-> IsInjectiveTransOnList([ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,
> 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12 ], x));
false

gap> o:=Orb(g[2], [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15 ], OnSets);;
gap> Enumerate(o);;
gap> ForAny(o, x-> IsInjectiveTransOnList([ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,
> 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12 ], x));
false

# Group of degree 23

gap> g:=PrimitiveGroup(23,5);; 

gap> o:=Orb(g, [ 1, 2, 3, 4, 5, 8, 11 ], OnSets);; Enumerate(o);;
gap> ForAny(o, x-> IsInjectiveTransOnList([ 1, 2, 3, 4, 5, 6, 7, 7, 7, 7, 7,
> 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7 ], x));
false