  
  [1X5 [33X[0;0YGroups and Homomorphisms[133X[101X
  
  [33X[0;0YIn  this  chapter  we  will show some computations with groups. The examples
  deal  mostly with permutation groups, because they are the easiest to input.
  The   functions   mentioned  here,  like  [2XGroup[102X  ([14XReference:  Groups[114X),  [2XSize[102X
  ([14XReference:  Size[114X) or [2XSylowSubgroup[102X ([14XReference: SylowSubgroup[114X), however, are
  the  same for all kinds of groups, although the algorithms which compute the
  information of course will be different in most cases.[133X
  
  
  [1X5.1 [33X[0;0YPermutation groups[133X[101X
  
  [33X[0;0YPermutation  groups  are  so  easy  to  input  because their elements, i.e.,
  permutations,  are  so  easy  to  type:  they  are  entered and displayed in
  disjoint cycle notation. So let's construct a permutation group:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs8 := Group( (1,2), (1,2,3,4,5,6,7,8) );[127X[104X
    [4X[28XGroup([ (1,2), (1,2,3,4,5,6,7,8) ])[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe   formed   the   group   generated   by   the   permutations   [10X(1,2)[110X  and
  [10X(1,2,3,4,5,6,7,8)[110X,  which  is  well  known  to be the symmetric group [22XS_8[122X on
  eight  points,  and  assigned  it to the identifier [10Xs8[110X. Now [22XS_8[122X contains the
  alternating  group  on  eight points which can be described in several ways,
  e.g.,  as  the  group  of  all  even  permutations  in [10Xs8[110X, or as its derived
  subgroup.  Once  we ask [5XGAP[105X to verify that the group is an alternating group
  acting  in  its  natural permutation representation, the system will display
  the group accordingly.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xa8 := DerivedSubgroup( s8 );[127X[104X
    [4X[28XGroup([ (1,2,3), (2,3,4), (2,4)(3,5), (2,6,4), (2,4)(5,7), [128X[104X
    [4X[28X  (2,8,6,4)(3,5) ])[128X[104X
    [4X[25Xgap>[125X [27XSize( a8 ); IsAbelian( a8 ); IsPerfect( a8 );[127X[104X
    [4X[28X20160[128X[104X
    [4X[28Xfalse[128X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsNaturalAlternatingGroup(a8);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xa8;[127X[104X
    [4X[28XAlt( [ 1 .. 8 ] )[128X[104X
  [4X[32X[104X
  
  [33X[0;0YOnce information about a group like [10Xs8[110X or [10Xa8[110X has been computed, it is stored
  in  the  group so that it can simply be looked up when it is required again.
  This holds for all pieces of information in the previous example. Namely, [10Xa8[110X
  stores  its  order  and that it is nonabelian and perfect, and [10Xs8[110X stores its
  derived  subgroup  [10Xa8[110X.  Had  we computed [10Xa8[110X as [10XCommutatorSubgroup( s8, s8 )[110X,
  however,  it  would  not  have  been stored, because it would then have been
  computed  as  a function of [13Xtwo[113X arguments, and hence one could not attribute
  it  to  just  one  of  them.  (Of  course  the  function  [2XCommutatorSubgroup[102X
  ([14XReference:  CommutatorSubgroup[114X)  can compute the commutator subgroup of [13Xtwo[113X
  arbitrary   subgroups.)   The   situation  is  a  bit  different  for  Sylow
  [22Xp[122X-subgroups:  The  function  [2XSylowSubgroup[102X  ([14XReference:  SylowSubgroup[114X) also
  requires  two  arguments,  namely  a  group and a prime [22Xp[122X, but the result is
  stored  in the group –namely together with the prime [22Xp[122X in a list that can be
  accessed  with  [10XComputedSylowSubgroups[110X,  but  we  won't dwell on the details
  here.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xsyl2 := SylowSubgroup( a8, 2 );; Size( syl2 );[127X[104X
    [4X[28X64[128X[104X
    [4X[25Xgap>[125X [27XNormalizer( a8, syl2 ) = syl2;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xcent := Centralizer( a8, Centre( syl2 ) );; Size( cent );[127X[104X
    [4X[28X192[128X[104X
    [4X[25Xgap>[125X [27XDerivedSeries( cent );; List( last, Size );[127X[104X
    [4X[28X[ 192, 96, 32, 2, 1 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  have  typed double semicolons after some commands to avoid the output of
  the  groups (which would be printed by their generator lists). Nevertheless,
  the  beginner is encouraged to type a single semicolon instead and study the
  full output. This remark also applies for the rest of this tutorial.[133X
  
  [33X[0;0YWith  the  next  examples,  we  want to calculate a subgroup of [10Xa8[110X, then its
  normalizer and finally determine the structure of the extension. We begin by
  forming  a  subgroup  generated  by  three  commuting  involutions,  i.e., a
  subgroup isomorphic to the additive group of the vector space [22X2^3[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xelab := Group( (1,2)(3,4)(5,6)(7,8), (1,3)(2,4)(5,7)(6,8),[127X[104X
    [4X[25X>[125X [27X                  (1,5)(2,6)(3,7)(4,8) );;[127X[104X
    [4X[25Xgap>[125X [27XSize( elab );[127X[104X
    [4X[28X8[128X[104X
    [4X[25Xgap>[125X [27XIsElementaryAbelian( elab );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YAs  usual,  [5XGAP[105X  prints  the group by giving all its generators. This can be
  annoying,  especially  if  there  are  many  of  them or if they are of huge
  degree.  It  also  makes  it  difficult to recognize a particular group when
  there are already several around. Note that although it is no problem for [13Xus[113X
  to  specify a particular group to [5XGAP[105X, by using well-chosen identifiers such
  as  [10Xa8[110X  and  [10Xelab[110X,  it  is  impossible for [5XGAP[105X to use these identifiers when
  printing a group for us, because the group does not know which identifier(s)
  point  to  it,  in fact there can be several. In order to give a name to the
  group  itself  (rather  than  to  the  identifier), you can use the function
  [2XSetName[102X  ([14XReference:  SetName[114X).  We  do  this  with  the name [10X2^3[110X here which
  reflects the mathematical properties of the group. From now on, [5XGAP[105X will use
  this  name when printing the group for us, but we still cannot use this name
  to  specify  the group to [5XGAP[105X, because the name does not know to which group
  it  was  assigned  (after  all,  you  could  assign the same name to several
  groups). When talking to the computer, you must always use identifiers.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSetName( elab, "<group of type 2^3>" ); elab;[127X[104X
    [4X[28X<group of type 2^3>[128X[104X
    [4X[25Xgap>[125X [27Xnorm := Normalizer( a8, elab );; Size( norm );[127X[104X
    [4X[28X1344[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow  that  we have the subgroup [10Xnorm[110X of order 1344 and its subgroup [10Xelab[110X, we
  want  to  look at its factor group. But since we also want to find preimages
  of  factor  group  elements  in  [10Xnorm[110X, we really want to look at the [13Xnatural
  homomorphism[113X  defined on [10Xnorm[110X with kernel [10Xelab[110X and whose image is the factor
  group.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xhom := NaturalHomomorphismByNormalSubgroup( norm, elab );[127X[104X
    [4X[28X<action epimorphism>[128X[104X
    [4X[25Xgap>[125X [27Xf := Image( hom );[127X[104X
    [4X[28XGroup([ (), (), (), (4,5)(6,7), (4,6)(5,7), (2,3)(6,7), (2,4)(3,5), [128X[104X
    [4X[28X (1,2)(5,6) ])[128X[104X
    [4X[25Xgap>[125X [27XSize( f );[127X[104X
    [4X[28X168[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  factor  group  is  again  represented as a permutation group (its first
  three generators are trivial, meaning that the first three generators of the
  preimage  are  in  the  kernel  of  [10Xhom[110X). However, the action domain of this
  factor  group  has  nothing  to  do with the action domain of [10Xnorm[110X. (It only
  happens  that  both  are  subsets  of  the natural numbers.) We can now form
  images and preimages under the natural homomorphism. The set of preimages of
  an  element  under [10Xhom[110X is a coset modulo [10Xelab[110X. We use the function [2XPreImages[102X
  ([14XReference: PreImages[114X) here because [10Xhom[110X is not a bijection, so an element of
  the range can have several preimages or none at all.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xker:= Kernel( hom );[127X[104X
    [4X[28X<group of type 2^3>[128X[104X
    [4X[25Xgap>[125X [27Xx := (1,8,3,5,7,6,2);; Image( hom, x );[127X[104X
    [4X[28X(1,7,5,6,2,3,4)[128X[104X
    [4X[25Xgap>[125X [27Xcoset := PreImages( hom, last );[127X[104X
    [4X[28XRightCoset(<group of type 2^3>,(2,8,6,7,3,4,5))[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNote  that  [5XGAP[105X  is  free  to  choose  any  representative  for the coset of
  preimages.  Of course the quotient of two representatives lies in the kernel
  of the homomorphism.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xrep:= Representative( coset );[127X[104X
    [4X[28X(2,8,6,7,3,4,5)[128X[104X
    [4X[25Xgap>[125X [27Xx * rep^-1 in ker;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  factor  group  [10Xf[110X  is a simple group, i.e., it has no non-trivial normal
  subgroups.  [5XGAP[105X  can detect this fact, and it can then also find the name by
  which  this  simple group is known among group theorists. (Such names are of
  course not available for non-simple groups.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsSimple( f ); IsomorphismTypeInfoFiniteSimpleGroup( f );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28Xrec( [128X[104X
    [4X[28X  name := "A(1,7) = L(2,7) ~ B(1,7) = O(3,7) ~ C(1,7) = S(2,7) ~ 2A(1,\[128X[104X
    [4X[28X7) = U(2,7) ~ A(2,2) = L(3,2)", parameter := [ 2, 7 ], series := "L" )[128X[104X
    [4X[25Xgap>[125X [27XSetName( f, "L_3(2)" );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  give  [10Xf[110X the name [10XL_3(2)[110X because the last part of the name string reveals
  that  it  is  isomorphic  to  the  simple  linear  group [22XL_3(2)[122X. This group,
  however,  also  has  a lot of other names. Names that are connected with a [10X=[110X
  sign  are different names for the same matrix group, e.g., [10XA(2,2)[110X is the Lie
  type  notation  for  the classical notation [10XL(3,2)[110X. Other pairs of names are
  connected  via  [10X~[110X,  these  then  specify  other  classical  groups  that are
  isomorphic  to  that  linear group (e.g., the symplectic group [10XS(2,7)[110X, whose
  Lie type notation would be [10XC(1,7)[110X).[133X
  
  [33X[0;0YThe  group  [10Xnorm[110X  acts  on the eight elements of its normal subgroup [10Xelab[110X by
  conjugation,  yielding  a  representation  of  [22XL_3(2)[122X in [10Xs8[110X which leaves one
  point  fixed  (namely  point [10X1[110X).  The  image  of  this representation can be
  computed  with  the function [2XAction[102X ([14XReference: Action homomorphisms[114X); it is
  even contained in the group [10Xnorm[110X and we can show that [10Xnorm[110X is indeed a split
  extension of the elementary abelian group [22X2^3[122X with this image of [22XL_3(2)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xop := Action( norm, elab );[127X[104X
    [4X[28XGroup([ (), (), (), (5,6)(7,8), (5,7)(6,8), (3,4)(7,8), (3,5)(4,6), [128X[104X
    [4X[28X  (2,3)(6,7) ])[128X[104X
    [4X[25Xgap>[125X [27XIsSubgroup( a8, op ); IsSubgroup( norm, op );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsTrivial( Intersection( elab, op ) );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XSetName( norm, "2^3:L_3(2)" );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YBy  the  way,  you  should  not  try  the operator [10X<[110X instead of the function
  [2XIsSubgroup[102X ([14XReference: IsSubgroup[114X). Something like[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xelab < a8;[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [33X[0;0Ywill  not cause an error, but the result does not signify anything about the
  inclusion  of  one group in another; [10X<[110X tests which of the two groups is less
  in some total order. On the other hand, the equality operator [10X=[110X in fact does
  test the equality of its arguments.[133X
  
  [33X[0;0Y[13XSummary.[113X  In  this  section  we  have used the elementary group functions to
  determine  the  structure  of  a  normalizer.  We have assigned names to the
  involved  groups  which  reflect  their  mathematical structure and [5XGAP[105X uses
  these names when printing the groups.[133X
  
  
  [1X5.2 [33X[0;0YActions of Groups[133X[101X
  
  [33X[0;0YIn  order  to  get another representation of [10Xa8[110X, we consider another action,
  namely that on the elements of a certain conjugacy class by conjugation.[133X
  
  [33X[0;0YIn  the following example we temporarily increase the line length limit from
  its default value 80 to 82 in order to make the long expression fit into one
  line.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xccl := ConjugacyClasses( a8 );; Length( ccl );[127X[104X
    [4X[28X14[128X[104X
    [4X[25Xgap>[125X [27XList( ccl, c -> Order( Representative( c ) ) );[127X[104X
    [4X[28X[ 1, 2, 2, 3, 6, 3, 4, 4, 5, 15, 15, 6, 7, 7 ][128X[104X
    [4X[25Xgap>[125X [27XList( ccl, Size );[127X[104X
    [4X[28X[ 1, 210, 105, 112, 1680, 1120, 2520, 1260, 1344, 1344, 1344, 3360, [128X[104X
    [4X[28X  2880, 2880 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YNote  the  difference  between  [2XOrder[102X  ([14XReference:  Order[114X)  (which means the
  element  order),  [2XSize[102X  ([14XReference:  Size[114X)  (which  means  the  size  of the
  conjugacy class) and [2XLength[102X ([14XReference: Length[114X) (which means the length of a
  list). We choose to let [10Xa8[110X operate on the class of length 112.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xclass := First( ccl, c -> Size(c) = 112 );;[127X[104X
    [4X[25Xgap>[125X [27Xop := Action( a8, AsList( class ),OnPoints );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YWe use [2XAsList[102X ([14XReference: AsList[114X) here to convert the conjugacy class into a
  list  of  its elements whereas we wrote [10XAction( norm, elab )[110X directly in the
  previous  section.  The reason is that the elementary abelian group [10Xelab[110X can
  be  quickly  enumerated  by  [5XGAP[105X whereas the standard enumeration method for
  conjugacy  classes is slower than just explicit calculation of the elements.
  However,  [5XGAP[105X  is reluctant to construct explicit element lists, because for
  really large groups this direct method is infeasible.[133X
  
  [33X[0;0YNote  also  the  function  [2XFirst[102X  ([14XReference: First[114X), used to find the first
  element in a list which passes some test.[133X
  
  [33X[0;0YIn  this example, we have specified the action function [2XOnPoints[102X ([14XReference:
  OnPoints[114X)  in  this  example,  which is defined as [10XOnPoints( [110X[22Xd[122X[10X, [110X[22Xg[122X[10X ) = [110X[22Xd[122X[10X ^ [110X[22Xg[122X.
  This [21Xcaret[121X operator denotes conjugation in a group if both arguments [22Xd[122X and [22Xg[122X
  are  group  elements  (contained in a common group), but it also denotes the
  natural  action  of permutations on positive integers (and exponentiation of
  integers  as  well, of course). It is in fact the default action and will be
  supplied  by  the  system if not given. Another common action is for example
  always   assumes   [2XOnRight[102X   ([14XReference:   OnRight[114X),   which   means   right
  multiplication,  defined as [22Xd[122X[10X * [110X[22Xg[122X. (Group actions in [5XGAP[105X are always from the
  right.)[133X
  
  [33X[0;0YWe now have a permutation representation [10Xop[110X on 112 points, which we test for
  primitivity.  If  it  is not primitive, we can obtain a minimal block system
  (i.e.,  one  where  the  blocks  have minimal length) by the function [2XBlocks[102X
  ([14XReference: Blocks[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsPrimitive( op, [ 1 .. 112 ] );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xblocks := Blocks( op, [ 1 .. 112 ] );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YNote that we must specify the domain of the action. You might think that the
  functions   [2XIsPrimitive[102X  ([14XReference:  IsPrimitive[114X)  and  [2XBlocks[102X  ([14XReference:
  Blocks[114X) could use [10X[ 1 .. 112 ][110X as default domain if no domain was given. But
  this  is not so easy, for example would the default domain of [10XGroup( (2,3,4)
  )[110X  be  [10X[  1  ..  4 ][110X or [10X[ 2 .. 4 ][110X? To avoid confusion, all action functions
  require  that  you  specify the domain of action. If we had specified [10X[ 1 ..
  113  ][110X  in  the primitivity test above, point 113 would have been a fixpoint
  (and the action would not even have been transitive).[133X
  
  [33X[0;0YNow  [10Xblocks[110X  is  a  list  of blocks (i.e., a list of lists), which we do not
  print  here  for the sake of saving paper (try it for yourself). In fact all
  we want to know is the size of the blocks, or rather how many there are (the
  product  of  these  two numbers must of course be 112). Then we can obtain a
  new permutation group of the corresponding degree by letting [10Xop[110X act on these
  blocks setwise.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLength( blocks[1] );  Length( blocks );[127X[104X
    [4X[28X2[128X[104X
    [4X[28X56[128X[104X
    [4X[25Xgap>[125X [27Xop2 := Action( op, blocks, OnSets );;[127X[104X
    [4X[25Xgap>[125X [27XIsPrimitive( op2, [ 1 .. 56 ] );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNote  that  we give a third argument (the action function [2XOnSets[102X ([14XReference:
  OnSets[114X)) to indicate that the action is not the default action on points but
  an  action  on  sets of elements given as sorted lists. (Section [14X'Reference:
  Basic Actions'[114X lists all actions that are pre-defined by [5XGAP[105X.)[133X
  
  [33X[0;0YThe  action  of [10Xop[110X on the given block system gave us a new representation on
  56  points  which  is  primitive,  i.e.,  the  point stabilizer is a maximal
  subgroup.  We  compute  its  preimage  in the representation on eight points
  using  the associated action homomorphisms (which of course in this case are
  monomorphisms). We construct the composition of two homomorphisms with the [10X*[110X
  operator, reading left-to-right.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xophom := ActionHomomorphism( a8, op );;[127X[104X
    [4X[25Xgap>[125X [27Xophom2 := ActionHomomorphism( op, op2 );;[127X[104X
    [4X[25Xgap>[125X [27Xcomposition := ophom * ophom2;;[127X[104X
    [4X[25Xgap>[125X [27Xstab := Stabilizer( op2, 2 );;[127X[104X
    [4X[25Xgap>[125X [27Xpreim := PreImages( composition, stab );[127X[104X
    [4X[28XGroup([ (1,4,2), (3,6,7), (3,8,5,7,6), (1,4)(7,8) ])[128X[104X
  [4X[32X[104X
  
  [33X[0;0YAlternatively,  it  is  possible  to create action homomorphisms immediately
  (without  creating  the action first) by giving the same set of arguments to
  [2XActionHomomorphism[102X ([14XReference: ActionHomomorphism[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xnophom := ActionHomomorphism( a8, AsList(class) );[127X[104X
    [4X[28X<action homomorphism>[128X[104X
    [4X[25Xgap>[125X [27XIsSurjective(nophom);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XImage(nophom,(1,2,3));[127X[104X
    [4X[28X(2,43,14)(3,44,20)(4,45,26)(5,46,32)(6,47,38)(8,13,48)(9,19,53)(10,25,[128X[104X
    [4X[28X58)(11,31,63)(12,37,68)(15,49,73)(16,50,74)(17,51,75)(18,52,76)(21,54,[128X[104X
    [4X[28X77)(22,55,78)(23,56,79)(24,57,80)(27,59,81)(28,60,82)(29,61,83)(30,62,[128X[104X
    [4X[28X84)(33,64,85)(34,65,86)(35,66,87)(36,67,88)(39,69,89)(40,70,90)(41,71,[128X[104X
    [4X[28X91)(42,72,92)[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn this situation, however (for performance reasons, avoiding computation an
  image that might never be needed) the homomorphism is defined to be not into
  the  [13XImage[113X  of the action, but into the [13Xfull symmetric group[113X, i.e. it is not
  automatically  surjective. Surjectivity can be enforced by giving the string
  [10X"surjective"[110X as an extra last argument. The [10XImage[110X of the action homomorphism
  of course is the same group in either case.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSize(Range(nophom));[127X[104X
    [4X[28X1974506857221074023536820372759924883412778680349753377966562950949028\[128X[104X
    [4X[28X5896977181144089422435502777936659795733823785363827233491968638562181\[128X[104X
    [4X[28X1850780464277094400000000000000000000000000[128X[104X
    [4X[25Xgap>[125X [27XSize(Range(ophom));[127X[104X
    [4X[28X20160[128X[104X
    [4X[25Xgap>[125X [27Xnophom := ActionHomomorphism( a8, AsList(class),"surjective" );[127X[104X
    [4X[28X<action epimorphism>[128X[104X
    [4X[25Xgap>[125X [27XSize(Range(nophom));[127X[104X
    [4X[28X20160[128X[104X
  [4X[32X[104X
  
  [33X[0;0YContinuing  the example, the normalizer of an element in the conjugacy class
  [10Xclass[110X  is  a  group  of  order  360,  too. In fact, it is a conjugate of the
  maximal  subgroup  we  had  found before, and a conjugating element in [10Xa8[110X is
  found      by      the     function     [2XRepresentativeAction[102X     ([14XReference:
  RepresentativeAction[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xsgp := Normalizer( a8, Subgroup(a8,[Representative(class)]) );;[127X[104X
    [4X[25Xgap>[125X [27XSize( sgp );[127X[104X
    [4X[28X360[128X[104X
    [4X[25Xgap>[125X [27XRepresentativeAction( a8, sgp, preim );[127X[104X
    [4X[28X(2,4,3)[128X[104X
  [4X[32X[104X
  
  [33X[0;0YOne of the most prominent actions of a group is on the cosets of a subgroup.
  Naïvely  this  can  be done by constructing the cosets and acting on them by
  right multiplication.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xcosets:=RightCosets(a8,norm);;[127X[104X
    [4X[25Xgap>[125X [27Xop:=Action(a8,cosets,OnRight);[127X[104X
    [4X[28XGroup([ (1,2,3)(4,6,5)(7,8,9)(10,12,11)(13,14,15), [128X[104X
    [4X[28X  (1,3,2)(4,9,13)(5,11,7)(6,15,10)(8,12,14), [128X[104X
    [4X[28X  (1,13)(2,7)(3,10)(4,11)(5,15)(6,9), [128X[104X
    [4X[28X  (1,8,13)(2,7,12)(3,9,5)(4,14,11)(6,10,15), [128X[104X
    [4X[28X  (2,3)(4,14)(5,7)(8,13)(9,12)(10,15), [128X[104X
    [4X[28X  (1,8)(2,3,11,6)(4,12,10,15)(5,7,14,9) ])[128X[104X
    [4X[25Xgap>[125X [27XNrMovedPoints(op);[127X[104X
    [4X[28X15[128X[104X
  [4X[32X[104X
  
  [33X[0;0YA  problem  with this approach is that creating (and storing) all cosets can
  be  very memory intensive if the subgroup index gets large. Because of this,
  [5XGAP[105X  provides  special objects which act like a list of elements, but do not
  actually store elements but compute them on the go. Such a simulated list is
  called  an [13Xenumerator[113X. The easiest example of this concept is the [2XEnumerator[102X
  ([14XReference:  Enumerator[114X)  of  a  group.  While  it  behaves  like  a list of
  elements,  it  requires  far  less storage, and is applicable to potentially
  huge  groups  for  which it would be completely infeasible to write down all
  elements:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xenum:=Enumerator(SymmetricGroup(20));[127X[104X
    [4X[28X<enumerator of perm group>[128X[104X
    [4X[25Xgap>[125X [27XLength(enum);[127X[104X
    [4X[28X2432902008176640000[128X[104X
    [4X[25Xgap>[125X [27Xenum[123456789012345];[127X[104X
    [4X[28X(1,4,15,3,14,11,8,17,6,18,5,7,20,13,10,9,2,12)[128X[104X
    [4X[25Xgap>[125X [27XPosition(enum,(1,2,3,4,5,6,7,8,9,10));[127X[104X
    [4X[28X71948729603[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  the  action  on  cosets  the object of interest is the [2XRightTransversal[102X
  ([14XReference:  RightTransversal[114X)  of  a subgroup. Again, it does not write out
  actual elements and thus can be created even for subgroups of large index.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:=RightTransversal(a8,norm);[127X[104X
    [4X[28XRightTransversal(Alt( [ 1 .. 8 ] ),2^3:L_3(2))[128X[104X
    [4X[25Xgap>[125X [27Xt[7];[127X[104X
    [4X[28X(4,6,5)[128X[104X
    [4X[25Xgap>[125X [27XPosition(t,(4,6,7,8,5));[127X[104X
    [4X[28X8[128X[104X
    [4X[25Xgap>[125X [27XPosition(t,(1,2,3));[127X[104X
    [4X[28Xfail[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  the  action  on  cosets  there is the added complication that not every
  group  element  is  in  the  transversal (as the last example shows) but the
  action  on  cosets  of  a subgroup usually will not preserve a chosen set of
  coset  representatives.  Because  of  this  issue,  all action functionality
  actually  uses  [2XPositionCanonical[102X  ([14XReference: PositionCanonical[114X) instead of
  [2XPosition[102X  ([14XReference:  Position[114X).  In  general,  for elements contained in a
  list,  [2XPositionCanonical[102X  ([14XReference: PositionCanonical[114X) returns the same as
  [10XPosition[110X.  If  the  element  is  not  contained in the list (and for special
  lists,  such  as  transversals),  [10XPositionCanonical[110X returns the list element
  representing the same objects, e.g. the transversal element representing the
  same coset.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XPositionCanonical(t,(1,2,3));[127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27Xt[2];[127X[104X
    [4X[28X(6,7,8)[128X[104X
    [4X[25Xgap>[125X [27Xt[2]/(1,2,3);[127X[104X
    [4X[28X(1,3,2)(6,7,8)[128X[104X
    [4X[25Xgap>[125X [27Xlast in norm;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThus,  acting on a [10XRightTransversal[110X with the [10XOnRight[110X action will in fact (in
  a  slight abuse of definitions) produce the action of a group on cosets of a
  subgroup and is in general the most efficient way of creating this action.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XAction(a8,RightTransversal(a8,norm),OnRight);[127X[104X
    [4X[28XGroup([ (1,2,3)(4,6,5)(7,8,9)(10,12,11)(13,14,15), [128X[104X
    [4X[28X  (1,3,2)(4,9,13)(5,11,7)(6,15,10)(8,12,14), [128X[104X
    [4X[28X  (1,13)(2,7)(3,10)(4,11)(5,15)(6,9), [128X[104X
    [4X[28X  (1,8,13)(2,7,12)(3,9,5)(4,14,11)(6,10,15), [128X[104X
    [4X[28X  (2,3)(4,14)(5,7)(8,13)(9,12)(10,15), [128X[104X
    [4X[28X  (1,8)(2,3,11,6)(4,12,10,15)(5,7,14,9) ])[128X[104X
  [4X[32X[104X
  
  [33X[0;0Y[13XSummary.[113X  In  this  section  we  have  learned how groups can operate on [5XGAP[105X
  objects such as integers and group elements. We have used [2XActionHomomorphism[102X
  ([14XReference:    ActionHomomorphism[114X),   among   others,   to   construct   the
  corresponding  actions  and homomorphisms and have seen how transversals can
  be used to create the action on cosets of a subgroup.[133X
  
  
  [1X5.3 [33X[0;0YSubgroups as Stabilizers[133X[101X
  
  [33X[0;0YAction  functions  can  also  be used without constructing external sets. We
  will try to find several subgroups in [10Xa8[110X as stabilizers of such actions. One
  subgroup  is  immediately available, namely the stabilizer of one point. The
  index  of the stabilizer must of course be equal to the length of the orbit,
  i.e., 8.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xu8 := Stabilizer( a8, 1 );[127X[104X
    [4X[28XGroup([ (2,3,4,5,6,7,8), (2,4,5,6,7,8,3) ])[128X[104X
    [4X[25Xgap>[125X [27XIndex( a8, u8 );[127X[104X
    [4X[28X8[128X[104X
    [4X[25Xgap>[125X [27XOrbit( a8, 1 ); Length( last );[127X[104X
    [4X[28X[ 1, 3, 2, 4, 5, 6, 7, 8 ][128X[104X
    [4X[28X8[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThis  gives  us  a  hint how to find further subgroups. Each subgroup is the
  stabilizer of a point of an appropriate transitive action (namely the action
  on  the cosets of that subgroup or another action that is equivalent to this
  action).  So the question is how to find other actions. The obvious thing is
  to  operate  on  pairs  of  points. So using the function [2XTuples[102X ([14XReference:
  Tuples[114X) we first generate a list of all pairs.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xpairs := Tuples( [1..8], 2 );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YNow  we  would like to have [10Xa8[110X operate on this domain. But we cannot use the
  default  action  [2XOnPoints[102X ([14XReference: OnPoints[114X) because powering a list by a
  permutation  via  the  caret  operator [10X^[110X is not defined. So we must tell the
  functions  from  the  actions  package how the group elements operate on the
  elements  of  the domain (here and below, the word [21Xpackage[121X refers to the [5XGAP[105X
  functionality  for  group  actions, not to a [5XGAP[105X package). In our example we
  can  do  this  by simply passing [2XOnPairs[102X ([14XReference: OnPairs[114X) as an optional
  last  argument.  All  functions  from  the  actions  package  accept such an
  optional  argument  that  describes  the action. One example is [2XIsTransitive[102X
  ([14XReference: IsTransitive[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsTransitive( a8, pairs, OnPairs );[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  action is of course not transitive, since the pairs [10X[ 1, 1 ][110X and [10X[ 1, 2
  ][110X  cannot lie in the same orbit. So we want to find out what the orbits are.
  The  function [2XOrbits[102X ([14XReference: Orbits[114X) does that for us. It returns a list
  of  all the orbits. We look at the orbit lengths and representatives for the
  orbits.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xorbs := Orbits( a8, pairs, OnPairs );; Length( orbs );[127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27XList( orbs, Length );[127X[104X
    [4X[28X[ 8, 56 ][128X[104X
    [4X[25Xgap>[125X [27XList( orbs, o -> o[1] );[127X[104X
    [4X[28X[ [ 1, 1 ], [ 1, 2 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  action  of [10Xa8[110X on the first orbit (this is the one containing [10X[1,1][110X, try
  [10X[1,1]  in  orbs[1][110X)  is  of  course equivalent to the original action, so we
  ignore it and work with the second orbit.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xu56 := Stabilizer( a8, orbs[2][1], OnPairs );; Index( a8, u56 );[127X[104X
    [4X[28X56[128X[104X
  [4X[32X[104X
  
  [33X[0;0YSo now we have found a second subgroup. To make the following computations a
  little bit easier and more efficient we would now like to work on the points
  [10X[  1  ..  56 ][110X instead of the list of pairs. The function [2XActionHomomorphism[102X
  ([14XReference: ActionHomomorphism[114X) does what we need. It creates a homomorphism
  defined  on  [10Xa8[110X  whose  image is a new group that acts on [10X[ 1 .. 56 ][110X in the
  same way that [10Xa8[110X acts on the second orbit.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xh56 := ActionHomomorphism( a8, orbs[2], OnPairs );;[127X[104X
    [4X[25Xgap>[125X [27Xa8_56 := Image( h56 );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  would  now like to know if the subgroup [10Xu56[110X of index 56 that we found is
  maximal  or  not.  As  we  have  used  already in Section [14X5.2[114X, a subgroup is
  maximal  if  and  only  if  the  action  on  the  cosets of this subgroup is
  primitive.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsPrimitive( a8_56, [1..56] );[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [33X[0;0YRemember  that we can leave out the function if we mean [2XOnPoints[102X ([14XReference:
  OnPoints[114X)  but  that  we  have  to  specify the action domain for all action
  functions.[133X
  
  [33X[0;0YWe  see that [10Xa8_56[110X is not primitive. This means of course that the action of
  [10Xa8[110X  on [10Xorb[2][110X is not primitive, because those two actions are equivalent. So
  the  stabilizer  [10Xu56[110X  is not maximal. Let us try to find its supergroups. We
  use  the  function  [2XBlocks[102X  ([14XReference:  Blocks[114X) to find a block system. The
  (optional)  third argument in the following example tells [2XBlocks[102X ([14XReference:
  Blocks[114X) that we want a block system where 1 and 3 lie in one block.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xblocks := Blocks( a8_56, [1..56], [1,3] );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  result  is  a  list of sets, such that [10Xa8_56[110X acts on those sets. Now we
  would  like  the  stabilizer  of this action on the sets. Because we want to
  operate  on  the  sets  we  have to pass [2XOnSets[102X ([14XReference: OnSets[114X) as third
  argument.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xu8_56 := Stabilizer( a8_56, blocks[1], OnSets );;[127X[104X
    [4X[25Xgap>[125X [27XIndex( a8_56, u8_56 );[127X[104X
    [4X[28X8[128X[104X
    [4X[25Xgap>[125X [27Xu8b := PreImages( h56, u8_56 );; Index( a8, u8b );[127X[104X
    [4X[28X8[128X[104X
    [4X[25Xgap>[125X [27XIsConjugate( a8, u8, u8b );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YSo  we have found a supergroup of [10Xu56[110X that is conjugate in [10Xa8[110X to [10Xu8[110X. This is
  not  surprising,  since  [10Xu8[110X  is  a  point stabilizer, and [10Xu56[110X is a two point
  stabilizer in the natural action of [10Xa8[110X on eight points.[133X
  
  [33X[0;0YHere  is  a  [13Xwarning[113X:  If  you  specify  [2XOnSets[102X ([14XReference: OnSets[114X) as third
  argument to a function like [2XStabilizer[102X ([14XReference: Stabilizers[114X), you have to
  make  sure  that  the  point  (i.e.  the  second  argument) is indeed a set.
  Otherwise  you  will  get a puzzling error message or even wrong results! In
  the  above  example,  the  second  argument [10Xblocks[1][110X came from the function
  [2XBlocks[102X  ([14XReference: Blocks[114X), which returns a list of sets, so everything was
  OK.[133X
  
  [33X[0;0YActually  there  is a third block system of [10Xa8_56[110X that gives rise to a third
  subgroup.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xblocks := Blocks( a8_56, [1..56], [1,13] );;[127X[104X
    [4X[25Xgap>[125X [27Xu28_56 := Stabilizer( a8_56, [1,13], OnSets );;[127X[104X
    [4X[25Xgap>[125X [27Xu28 := PreImages( h56, u28_56 );;[127X[104X
    [4X[25Xgap>[125X [27XIndex( a8, u28 );[127X[104X
    [4X[28X28[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  know  that the subgroup [10Xu28[110X of index 28 is maximal, because we know that
  [10Xa8[110X  has no subgroups of index 2, 4, or 7. However we can also quickly verify
  this by checking that [10Xa8_56[110X acts primitively on the 28 blocks.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsPrimitive( a8_56, blocks, OnSets );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0Y[2XStabilizer[102X ([14XReference: Stabilizers[114X) is not only applicable to groups like [10Xa8[110X
  but  also  to  their  subgroups  like  [10Xu56[110X.  So another method to find a new
  subgroup is to compute the stabilizer of another point in [10Xu56[110X. Note that [10Xu56[110X
  already leaves 1 and 2 fixed.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xu336 := Stabilizer( u56, 3 );;[127X[104X
    [4X[25Xgap>[125X [27XIndex( a8, u336 );[127X[104X
    [4X[28X336[128X[104X
  [4X[32X[104X
  
  [33X[0;0YOther  functions  are also applicable to subgroups. In the following we show
  that  [10Xu336[110X  acts  regularly on the 60 triples of [10X[ 4 .. 8 ][110X which contain no
  element  twice.  We construct the list of these 60 triples with the function
  [2XOrbit[102X  ([14XReference:  Orbit[114X)  (using  [2XOnTuples[102X  ([14XReference:  OnTuples[114X)  as the
  natural  generalization of [2XOnPairs[102X ([14XReference: OnPairs[114X)) and then pass it as
  action domain to the function [2XIsRegular[102X ([14XReference: IsRegular[114X). The positive
  result  of  the  regularity test means that this action is equivalent to the
  actions of [10Xu336[110X on its 60 elements from the right.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsRegular( u336, Orbit( u336, [4,5,6], OnTuples ), OnTuples );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YJust  as  we  did  in  the  case  of  the  action on the pairs above, we now
  construct  a new permutation group that acts on [10X[ 1 .. 336 ][110X in the same way
  that  [10Xa8[110X  acts  on  the cosets of [10Xu336[110X. But this time we let [10Xa8[110X operate on a
  right transversal, just like [10Xnorm[110X did in the natural homomorphism above.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt := RightTransversal( a8, u336 );;[127X[104X
    [4X[25Xgap>[125X [27Xa8_336 := Action( a8, t, OnRight );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YTo find subgroups above [10Xu336[110X we again look for nontrivial block systems.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xblocks := Blocks( a8_336, [1..336] );; blocks[1];[127X[104X
    [4X[28X[ 1, 43, 85 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe see that the union of [10Xu336[110X with its 43rd and its 85th coset is a subgroup
  in  [10Xa8_336[110X, its index is 112. We can obtain it as the closure of [10Xu336[110X with a
  representative  of the 43rd coset, which can be found as the 43rd element of
  the  transversal [10Xt[110X.  Note  that  in the representation [10Xa8_336[110X on 336 points,
  this subgroup corresponds to the stabilizer of the block [10X[ 1, 43, 85 ][110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xu112 := ClosureGroup( u336, t[43] );;[127X[104X
    [4X[25Xgap>[125X [27XIndex( a8, u112 );[127X[104X
    [4X[28X112[128X[104X
  [4X[32X[104X
  
  [33X[0;0YAbove  this  subgroup of index 112 lies a subgroup of index 56, which is not
  conjugate to [10Xu56[110X. In fact, unlike [10Xu56[110X it is maximal. We obtain this subgroup
  in  the same way that we obtained [10Xu112[110X, this time forcing two points, namely
  7 and 43 into the first block.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xblocks := Blocks( a8_336, [1..336], [1,7,43] );;[127X[104X
    [4X[25Xgap>[125X [27XLength( blocks );[127X[104X
    [4X[28X56[128X[104X
    [4X[25Xgap>[125X [27Xu56b := ClosureGroup( u112, t[7] );; Index( a8, u56b );[127X[104X
    [4X[28X56[128X[104X
    [4X[25Xgap>[125X [27XIsPrimitive( a8_336, blocks, OnSets );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe already mentioned in Section [14X5.2[114X that there is another standard action of
  permutations,  namely  the  conjugation.  E.g.,  since  no  other  action is
  specified  in  the  following  example, [2XOrbitLength[102X ([14XReference: OrbitLength[114X)
  simply  acts via [2XOnPoints[102X ([14XReference: OnPoints[114X), and because [3Xperm_1[103X[10X ^ [110X[3Xperm_2[103X
  is  defined  as  the conjugation of [3Xperm_2[103X on [3Xperm_1[103X, in fact we compute the
  length of the conjugacy class of [10X(1,2)(3,4)(5,6)(7,8)[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XOrbitLength( a8, (1,2)(3,4)(5,6)(7,8) );[127X[104X
    [4X[28X105[128X[104X
    [4X[25Xgap>[125X [27Xorb := Orbit( a8, (1,2)(3,4)(5,6)(7,8) );;[127X[104X
    [4X[25Xgap>[125X [27Xu105 := Stabilizer( a8, (1,2)(3,4)(5,6)(7,8) );; Index( a8, u105 );[127X[104X
    [4X[28X105[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNote  that  although the length of a conjugacy class of any element [22Xg[122X in any
  finite  group  [22XG[122X  can  be computed as [10XOrbitLength( [110X[22XG[122X[10X, [110X[22Xg[122X[10X )[110X, the command [10XSize(
  ConjugacyClass( [110X[22XG[122X[10X, [110X[22Xg[122X[10X ) )[110X is probably more efficient.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSize( ConjugacyClass( a8, (1,2)(3,4)(5,6)(7,8) ) );[127X[104X
    [4X[28X105[128X[104X
  [4X[32X[104X
  
  [33X[0;0YOf  course  the  stabilizer  [10Xu105[110X  is in fact the centralizer of the element
  [10X(1,2)(3,4)(5,6)(7,8)[110X.  [2XStabilizer[102X  ([14XReference: Stabilizers[114X) notices that and
  computes  the  stabilizer  using  the  centralizer algorithm for permutation
  groups. In the usual way we now look for the subgroups above [10Xu105[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xblocks := Blocks( a8, orb );; Length( blocks );[127X[104X
    [4X[28X15[128X[104X
    [4X[25Xgap>[125X [27Xblocks[1];[127X[104X
    [4X[28X[ (1,2)(3,4)(5,6)(7,8), (1,3)(2,4)(5,7)(6,8), (1,4)(2,3)(5,8)(6,7), [128X[104X
    [4X[28X  (1,5)(2,6)(3,7)(4,8), (1,6)(2,5)(3,8)(4,7), (1,7)(2,8)(3,5)(4,6), [128X[104X
    [4X[28X  (1,8)(2,7)(3,6)(4,5) ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YTo  find  the  subgroup  of  index 15 we again use closure. Now we must be a
  little   bit   careful  to  avoid  confusion.  [10Xu105[110X  is  the  stabilizer  of
  [10X(1,2)(3,4)(5,6)(7,8)[110X.  We  know  that  there is a correspondence between the
  points  of  the orbit and the cosets of [10Xu105[110X. The point [10X(1,2)(3,4)(5,6)(7,8)[110X
  corresponds to [10Xu105[110X. To get the subgroup above [10Xu105[110X that has index 15 in [10Xa8[110X,
  we  must  form  the  closure  of  [10Xu105[110X  with  an  element  of the coset that
  corresponds  to  any  other point in the first block. If we choose the point
  [10X(1,3)(2,4)(5,8)(6,7)[110X,   we   must   use   an   element   of   [10Xa8[110X  that  maps
  [10X(1,2)(3,4)(5,6)(7,8)[110X      to      [10X(1,3)(2,4)(5,8)(6,7)[110X.     The     function
  [2XRepresentativeAction[102X ([14XReference: RepresentativeAction[114X) does what we need. It
  takes  a  group and two points and returns an element of the group that maps
  the  first  point  to  the second. In fact it also allows you to specify the
  action  as  an  optional  fourth  argument as usual, but we do not need this
  here. If no such element exists in the group, i.e., if the two points do not
  lie   in   one  orbit  under  the  group,  [2XRepresentativeAction[102X  ([14XReference:
  RepresentativeAction[114X) returns [9Xfail[109X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xrep := RepresentativeAction( a8, (1,2)(3,4)(5,6)(7,8),[127X[104X
    [4X[25X>[125X [27X                                       (1,3)(2,4)(5,8)(6,7) );[127X[104X
    [4X[28X(2,3)(6,8)[128X[104X
    [4X[25Xgap>[125X [27Xu15 := ClosureGroup( u105, rep );; Index( a8, u15 );[127X[104X
    [4X[28X15[128X[104X
  [4X[32X[104X
  
  [33X[0;0Y[10Xu15[110X  is of course a maximal subgroup, because [10Xa8[110X has no subgroups of index 3
  or 5.  There  is  in  fact another class of subgroups of index 15 above [10Xu105[110X
  that we get by adding [10X(2,3)(6,7)[110X to [10Xu105[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xu15b := ClosureGroup( u105, (2,3)(6,7) );; Index( a8, u15b );[127X[104X
    [4X[28X15[128X[104X
    [4X[25Xgap>[125X [27XRepresentativeAction( a8, u15, u15b );[127X[104X
    [4X[28Xfail[128X[104X
  [4X[32X[104X
  
  [33X[0;0Y[2XRepresentativeAction[102X  ([14XReference:  RepresentativeAction[114X) tells us that there
  is  no  element [22Xg[122X in [10Xa8[110X such that [10Xu15 ^ [110X[22Xg[122X[10X = u15b[110X. Because [10X^[110X also denotes the
  conjugation of subgroups this tells us that [10Xu15[110X and [10Xu15b[110X are not conjugate.[133X
  
  [33X[0;0Y[13XSummary.[113X  In  this  section  we  have  demonstrated  some functions from the
  actions  package.  There  is  a  whole  class  of  functions that we did not
  mention, namely those that take a single element instead of a whole group as
  first  argument,  e.g., [2XCycle[102X ([14XReference: Cycle[114X) and [2XPermutation[102X ([14XReference:
  Permutation[114X).  These  are  fully  described  in  Chapter  [14X'Reference:  Group
  Actions'[114X.[133X
  
  
  [1X5.4 [33X[0;0YGroup Homomorphisms by Images[133X[101X
  
  [33X[0;0YWe  have  already seen examples of group homomorphisms in the last sections,
  namely  natural  homomorphisms  and action homomorphisms. In this section we
  will  show  how  to  construct  a  group  homomorphism [22XG → H[122X by specifying a
  generating  set  for  [22XG[122X  and the images of these generators in [22XH[122X. We use the
  function  [10XGroupHomomorphismByImages(  [3XG[103X[10X,  [3XH[103X[10X,  [3Xgens[103X[10X,  [3Ximgs[103X[10X  )[110X where [3Xgens[103X is a
  generating  set  for  [3XG[103X  and  [3Ximgs[103X is a list whose [22Xi[122Xth entry is the image of
  [22X[3Xgens[103X[i][122X under the homomorphism.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs4 := Group((1,2,3,4),(1,2));; s3 := Group((1,2,3),(1,2));;[127X[104X
    [4X[25Xgap>[125X [27Xhom := GroupHomomorphismByImages( s4, s3,[127X[104X
    [4X[25X>[125X [27X          GeneratorsOfGroup(s4), [(1,2),(2,3)] );[127X[104X
    [4X[28X[ (1,2,3,4), (1,2) ] -> [ (1,2), (2,3) ][128X[104X
    [4X[25Xgap>[125X [27XKernel( hom );[127X[104X
    [4X[28XGroup([ (1,4)(2,3), (1,3)(2,4) ])[128X[104X
    [4X[25Xgap>[125X [27XImage( hom, (1,2,3) );[127X[104X
    [4X[28X(1,2,3)[128X[104X
    [4X[25Xgap>[125X [27XSize( Image( hom, DerivedSubgroup(s4) ) );[127X[104X
    [4X[28X3[128X[104X
  [4X[32X[104X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XPreImage( hom, (1,2,3) );[127X[104X
    [4X[28XError, <map> must be an inj. and surj. mapping called from[128X[104X
    [4X[28X<function "PreImage">( <arguments> )[128X[104X
    [4X[28X called from read-eval loop at line 4 of *stdin*[128X[104X
    [4X[28Xyou can 'quit;' to quit to outer loop, or[128X[104X
    [4X[28Xyou can 'return;' to continue[128X[104X
    [4X[26Xbrk>[126X [27Xquit;[127X[104X
  [4X[32X[104X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XPreImagesRepresentative( hom, (1,2,3) );[127X[104X
    [4X[28X(1,4,2)[128X[104X
    [4X[25Xgap>[125X [27XPreImage( hom, TrivialSubgroup(s3) );  # the kernel[127X[104X
    [4X[28XGroup([ (1,4)(2,3), (1,3)(2,4) ])[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThis  homomorphism  from  [22XS_4[122X  onto  [22XS_3[122X is well known from elementary group
  theory.  Images  of  elements and subgroups under [10Xhom[110X can be calculated with
  the  function  [2XImage[102X  ([14XReference:  Image[114X).  But since the mapping [10Xhom[110X is not
  bijective,  we  cannot  use  the function [2XPreImage[102X ([14XReference: PreImage[114X) for
  preimages of elements (they can have several preimages). Instead, we have to
  use   [2XPreImagesRepresentative[102X  ([14XReference:  PreImagesRepresentative[114X),  which
  returns  one  preimage if at least one exists (and would return [9Xfail[109X if none
  exists,  which  cannot  occur for our surjective [10Xhom[110X). On the other hand, we
  can  use  [2XPreImage[102X  ([14XReference:  PreImage[114X)  for the preimage of a set (which
  always exists, even if it is empty).[133X
  
  [33X[0;0YSuppose  we  mistype  the  input  when trying to construct a homomorphism as
  below.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XGroupHomomorphismByImages( s4, s3,[127X[104X
    [4X[25X>[125X [27X       GeneratorsOfGroup(s4), [(1,2,3),(2,3)] );[127X[104X
    [4X[28Xfail[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThere is no such homomorphism, hence [9Xfail[109X is returned. But note that because
  of  this,  [2XGroupHomomorphismByImages[102X  ([14XReference: GroupHomomorphismByImages[114X)
  must  do  some checks, and this was also done for the mapping [10Xhom[110X above. One
  can  avoid  these checks if one is sure that the desired homomorphism really
  exists.  For  that,  the  function  [2XGroupHomomorphismByImagesNC[102X  ([14XReference:
  GroupHomomorphismByImagesNC[114X) can be used; the [10XNC[110X stands for [21Xno check[121X.[133X
  
  [33X[0;0YBut  note  that  horrible  things  can happen if [2XGroupHomomorphismByImagesNC[102X
  ([14XReference:  GroupHomomorphismByImagesNC[114X)  is  used  when the input does not
  describe a homomorphism.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xhom2 := GroupHomomorphismByImagesNC( s4, s3,[127X[104X
    [4X[25X>[125X [27X           GeneratorsOfGroup(s4), [(1,2,3),(2,3)] );[127X[104X
    [4X[28X[ (1,2,3,4), (1,2) ] -> [ (1,2,3), (2,3) ][128X[104X
    [4X[25Xgap>[125X [27XSize( Kernel(hom2) );[127X[104X
    [4X[28X24[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  other  words,  [5XGAP[105X  claims  that  the  kernel  is  the full [10Xs4[110X, yet [10Xhom2[110X
  obviously  has  some non-trivial images! Clearly there is no such thing as a
  homomorphism  which  maps  an  element  of order 4 (namely, (1,2,3,4)) to an
  element   of   order 3  (namely,  (1,2,3)).  [13XBut  if  you  use  the  command
  [2XGroupHomomorphismByImagesNC[102X  ([14XReference:  GroupHomomorphismByImagesNC[114X),  [5XGAP[105X
  trusts you.[113X[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsGroupHomomorphism( hom2 );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YAnd then it produces serious nonsense if the thing is not a homomorphism, as
  seen above![133X
  
  [33X[0;0YBesides    the    safe    command    [2XGroupHomomorphismByImages[102X   ([14XReference:
  GroupHomomorphismByImages[114X), which returns [9Xfail[109X if the requested homomorphism
  does   not   exist,   there   is  the  function  [2XGroupGeneralMappingByImages[102X
  ([14XReference:  GroupGeneralMappingByImages[114X),  which  returns a general mapping
  (that  is,  a  possibly  multi-valued  mapping)  that  can  be  tested  with
  [2XIsGroupHomomorphism[102X ([14XReference: IsGroupHomomorphism[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xhom2 := GroupGeneralMappingByImages( s4, s3,[127X[104X
    [4X[25X>[125X [27X           GeneratorsOfGroup(s4), [(1,2,3),(2,3)] );;[127X[104X
    [4X[25Xgap>[125X [27XIsGroupHomomorphism( hom2 );[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [33X[0;0YBut  the  possibility  of  testing  for being a homomorphism is not the only
  reason  why  [5XGAP[105X  offers  [13Xgroup  general mappings[113X. Another (more important?)
  reason  is  that their existence allows [21Xreversal of arrows[121X in a homomorphism
  such  as  our  original  [10Xhom[110X.  By this we mean the [2XGroupHomomorphismByImages[102X
  ([14XReference:  GroupHomomorphismByImages[114X) with left and right sides exchanged,
  in   which  case  it  is  of  course  merely  a  [2XGroupGeneralMappingByImages[102X
  ([14XReference: GroupGeneralMappingByImages[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xrev := GroupGeneralMappingByImages( s3, s4,[127X[104X
    [4X[25X>[125X [27X          [(1,2),(2,3)], GeneratorsOfGroup(s4) );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YNow  [22Xhom[122X  maps [22Xa[122X to [22Xb[122X if and only if [22Xrev[122X maps [22Xb[122X to [22Xa[122X, for [22Xa ∈[122X [10Xs4[110X and [22Xb ∈[122X [10Xs3[110X.
  Since  every  such  [22Xb[122X  has  four preimages under [10Xhom[110X, it now has four images
  under  [10Xrev[110X. Just as the four preimages form a coset of the kernel [22XV_4 ≤[122X[10Xs4[110X of
  [10Xhom[110X,  they  also  form  a coset of the [13Xcokernel[113X [22XV_4 ≤[122X[10Xs4[110X of [10Xrev[110X. The cokernel
  itself  is  the  set of all images of [10XOne( s3 )[110X. (It is a normal subgroup in
  the  group  of  all  images  under  [10Xrev[110X.) The operation [2XOne[102X ([14XReference: One[114X)
  returns  the  identity  element  of  a  group.  And this is why [5XGAP[105X wants to
  perform   such  a  reversal  of  arrows:  it  calculates  the  kernel  of  a
  homomorphism  like [10Xhom[110X as the cokernel of the reversed group general mapping
  (here [10Xrev[110X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XCoKernel( rev );[127X[104X
    [4X[28XGroup([ (1,4)(2,3), (1,3)(2,4) ])[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  reason  why  [10Xrev[110X  is not a homomorphism is that it is not single-valued
  (because [10Xhom[110X was not injective). But there is another critical condition: If
  we  reverse  the  arrows of a non-surjective homomorphism, we obtain a group
  general  mapping  which  is not defined everywhere, i.e., which is not total
  (although   it  will  be  single-valued  if  the  original  homomorphism  is
  injective). [5XGAP[105X requires that a group homomorphism be both single-valued and
  total,  so  you  will  get  [9Xfail[109X if you say [10XGroupHomomorphismByImages( [3XG[103X[10X, [3XH[103X[10X,
  [3Xgens[103X[10X,  [3Ximgs[103X[10X  )[110X  where  [3Xgens[103X  does  not generate [3XG[103X (even if this would give a
  decent  homomorphism  on  the  subgroup  generated  by  [3Xgens[103X).  For  a  full
  description, see Chapter [14X'Reference: Group Homomorphisms'[114X.[133X
  
  [33X[0;0YThe  last  example  of  this  section  shows  that  the notion of kernel and
  cokernel  naturally  extends  even  to  the  case where neither [10Xhom2[110X nor its
  inverse general mapping (with arrows reversed) is a homomorphism.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XCoKernel( hom2 );  Kernel( hom2 );[127X[104X
    [4X[28XGroup([ (2,3), (1,3) ])[128X[104X
    [4X[28XGroup([ (3,4), (2,3,4), (1,2,4) ])[128X[104X
    [4X[25Xgap>[125X [27XIsGroupHomomorphism( InverseGeneralMapping( hom2 ) );[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [33X[0;0Y[13XSummary.[113X  In  this  section  we have constructed homomorphisms by specifying
  images for a set of generators. We have seen that by reversing the direction
  of   the  mapping,  we  get  group  general  mappings,  which  need  not  be
  single-valued  (unless  the  mapping  was  injective)  nor total (unless the
  mapping was surjective).[133X
  
  
  [1X5.5 [33X[0;0YNice Monomorphisms[133X[101X
  
  [33X[0;0YFor  some  types  of  groups,  the best method to calculate in an isomorphic
  group  in  a  [21Xbetter[121X  representation  (say, a permutation group). We call an
  injective  homomorphism,  that  will  give  such  an isomorphic image a [21Xnice
  monomorphism[121X.[133X
  
  [33X[0;0YFor  example  in  the  case  of a matrix group we can take the action on the
  underlying   vector   space   (or  a  suitable  subset)  to  obtain  such  a
  monomorphism:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgrp:=GL(2,3);;[127X[104X
    [4X[25Xgap>[125X [27Xdom:=GF(3)^2;;[127X[104X
    [4X[25Xgap>[125X [27Xhom := ActionHomomorphism( grp, dom );; IsInjective( hom );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xp := Image( hom,grp );[127X[104X
    [4X[28XGroup([ (4,7)(5,8)(6,9), (2,7,6)(3,4,8) ])[128X[104X
  [4X[32X[104X
  
  [33X[0;0YTo demonstrate the technique of nice monomorphisms, we compute the conjugacy
  classes  of  the  permutation group and lift them back into the matrix group
  with  the monomorphism [10Xhom[110X. Lifting back a conjugacy class means finding the
  preimage  of the representative and of the centralizer; the latter is called
  [2XStabilizerOfExternalSet[102X ([14XReference: StabilizerOfExternalSet[114X) in [5XGAP[105X (because
  conjugacy  classes are represented as external sets, see Section [14X'Reference:
  Conjugacy Classes'[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xpcls := ConjugacyClasses( p );; gcls := [  ];;[127X[104X
    [4X[25Xgap>[125X [27Xfor pc  in pcls  do[127X[104X
    [4X[25X>[125X [27X     gc:=ConjugacyClass(grp,[127X[104X
    [4X[25X>[125X [27X                  PreImagesRepresentative(hom,Representative(pc)));[127X[104X
    [4X[25X>[125X [27X     SetStabilizerOfExternalSet(gc,PreImage(hom,[127X[104X
    [4X[25X>[125X [27X                                     StabilizerOfExternalSet(pc)));[127X[104X
    [4X[25X>[125X [27X     Add( gcls, gc );[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[25Xgap>[125X [27XList( gcls, Size );[127X[104X
    [4X[28X[ 1, 8, 12, 1, 8, 6, 6, 6 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YAll  the  steps we have made above are automatically performed by [5XGAP[105X if you
  simply ask for [10XConjugacyClasses( grp )[110X, provided that [5XGAP[105X already knows that
  [10Xgrp[110X  is  finite (e.g., because you asked [10XIsFinite( grp )[110X before). The reason
  for  this  is  that  a  finite  matrix  group  like [10Xgrp[110X is [21Xhandled by a nice
  monomorphism[121X.  For  such  groups,  [5XGAP[105X  uses  the  command  [2XNiceMonomorphism[102X
  ([14XReference:  NiceMonomorphism[114X)  to construct a monomorphism (such as the [10Xhom[110X
  in the previous example) and then proceeds as we have done above.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgrp:=GL(2,3);;[127X[104X
    [4X[25Xgap>[125X [27XIsHandledByNiceMonomorphism( grp );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xhom := NiceMonomorphism( grp );[127X[104X
    [4X[28X<action isomorphism>[128X[104X
    [4X[25Xgap>[125X [27Xp :=Image(hom,grp);[127X[104X
    [4X[28XGroup([ (4,7)(5,8)(6,9), (2,7,6)(3,4,8) ])[128X[104X
    [4X[25Xgap>[125X [27Xcc := ConjugacyClasses( grp );; ForAll(cc, x-> x in gcls); [127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XForAll(gcls, x->x in cc); # cc and gcls might be ordered differently[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNote  that  a  nice monomorphism might be defined on a larger group than [10Xgrp[110X
  –so we have to use [10XImage( hom, grp )[110X and not only [10XImage( hom )[110X.[133X
  
  [33X[0;0YNice  monomorphisms  are not only used for matrix groups, but also for other
  kinds  of  groups  in  which  one cannot calculate easily enough. As another
  example,  let us show that the automorphism group of the quaternion group of
  order 8  is  isomorphic  to the symmetric group of degree 4 by examining the
  [21Xnice object[121X associated with that automorphism group.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xp:=Group((1,7,6,8)(2,5,3,4), (1,2,6,3)(4,8,5,7));;[127X[104X
    [4X[25Xgap>[125X [27Xaut := AutomorphismGroup( p );; NiceMonomorphism(aut);;[127X[104X
    [4X[25Xgap>[125X [27Xniceaut := NiceObject( aut );[127X[104X
    [4X[28XGroup([ (1,4,2,3), (1,5,4)(2,6,3), (1,2)(3,4), (3,4)(5,6) ])[128X[104X
    [4X[25Xgap>[125X [27XIsomorphismGroups( niceaut, SymmetricGroup( 4 ) );[127X[104X
    [4X[28X[ (1,4,2,3), (1,5,4)(2,6,3), (1,2)(3,4), (3,4)(5,6) ] -> [128X[104X
    [4X[28X[ (1,4,3,2), (1,3,2), (1,3)(2,4), (1,2)(3,4) ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  range  of  a  nice  monomorphism  is in most cases a permutation group,
  because  nice  monomorphisms are mostly action homomorphisms. In some cases,
  like  in  our  last example, the group is solvable and you might prefer a pc
  group  as  nice  object.  You  cannot  change  the  nice monomorphism of the
  automorphism   group   (because   it   is   the   value   of  the  attribute
  [2XNiceMonomorphism[102X ([14XReference: NiceMonomorphism[114X)), but you can compose it with
  an  isomorphism  from  the  permutation  group  to a pc group to obtain your
  personal  nicer monomorphism. If you reconstruct the automorphism group, you
  can  even  prescribe  it  this  nicer  monomorphism  as its [2XNiceMonomorphism[102X
  ([14XReference:  NiceMonomorphism[114X),  because  a newly-constructed group will not
  yet have a [2XNiceMonomorphism[102X ([14XReference: NiceMonomorphism[114X) set.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xnicer := NiceMonomorphism(aut) * IsomorphismPcGroup(niceaut);;[127X[104X
    [4X[25Xgap>[125X [27Xaut2 := GroupByGenerators( GeneratorsOfGroup( aut ) );;[127X[104X
    [4X[25Xgap>[125X [27XSetIsHandledByNiceMonomorphism( aut2, true );[127X[104X
    [4X[25Xgap>[125X [27XSetNiceMonomorphism( aut2, nicer );[127X[104X
    [4X[25Xgap>[125X [27XNiceObject( aut2 );  # a pc group[127X[104X
    [4X[28XGroup([ f1*f2, f2^2*f3, f4, f3 ])[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe star [10X*[110X denotes composition of mappings from the left to the right, as we
  have seen in Section [14X5.2[114X above. Reconstructing the automorphism group may of
  course  result  in  the  loss of other information [5XGAP[105X had already gathered,
  besides the (not-so-)nice monomorphism.[133X
  
  [33X[0;0Y[13XSummary.[113X  In  this  section  we  have seen how calculations in groups can be
  carried  out  in  isomorphic  images  in nicer groups. We have seen that [5XGAP[105X
  pursues  this  technique  automatically for certain classes of groups, e.g.,
  for matrix groups that are known to be finite.[133X
  
  
  [1X5.6 [33X[0;0YFurther Information about Groups and Homomorphisms[133X[101X
  
  [33X[0;0YGroups  and  the  functions  for  groups  are treated in Chapter [14X'Reference:
  Groups'[114X.  There  are  several  chapters  dealing  with  groups  in  specific
  representations,  for  example  Chapter [14X'Reference:  Permutation  Groups'[114X on
  permutation  groups, [14X'Reference: Polycyclic Groups'[114X on polycyclic (including
  finite  solvable)  groups,  [14X'Reference:  Matrix Groups'[114X on matrix groups and
  [14X'Reference:   Finitely  Presented  Groups'[114X  on  finitely  presented  groups.
  Chapter [14X'Reference:   Group   Actions'[114X   deals  with  group  actions.  Group
  homomorphisms are the subject of Chapter [14X'Reference: Group Homomorphisms'[114X.[133X
  
