5 Groups and Homomorphisms

Sections

1. Permutation groups
2. Actions of Groups
3. Subgroups as Stabilizers
4. Group Homomorphisms by Images
5. Nice Monomorphisms
6. Further Information about Groups and Homomorphisms

In 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 Group, Size or SylowSubgroup, however, are the same for all kinds of groups, although the algorithms which compute the information of course will be different in most cases.

5.1 Permutation groups

Permutation 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:

gap> s8 := Group( (1,2), (1,2,3,4,5,6,7,8) );
Group([ (1,2), (1,2,3,4,5,6,7,8) ])

We formed the group generated by the permutations (1,2) and (1,2,3,4,5,6,7,8), which is well known to be the symmetric group on eight points, and assigned it to the identifier s8. Now the group S₈ contains the alternating group on eight points which can be described in several ways, e.g., as the group of all even permutations in s8, or as its derived subgroup.

gap> a8 := DerivedSubgroup( s8 );
Group([ (1,2,3), (2,3,4), (2,4)(3,5), (2,6,4), (2,4)(5,7), (2,8,6,4)(3,5) ])
gap> Size( a8 ); IsAbelian( a8 ); IsPerfect( a8 );
20160
false
true

Once information about a group like s8 or a8 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, a8 stores its order and that it is nonabelian and perfect, and s8 stores its derived subgroup a8. Had we computed a8 as CommutatorSubgroup( s8, s8 ), however, it would not have been stored, because it would then have been computed as a function of two arguments, and hence one could not attribute it to just one of them. (Of course the function CommutatorSubgroup can compute the commutator subgroup of two arbitrary subgroups.) The situation is a bit different for Sylow p-subgroups: The function SylowSubgroup also requires two arguments, namely a group and a prime p, but the result is stored in the group --- namely together with the prime p in a list called ComputedSylowSubgroups, but we won't dwell on the details here.

gap> syl2 := SylowSubgroup( a8, 2 );; Size( syl2 );
64
gap> Normalizer( a8, syl2 ) = syl2;
true
gap> cent := Centralizer( a8, Centre( syl2 ) );; Size( cent );
192
gap> DerivedSeries( cent );; List( last, Size );
[ 192, 96, 32, 2, 1 ]

We 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.

With the next examples, we want to calculate a subgroup of a8, 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 2³.

gap> elab := Group( (1,2)(3,4)(5,6)(7,8), (1,3)(2,4)(5,7)(6,8),
>                   (1,5)(2,6)(3,7)(4,8) );;
gap> Size( elab );
8
gap> IsElementaryAbelian( elab );
true

As usual, GAP 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 already several around. Note that although it is no problem for us to specify a particular group to GAP, by using well-chosen identifiers such as a8 and elab, it is impossible for GAP 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 have to use the function SetName. We do this with the name 2^3 here which reflects the mathematical properties of the group. From now on, GAP will use this name when printing the group for us, but we still cannot use this name to specify the group to GAP, 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.

gap> SetName( elab, "2^3" ); elab;
2^3
gap> norm := Normalizer( a8, elab );; Size( norm );
1344

Now that we have the subgroup norm of order 1344 and its subgroup elab, we want to look at its factor group. But since we also want to find preimages of factor group elements in norm, we really want to look at the natural homomorphism defined on norm with kernel elab and whose image is the factor group.

gap> hom := NaturalHomomorphismByNormalSubgroup( norm, elab );
<action epimorphism>
gap> f := Image( hom );
Group([ (), (), (), (4,5)(6,7), (4,6)(5,7), (2,3)(6,7), (2,4)(3,5), 
  (1,2)(5,6) ])
gap> Size( f );
168

The factor group is again represented as a permutation group. However, the action domain of this factor group has nothing to do with the action domain of norm. (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 hom is a coset modulo elab. We use the function PreImages here because hom is not a bijection, so an element of the range can have several preimages or none at all.

gap> ker:= Kernel( hom );
2^3
gap> x := (1,8,3,5,7,6,2);; Image( hom, x );
(1,7,5,6,2,3,4)
gap> coset := PreImages( hom, last );
RightCoset(2^3,(2,8,6,7,3,4,5))

Note that GAP 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.

gap> rep:= Representative( coset );
(2,8,6,7,3,4,5)
gap> x * rep^-1 in ker;
true

The factor group f is a simple group, i.e., it has no non-trivial normal subgroups. GAP 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.)

gap> IsSimple( f ); IsomorphismTypeInfoFiniteSimpleGroup( f );
true
rec( series := "L", parameter := [ 2, 7 ], 
  name := "A(1,7) = L(2,7) ~ B(1,7) = O(3,7) ~ C(1,7) = S(2,7) ~ 2A(1,7) = U(2\
,7) ~ A(2,2) = L(3,2)" )
gap> SetName( f, "L_3(2)" );

We give f the name L_3(2) because the last part of the name string reveals that it is isomorphic to the simple linear group L₃(2). This group, however, also has a lot of other names. Names that are connected with a = sign are different names for the same matrix group, e.g., A(2,2) is the Lie type notation for the classical notation L(3,2). Other pairs of names are connected via  , these then specify other classical groups that are isomorphic to that linear group (e.g., the symplectic group S(2,7), whose Lie type notation would be C(1,7)).

The group norm acts on the eight elements of its normal subgroup elab by conjugation, yielding a representation of L₃(2) in s8 which leaves one point fixed (namely point 1). The image of this representation can be computed with the function Action; it is even contained in the group norm and we can show that norm is indeed a split extension of the elementary abelian group 2³ with this image of L₃(2).

gap> op := Action( norm, elab );
Group([ (), (), (), (5,6)(7,8), (5,7)(6,8), (3,4)(7,8), (3,5)(4,6), 
  (2,3)(6,7) ])
gap> IsSubgroup( a8, op ); IsSubgroup( norm, op );
true
true
gap> IsTrivial( Intersection( elab, op ) );
true
gap> SetName( norm, "2^3:L_3(2)" );

By the way, you should not try the operator < instead of the function IsSubgroup. Something like

gap> elab < a8;
false

will not cause an error, but the result does not signify anything about the inclusion of one group in another; < tests which of the two groups is less in some total order. On the other hand, the equality operator = in fact does test the equality of its arguments.

5.2 Actions of Groups

In order to get another representation of a8, we consider another action, namely that on the elements of a certain conjugacy class by conjugation.

In 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.

gap> ccl := ConjugacyClasses( a8 );; Length( ccl );
14
gap> List( ccl, c -> Order( Representative( c ) ) );
[ 1, 2, 2, 3, 6, 3, 4, 4, 5, 15, 15, 6, 7, 7 ]
gap> SizeScreen([ 82, ]);;
gap> List( ccl, Size );
[ 1, 210, 105, 112, 1680, 1120, 2520, 1260, 1344, 1344, 1344, 3360, 2880, 2880 ]
gap> SizeScreen([ 80, ]);;

Note the difference between Order (which means the element order), Size (which means the size of the conjugacy class) and Length (which means the length of a list). We choose to let a8 operate on the class of length 112.

gap> class := First( ccl, c -> Size(c) = 112 );;
gap> op := Action( a8, AsList( class ) );;

We use AsList here to convert the conjugacy class into a list of its elements whereas we wrote Action( norm, elab ) directly in the previous section. The reason is that the elementary abelian group elab can be quickly enumerated by GAP whereas the standard enumeration method for conjugacy classes is slower than just explicit calculation of the elements. However, GAP is reluctant to construct explicit element lists, because for really large groups this direct method is infeasible.

Note also the function First, used to find the first element in a list which passes some test. See First in the reference manual for more details.

We now have a permutation representation op 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 Blocks.

gap> IsPrimitive( op, [ 1 .. 112 ] );
false
gap> blocks := Blocks( op, [ 1 .. 112 ] );;

Note that we must specify the domain of the action. You might think that the functions IsPrimitive and Blocks could use [1..112] as default domain if no domain was given. But this is not so easy, for example would the default domain of Group( (2,3,4) ) be [1..4] or [2..4]? To avoid confusion, all action functions require that you specify the domain of action. If we had specified [1..113] in the primitivity test above, point 113 would have been a fixpoint (and the action would not even have been transitive).

Now blocks 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 op act on these blocks setwise.

gap> Length( blocks[1] );  Length( blocks );
2
56
gap> op2 := Action( op, blocks, OnSets );;
gap> IsPrimitive( op2, [ 1 .. 56 ] );
true

Note that we give a third argument (the action function OnSets) to indicate that the action is not the default action on points but an action on sets of elements given as sorted lists. (Section Basic Actions of the reference manual lists all actions that are pre-defined by GAP.)

The action of op 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 are monomorphisms). We construct the composition of two homomorphisms with the * operator, reading left-to-right.

gap> ophom := ActionHomomorphism( a8, op );;
gap> ophom2 := ActionHomomorphism( op, op2 );;
gap> composition := ophom * ophom2;;
gap> stab := Stabilizer( op2, 2 );;
gap> preim := PreImages( composition, stab );
Group([ (1,4,2), (3,6,7), (3,8,5,7,6), (1,4)(7,8) ])

The normalizer of an element in the conjugacy class class 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 a8 is found by the function RepresentativeAction.

gap> sgp := Normalizer( a8, Subgroup(a8,[Representative(class)]) );;
gap> Size( sgp );
360
gap> RepresentativeAction( a8, sgp, preim );
(3,4)(7,8)

So far we have seen a few applications of the functions Action and ActionHomomorphism. But perhaps even more interesting is the fact that the natural homomorphism hom constructed above is also an action homomorphism; this is also the reason why its image is represented as a permutation group: it is the natural representation for actions. We will now look at this action homomorphism again to find out on what objects it operates. These objects form the so-called external set which is associated with every action homomorphism. We will mention external sets only superficially in this tutorial, for details see External Sets in the reference manual. For the moment, we need only know that the external set is obtained by the function UnderlyingExternalSet.

gap> t := UnderlyingExternalSet( hom );
<xset:RightTransversal(2^3:L_3(2),Group(
[ (1,5)(2,6)(3,7)(4,8), (1,3)(2,4)(5,7)(6,8), (1,2)(3,4)(5,6)(7,8), 
  (5,6)(7,8), (5,7)(6,8), (3,4)(7,8), (3,5)(4,6) ]))>

For the natural homomorphism hom the external set is a right transversal of a subgroup U in norm, and action on the right transversal really means action on the cosets of the subgroup U. When executing the function call NaturalHomomorphismByNormalSubgroup( norm, elab ), GAP has chosen a subgroup U for which the kernel of this action (i.e., the core of U in norm) is the desired normal subgroup elab. For the purpose of operating on the cosets, the right transversal t contains one representative from each coset of U. Regarded this way, a transversal is simply a list of group elements, and you can make GAP produce this list by AsList(t). (Try it.)

The image of such a representative from AsList(t) under right multiplication with an element from norm will in general not be in AsList(t), because it will not be among the chosen representatives again. Hence right multiplication is not an action on AsList(t). However, GAP uses a special trick to be discussed below to make this a well-defined action on the cosets represented by the elements of AsList(t). For now, it is important to know that the external set t is more than just the right transversal on which the group norm operates. Altogether three things are necessary to specify an action: a group G, a set D, and a function opr :D ×G → D . We can access these ingredients with the following functions:

gap> ActingDomain(t);  # the group
2^3:L_3(2)
gap> Enumerator(t);
RightTransversal(2^3:L_3(2),Group(
[ (1,5)(2,6)(3,7)(4,8), (1,3)(2,4)(5,7)(6,8), (1,2)(3,4)(5,6)(7,8), 
  (5,6)(7,8), (5,7)(6,8), (3,4)(7,8), (3,5)(4,6) ]))
gap> FunctionAction(t);
function( pnt, elm ) ... end
gap> NameFunction( last );
"OnRight"

The function which is named "OnRight" is also assigned to the identifier OnRight, and it means multiplication from the right; this is the usual way to operate on a right transversal. OnRight( d, g ) is defined as d * g.

Observe that the external set t and its Enumerator are printed the same way, but be aware that an external set also comprises the acting domain and the action function. The Enumerator itself, i.e., the right transversal, in turn comprises knowledge about the group norm and the subgroup U, and this is what allows the special trick promised above. As far as Position is concerned, the Enumerator behaves as an (immutable) list and you can ask for the position of an element in it.

gap> elm := (1,4)(2,7)(3,6)(5,8);;
gap> Position( Enumerator(t), elm );
fail
gap> PositionCanonical( Enumerator(t), elm );
5

The result fail means that the element was not found at all in the list: it is not among the chosen representatives. The difference between the functions Position and PositionCanonical is that the first simply looks whether elm is contained among the representatives which together form the right transversal t, whereas the second really looks for the position of the coset described by the representative elm. In other words, it first replaces elm by a canonical representative of the same coset (which must be contained in Enumerator(t)) and then looks for its position, hence the name. The function ActionHomomorphism (and its relatives) always use PositionCanonical when they calculate the images of the generators of the source group (here, norm) under the homomorphism (here, hom). Therefore they can give a well-defined action on an enumerator, even if the action would not be well-defined on AsList( enumerator ).

The image of the natural homomorphism is the permutation group f that results from the action of norm on the right transversal. It can be calculated by either of the following commands. The second of them shows that the external set t contains all information that is necessary for Action to do its work.

gap> Action( norm, Enumerator(t), OnRight ) = f;
true
gap> Action( t ) = f;
true

We have specified the action function OnRight in this example, but we have seen examples like Action( norm, elab ) earlier where this third argument was not given. If an action function is omitted, GAP always assumes OnPoints which is defined as OnPoints( d, g ) = d ^ g. This ``caret'' operator denotes conjugation in a group if both arguments d and g 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).

5.3 Subgroups as Stabilizers

Action functions can also be used without constructing external sets. We will try to find several subgroups in a8 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.

gap> u8 := Stabilizer( a8, 1 );
Group([ (2,3,4), (2,4)(3,5), (2,6,4), (2,4)(5,7), (2,8,6,4)(3,5) ])
gap> Index( a8, u8 );
8
gap> Orbit( a8, 1 ); Length( last );
[ 1, 3, 2, 4, 5, 6, 7, 8 ]
8

This 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 Tuples we first generate a list of all pairs.

gap> pairs := Tuples( [1..8], 2 );;

Now we would like to have a8 operate on this domain. But we cannot use the default action OnPoints because list ^ perm is not defined. So we must tell the functions from the actions package how the group elements operate on the elements of the domain. In our example we can do this by simply passing OnPairs as an optional last argument. All functions from the actions package accept such an optional argument that describes the action. One example is IsTransitive.

gap> IsTransitive( a8, pairs, OnPairs );
false

The action is of course not transitive, since the pairs [ 1, 1 ] and [ 1, 2 ] cannot lie in the same orbit. So we want to find out what the orbits are. The function Orbits does that for us. It returns a list of all the orbits. We look at the orbit lengths and representatives for the orbits.

gap> orbs := Orbits( a8, pairs, OnPairs );; Length( orbs );
2
gap> List( orbs, Length );
[ 8, 56 ]
gap> List( orbs, o -> o[1] );
[ [ 1, 1 ], [ 1, 2 ] ]

The action of a8 on the first orbit (this is the one containing [1,1], try [1,1] in orbs[1]) is of course equivalent to the original action, so we ignore it and work with the second orbit.

gap> u56 := Stabilizer( a8, orbs[2][1], OnPairs );; Index( a8, u56 );
56

So 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 [1..56] instead of the list of pairs. The function ActionHomomorphism does what we need. It creates a homomorphism defined on a8 whose image is a new group that operates on [1..56] in the same way that a8 operates on the second orbit.

gap> h56 := ActionHomomorphism( a8, orbs[2], OnPairs );;
gap> a8_56 := Image( h56 );;

We would now like to know if the subgroup u56 of index 56 that we found is maximal or not. As we have used already in Section Actions of groups, a subgroup is maximal if and only if the action on the cosets of this subgroup is primitive.

gap> IsPrimitive( a8_56, [1..56] );
false

Remember that we can leave out the function if we mean OnPoints but that we have to specify the action domain for all action functions.

We see that a8_56 is not primitive. This means of course that the action of a8 on orb[2] is not primitive, because those two actions are equivalent. So the stabilizer u56 is not maximal. Let us try to find its supergroups. We use the function Blocks to find a block system. The (optional) third argument in the following example tells Blocks that we want a block system where 1 and 14 lie in one block.

gap> blocks := Blocks( a8_56, [1..56], [1,14] );
[ [ 1, 3, 4, 5, 6, 14, 31 ], [ 2, 13, 15, 16, 17, 23, 24 ], 
  [ 7, 8, 22, 34, 37, 47, 49 ], [ 9, 11, 18, 20, 35, 38, 48 ], 
  [ 10, 25, 26, 27, 32, 39, 50 ], [ 12, 28, 29, 30, 33, 36, 40 ], 
  [ 19, 21, 42, 43, 45, 46, 55 ], [ 41, 44, 51, 52, 53, 54, 56 ] ]

The result is a list of sets, such that a8_56 operates 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 OnSets as third argument.

gap> u8_56 := Stabilizer( a8_56, blocks[1], OnSets );;
gap> Index( a8_56, u8_56 );
8
gap> u8b := PreImages( h56, u8_56 );; Index( a8, u8b );
8
gap> IsConjugate( a8, u8, u8b );
true

So we have found a supergroup of u56 that is conjugate in a8 to u8. This is not surprising, since u8 is a point stabilizer, and u56 is a two point stabilizer in the natural action of a8 on eight points.

Here is a warning: If you specify OnSets as third argument to a function like Stabilizer, 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 blocks[1] came from the function Blocks, which returns a list of sets, so everything was OK.

Actually there is a third block system of a8_56 that gives rise to a third subgroup.

gap> blocks := Blocks( a8_56, [1..56], [1,13] );;
gap> u28_56 := Stabilizer( a8_56, [1,13], OnSets );;
gap> u28 := PreImages( h56, u28_56 );;
gap> Index( a8, u28 );
28

We know that the subgroup u28 of index 28 is maximal, because we know that a8 has no subgroups of index 2, 4, or 7. However we can also quickly verify this by checking that a8_56 operates primitively on the 28 blocks.

gap> IsPrimitive( a8_56, blocks, OnSets );
true

Stabilizer is not only applicable to groups like a8 but also to their subgroups like u56. So another method to find a new subgroup is to compute the stabilizer of another point in u56. Note that u56 already leaves 1 and 2 fixed.

gap> u336 := Stabilizer( u56, 3 );;
gap> Index( a8, u336 );
336

Other functions are also applicable to subgroups. In the following we show that u336 operates regularly on the 60 triples of [4..8] which contain no element twice. We constuct the list of these 60 triples with the function Orbit (using OnTuples as the natural generalization of OnPairs) and then pass it as action domain to the function IsRegular. The positive result of the regularity test means that this action is equivalent to the actions of u336 on its 60 elements from the right.

gap> IsRegular( u336, Orbit( u336, [4,5,6], OnTuples ), OnTuples );
true

Just as we did in the case of the action on the pairs above, we now construct a new permutation group that operates on [1..336] in the same way that a8 operates on the cosets of u336. But this time we let a8 operate on a right transversal, just like norm did in the natural homomorphism above.

gap> t := RightTransversal( a8, u336 );;
gap> a8_336 := Action( a8, t, OnRight );;

To find subgroups above u336 we again look for nontrivial block systems.

gap> blocks := Blocks( a8_336, [1..336] );; blocks[1];
[ 1, 43, 85 ]

We see that the union of u336 with its 43rd and its 85th coset is a subgroup in a8_336, its index is 112. We can obtain it as the closure of u336 with a representative of the 43rd coset, which can be found as the 43rd element of the transversal t. Note that in the representation a8_336 on 336 points, this subgroup corresponds to the stabilizer of the block [ 1, 43, 85 ].

gap> u112 := ClosureGroup( u336, t[43] );;
gap> Index( a8, u112 );
112

Above this subgroup of index 112 lies a subgroup of index 56, which is not conjugate to u56. In fact, unlike u56 it is maximal. We obtain this subgroup in the same way that we obtained u112, this time forcing two points, namely 7 and 43 into the first block.

gap> blocks := Blocks( a8_336, [1..336], [1,7,43] );;
gap> Length( blocks );
56
gap> u56b := ClosureGroup( u112, t[7] );; Index( a8, u56b );
56
gap> IsPrimitive( a8_336, blocks, OnSets );
true

We already mentioned in Section Actions of groups that there is another standard action of permutations, namely the conjugation. E.g., since no other action is specified in the following example, OrbitLength simply operates via OnPoints, and because perm 1 ^ perm 2 is defined as the conjugation of perm₂ on perm₁, in fact we compute the length of the conjugacy class of (1,2)(3,4)(5,6)(7,8).

gap> OrbitLength( a8, (1,2)(3,4)(5,6)(7,8) );
105
gap> orb := Orbit( a8, (1,2)(3,4)(5,6)(7,8) );;
gap> u105 := Stabilizer( a8, (1,2)(3,4)(5,6)(7,8) );; Index( a8, u105 );
105

Note that although the length of a conjugacy class of any element elm in any finite group G can be computed as OrbitLength( G, elm ), the command Size( ConjugacyClass( G, elm ) ) is probably more efficient.

gap> Size( ConjugacyClass( a8, (1,2)(3,4)(5,6)(7,8) ) );
105

Of course the stabilizer u105 is in fact the centralizer of the element (1,2)(3,4)(5,6)(7,8). Stabilizer notices that and computes the stabilizer using the centralizer algorithm for permutation groups. In the usual way we now look for the subgroups above u105.

gap> blocks := Blocks( a8, orb );; Length( blocks );
15
gap> blocks[1];
[ (1,2)(3,4)(5,6)(7,8), (1,3)(2,4)(5,7)(6,8), (1,4)(2,3)(5,8)(6,7), 
  (1,5)(2,6)(3,7)(4,8), (1,6)(2,5)(3,8)(4,7), (1,7)(2,8)(3,5)(4,6), 
  (1,8)(2,7)(3,6)(4,5) ]

To find the subgroup of index 15 we again use closure. Now we must be a little bit careful to avoid confusion. u105 is the stabilizer of (1,2)(3,4)(5,6)(7,8). We know that there is a correspondence between the points of the orbit and the cosets of u105. The point (1,2)(3,4)(5,6)(7,8) corresponds to u105. To get the subgroup above u105 that has index 15 in a8, we must form the closure of u105 with an element of the coset that corresponds to any other point in the first block. If we choose the point (1,3)(2,4)(5,8)(6,7), we must use an element of a8 that maps (1,2)(3,4)(5,6)(7,8) to (1,3)(2,4)(5,8)(6,7). The function RepresentativeAction 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, RepresentativeAction returns fail.

gap> rep := RepresentativeAction( a8, (1,2)(3,4)(5,6)(7,8),
>                                        (1,3)(2,4)(5,8)(6,7) );
(2,3)(6,8)
gap> u15 := ClosureGroup( u105, rep );; Index( a8, u15 );
15

u15 is of course a maximal subgroup, because a8 has no subgroups of index 3 or 5. There is in fact another class of subgroups of index 15 above u105 that we get by adding (2,3)(6,7) to u105.

gap> u15b := ClosureGroup( u105, (2,3)(6,7) );; Index( a8, u15b );
15
gap> RepresentativeAction( a8, u15, u15b );
fail

RepresentativeAction tells us that there is no element g in a8 such that u15 ^ g = u15b. Because ^ also denotes the conjugation of subgroups this tells us that u15 and u15b are not conjugate.

5.4 Group Homomorphisms by Images

We 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 G→ H by specifying a generating set for G and the images of these generators in H. We use the function GroupHomomorphismByImages( G, H, gens, imgs ) where gens is a generating set for G and imgs is a list whose ith entry is the image of gens[ i ] under the homomorphism.

gap> s4 := Group((1,2,3,4),(1,2));; s3 := Group((1,2,3),(1,2));;
gap> hom := GroupHomomorphismByImages( s4, s3,
>           GeneratorsOfGroup(s4), [(1,2),(2,3)] );
[ (1,2,3,4), (1,2) ] -> [ (1,2), (2,3) ]
gap> Kernel( hom );
Group([ (1,4)(2,3), (1,3)(2,4) ])
gap> Image( hom, (1,2,3) );
(1,2,3)
gap> Image( hom, DerivedSubgroup(s4) );
Group([ (1,3,2), (1,3,2) ])

gap> PreImage( hom, (1,2,3) );
Error, <map> must be an inj. and surj. mapping called from
<function>( <arguments> ) called from read-eval-loop
Entering break read-eval-print loop ...
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue
brk> quit;

gap> PreImagesRepresentative( hom, (1,2,3) );
(1,4,2)
gap> PreImage( hom, TrivialSubgroup(s3) );  # the kernel
Group([ (1,4)(2,3), (1,3)(2,4) ])

This homomorphism from S₄ onto S₃ is well known from elementary group theory. Images of elements and subgroups under hom can be calculated with the function Image. But since the mapping hom is not bijective, we cannot use the function PreImage for preimages of elements (they can have several preimages). Instead, we have to use PreImagesRepresentative, which returns one preimage if at least one exists (and would return fail if none exists, which cannot occur for our surjective hom.) On the other hand, we can use PreImage for the preimage of a set (which always exists, even if it is empty).

Suppose we mistype the input when trying to construct a homomorphism, as in the following example.

gap> GroupHomomorphismByImages( s4, s3,
>        GeneratorsOfGroup(s4), [(1,2,3),(2,3)] );
fail

There is no such homomorphism, hence fail is returned. But note that because of this, GroupHomomorphismByImages must do some checks, and this was also done for the mapping hom above. One can avoid these checks if one is sure that the desired homomorphism really exists. For that, the function GroupHomomorphismByImagesNC can be used; the NC stands for ``no check''.

But note that horrible things can happen if GroupHomomorphismByImagesNC is used when the input does not describe a homomorphism.

gap> hom2 := GroupHomomorphismByImagesNC( s4, s3,
>            GeneratorsOfGroup(s4), [(1,2,3),(2,3)] );
[ (1,2,3,4), (1,2) ] -> [ (1,2,3), (2,3) ]
gap> Size( Kernel(hom2) );
24

In other words, GAP claims that the kernel is the full s4, yet hom2 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)). But if you use the command GroupHomomorphismByImagesNC, GAP trusts you.

gap> IsGroupHomomorphism( hom2 );
true

And then it produces serious nonsense if the thing is not a homomorphism, as seen above!

Besides the safe command GroupHomomorphismByImages, which returns fail if the requested homomorphism does not exist, there is the function GroupGeneralMappingByImages, which returns a general mapping (that is, a possibly multi-valued mapping) that can be tested with IsGroupHomomorphism.

gap> hom2 := GroupGeneralMappingByImages( s4, s3,
>            GeneratorsOfGroup(s4), [(1,2,3),(2,3)] );;
gap> IsGroupHomomorphism( hom2 );
false

But the possibility of testing for being a homomorphism is not the only reason why GAP offers group general mappings. Another (more important?) reason is that their existence allows ``reversal of arrows'' in a homomorphism such as our original hom. By this we mean the GroupHomomorphismByImages with left and right sides exchanged, in which case it is of course merely a GroupGeneralMappingByImages.

gap> rev := GroupGeneralMappingByImages( s3, s4,
>           [(1,2),(2,3)], GeneratorsOfGroup(s4) );;

Now we have

a hom →   b ⇔ b rev →   a

for a ∈ s4 and b ∈ s3. Since every such b has 4 preimages under hom, it now has 4 images under rev. Just as the 4 preimages form a coset of the kernel V₄ ≤ s4 of hom, they also form a coset of the cokernel V₄ ≤ s4 of rev. The cokernel itself is the set of all images of One( s3 ) (it is a normal subgroup in the group of all images under rev). The operation 'One' returns the identity element of a group, see One in the reference manual. And this is why GAP wants to perform such a reversal of arrows: it calculates the kernel of a homomorphism like hom as the cokernel of the reversed group general mapping (here rev).

gap> CoKernel( rev );
Group([ (1,4)(2,3), (1,3)(2,4) ])

The reason why rev is not a homomorphism is that it is not single-valued (because hom 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). GAP requires that a group homomorphism be both single-valued and total, so you will get fail if you say GroupHomomorphismByImages( G, H, gens, imgs ) where gens does not generate G (even if this would give a decent homomorphism on the subgroup generated by gens). For a full description, see Chapter Group Homomorphisms in the reference manual.

The last example of this section shows that the notion of kernel and cokernel naturally extends even to the case where neither hom2 nor its inverse general mapping (with arrows reversed) is a homomorphism.

gap> CoKernel( hom2 );  Kernel( hom2 );
Group([ (2,3), (1,3) ])
Group([ (3,4), (2,3,4), (1,2,4) ])
gap> IsGroupHomomorphism( InverseGeneralMapping( hom2 ) );
false

5.5 Nice Monomorphisms

For some types of groups, the best method to calculate in an isomorphic group in a ``better'' representation (say, a permutation group). We call an injective homomorphism, that will give such an isomorphic image a ``nice monomorphism''.

For 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:

gap> grp:=GL(2,3);;
gap> dom:=GF(3)^2;;
gap> hom := ActionHomomorphism( grp, dom );; IsInjective( hom );
true
gap> p := Image( hom,grp );
Group([ (4,7)(5,8)(6,9), (2,7,6)(3,4,8) ])

To 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 hom. Lifting back a conjugacy class means finding the preimage of the representative and of the centralizer; the latter is called StabilizerOfExternalSet in GAP (because conjugacy classes are represented as external sets, see Section Conjugacy Classes in the reference manual).

gap> pcls := ConjugacyClasses( p );; gcls := [  ];;
gap> for pc  in pcls  do
>      gc:=ConjugacyClass(grp,PreImagesRepresentative(hom,Representative(pc)));
>      SetStabilizerOfExternalSet(gc,PreImage(hom,
>                                 StabilizerOfExternalSet(pc)));
>      Add( gcls, gc );
>    od;
gap> List( gcls, Size );
[ 1, 8, 12, 1, 8, 6, 6, 6 ]

All the steps we have made above are automatically performed by GAP if you simply ask for ConjugacyClasses( grp ), provided that GAP already knows that grp is finite (e.g., because you asked IsFinite( grp ) before). The reason for this is that a finite matrix group like grp is ``handled by a nice monomorphism''. For such groups, GAP uses the command NiceMonomorphism to construct a monomorphism (such as the hom in the previous example) and then proceeds as we have done above.

gap> grp:=GL(2,3);;
gap> IsHandledByNiceMonomorphism( grp );
true
gap> hom := NiceMonomorphism( grp );
<action isomorphism>
gap> p :=Image(hom,grp);
Group([ (4,7)(5,8)(6,9), (2,7,6)(3,4,8) ])
gap> cc := ConjugacyClasses( grp );; ForAll(cc, x-> x in gcls); 
true
gap> ForAll(gcls, x->x in cc); # cc and gcls might be ordered differently
true

Note that a nice monomorphism might be defined on a larger group than grp -- so we have to use Image(hom,grp) and not only Image(hom).

Nice 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 ``nice object'' associated with that automorphism group.

gap> p:=Group((1,7,6,8)(2,5,3,4), (1,2,6,3)(4,8,5,7));;
gap> aut := AutomorphismGroup( p );; NiceMonomorphism(aut);;
gap> niceaut := NiceObject( aut );
Group([ (1,2)(3,4), (3,4)(5,6), (1,3)(2,4)(5,6), (1,6)(2,5)(3,4) ])
gap> IsomorphismGroups( niceaut, SymmetricGroup( 4 ) );
[ (1,2)(3,4), (3,4)(5,6), (1,3)(2,4)(5,6), (1,6)(2,5)(3,4) ] -> 
[ (1,3)(2,4), (1,4)(2,3), (2,4), (1,2) ]

The 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 NiceMonomorphism), 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 NiceMonomorphism, because a newly-constructed group will not yet have a NiceMonomorphism set.

gap> nicer := NiceMonomorphism(aut) * IsomorphismPcGroup(niceaut);;
gap> aut2 := GroupByGenerators( GeneratorsOfGroup( aut ) );;
gap> SetIsHandledByNiceMonomorphism( aut2, true );
gap> SetNiceMonomorphism( aut2, nicer );
gap> NiceObject( aut2 );  # a pc group
Group([ f4, f3, f1*f2^2*f3, f1*f2*f3*f4 ])

The star * denotes composition of mappings from the left to the right, as we have seen in Section Actions of groups above. Reconstructing the automorphism group may of course result in the loss of other information GAP had already gathered, besides the (not-so-)nice monomorphism.

5.6 Further Information about Groups and Homomorphisms

Groups and the functions for groups are treated in Chapter Groups. There are several chapters dealing with groups in specific representations, for example Chapter Permutation Groups on permutation groups, Polycyclic Groups on polycyclic (including finite solvable) groups, Matrix Groups on matrix groups and Finitely Presented Groups on finitely presented groups. Chapter Group Actions deals with group actions. Group homomorphisms are the subject of Chapter Group Homomorphisms.

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GAP 4 manual
March 2006