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 (Reference: Group), Size (Reference: Size) or SylowSubgroup (Reference: 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.
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 \(S_8\) on eight points, and assigned it to the identifier s8. Now \(S_8\) 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. Once we ask GAP to verify that the group is an alternating group acting in its natural permutation representation, the system will display the group accordingly.
gap> a8 := DerivedSubgroup( s8 ); Group([ (1,2,3), (2,4,3), (2,4,5), (2,5,6,3,4), (3,7,4), (2,6) (4,7,8,5) ]) gap> Size( a8 ); IsAbelian( a8 ); IsPerfect( a8 ); 20160 false true gap> IsNaturalAlternatingGroup(a8); true gap> a8; Alt( [ 1 .. 8 ] )
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 (Reference: CommutatorSubgroup) can compute the commutator subgroup of two arbitrary subgroups.) The situation is a bit different for Sylow \(p\)-subgroups: The function SylowSubgroup (Reference: 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 that can be accessed with 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 \(2^3\).
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 are 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 can use the function SetName (Reference: Name). 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, "<group of type 2^3>" ); elab; <group of type 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 ); [ (2,3)(6,7), (3,4)(7,8), (3,5)(4,6), (5,7)(6,8), (5,6)(7,8), (1,2)(3,4)(5,6)(7,8), (1,3)(2,4)(5,7)(6,8), (1,6)(2,5)(3,8)(4,7) ] -> [ (1,2)(5,6), (2,3)(6,7), (2,4)(3,5), (4,6)(5,7), (4,5)(6,7), (), (), () ] gap> f := Image( hom ); Group([ (1,2)(5,6), (2,3)(6,7), (2,4)(3,5), (4,6)(5,7), (4,5)(6,7), (), (), () ]) gap> Size( f ); 168
The factor group is again represented as a permutation group (its last three generators are trivial, meaning that the last three generators of the preimage are in the kernel of hom). 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 (Reference: 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 ); 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> x := (1,8,3,5,7,6,2);; Image( hom, x ); (1,7,5,6,2,3,4) gap> coset := PreImages( hom, last ); RightCoset(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) ]),(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 is a non-trivial group whose only normal subgroups are its trivial subgroup and itself. 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( 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)", parameter := [ 2, 7 ], series := "L", shortname := "L3(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 \(L_3(2)\). 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 \(L_3(2)\) in s8 which leaves one point fixed (namely point 1). The image of this representation can be computed with the function Action (Reference: Action homomorphisms); it is even contained in the group norm and we can show that norm is indeed a split extension of the elementary abelian group \(2^3\) with this image of \(L_3(2)\).
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 (Reference: 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.
Summary. 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 GAP uses these names when printing the 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.
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> List( ccl, Size ); [ 1, 210, 105, 112, 1680, 1120, 2520, 1260, 1344, 1344, 1344, 3360, 2880, 2880 ]
Note the difference between Order (Reference: Order) (which means the element order), Size (Reference: Size) (which means the size of the conjugacy class) and Length (Reference: 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 ),OnPoints );;
We use AsList (Reference: 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 (Reference: First), used to find the first element in a list which passes some test.
In this example, we have specified the action function OnPoints (Reference: OnPoints) in this example, 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). It is in fact the default action and will be supplied by the system if not given. Another common action is for example OnRight (Reference: OnRight), which means right multiplication, defined as \(d\) * \(g\). (Group actions in GAP are always from the right.)
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 (Reference: 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 (Reference: IsPrimitive) and Blocks (Reference: 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 fixed point (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 (Reference: OnSets)) to indicate that the action is not the default action on points but an action on sets of elements given as strictly sorted lists. (Section Reference: Basic Actions 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 in this case 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([ (1,2,4), (6,7,8), (3,6,8), (5,8,6), (1,2)(3,8) ])
Alternatively, it is possible to create action homomorphisms immediately (without creating the action first) by giving the same set of arguments to ActionHomomorphism (Reference: ActionHomomorphism).
gap> nophom := ActionHomomorphism( a8, AsList(class) ); <action homomorphism> gap> IsSurjective(nophom); false gap> Image(nophom,(1,2,3)); (2,43,14)(3,44,20)(4,45,26)(5,46,32)(6,47,38)(8,13,48)(9,19,53)(10,25, 58)(11,31,63)(12,37,68)(15,49,73)(16,50,74)(17,51,75)(18,52,76)(21,54, 77)(22,55,78)(23,56,79)(24,57,80)(27,59,81)(28,60,82)(29,61,83)(30,62, 84)(33,64,85)(34,65,86)(35,66,87)(36,67,88)(39,69,89)(40,70,90)(41,71, 91)(42,72,92)
In this situation, however (for performance reasons, avoiding computation an image that might never be needed) the homomorphism is defined to be not into the Image of the action, but into the full symmetric group, i.e. it is not automatically surjective. Surjectivity can be enforced by giving the string "surjective" as an extra last argument. The Image of the action homomorphism of course is the same group in either case.
gap> Size(Range(nophom)); 1974506857221074023536820372759924883412778680349753377966562950949028\ 5896977181144089422435502777936659795733823785363827233491968638562181\ 1850780464277094400000000000000000000000000 gap> Size(Range(ophom)); 20160 gap> nophom := ActionHomomorphism( a8, AsList(class),"surjective" ); <action epimorphism> gap> Size(Range(nophom)); 20160
Continuing the example, 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 (Reference: RepresentativeAction).
gap> sgp := Normalizer( a8, Subgroup(a8,[Representative(class)]) );; gap> Size( sgp ); 360 gap> RepresentativeAction( a8, sgp, preim ); (2,4,3)
One 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.
gap> cosets:=RightCosets(a8,norm);; gap> op:=Action(a8,cosets,OnRight); Group([ (1,2,3)(4,6,5)(7,8,9)(10,12,11)(13,14,15), (1,2,3)(4,13,9) (5,7,11)(6,10,15)(8,14,12), (1,11,6)(2,15,4)(3,9,5)(7,13,10) (8,12,14), (1,10,12,3,13)(2,7,15,14,5)(4,6,8,9,11), (1,12,15)(2,10,5) (3,11,8)(4,9,13)(6,14,7), (1,4,13,10)(2,15,14,3)(5,12,11,6)(7,9) ]) gap> NrMovedPoints(op); 15
A 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, GAP 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 enumerator. The easiest example of this concept is the Enumerator (Reference: Enumerator) 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:
gap> enum:=Enumerator(SymmetricGroup(20)); <enumerator of perm group> gap> Length(enum); 2432902008176640000 gap> enum[123456789012345]; (1,4,15,3,14,11,8,17,6,18,5,7,20,13,10,9,2,12) gap> Position(enum,(1,2,3,4,5,6,7,8,9,10)); 71948729603
For the action on cosets the object of interest is the RightTransversal (Reference: RightTransversal) of a subgroup. Again, it does not write out actual elements and thus can be created even for subgroups of large index.
gap> t:=RightTransversal(a8,norm); RightTransversal(Alt( [ 1 .. 8 ] ),2^3:L_3(2)) gap> t[7]; (4,6,5) gap> Position(t,(4,6,7,8,5)); 8 gap> Position(t,(1,2,3)); fail
For 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 PositionCanonical (Reference: PositionCanonical) instead of Position (Reference: Position). In general, for elements contained in a list, PositionCanonical (Reference: PositionCanonical) returns the same as Position. If the element is not contained in the list (and for special lists, such as transversals), PositionCanonical returns the list element representing the same objects, e.g. the transversal element representing the same coset.
gap> PositionCanonical(t,(1,2,3)); 2 gap> t[2]; (6,7,8) gap> t[2]/(1,2,3); (1,3,2)(6,7,8) gap> last in norm; true
Thus, acting on a RightTransversal with the OnRight 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.
gap> Action(a8,RightTransversal(a8,norm),OnRight); Group([ (1,2,3)(4,6,5)(7,8,9)(10,12,11)(13,14,15), (1,2,3)(4,13,9) (5,7,11)(6,10,15)(8,14,12), (1,11,6)(2,15,4)(3,9,5)(7,13,10) (8,12,14), (1,10,12,3,13)(2,7,15,14,5)(4,6,8,9,11), (1,12,15)(2,10,5) (3,11,8)(4,9,13)(6,14,7), (1,4,13,10)(2,15,14,3)(5,12,11,6)(7,9) ])
Summary. In this section we have learned how groups can operate on GAP objects such as integers and group elements. We have used ActionHomomorphism (Reference: ActionHomomorphism), 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.
Action functions can also be used to construct subgroups. 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([ (2,3,4,5,6,7,8), (2,4,5,6,7,8,3) ]) 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 (Reference: 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 (Reference: OnPoints) because powering a list by a permutation via the caret operator ^ 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 package
refers to the GAP functionality for group actions, not to a GAP package). In our example we can do this by simply passing OnPairs (Reference: 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 (Reference: 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 (Reference: 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 (Reference: ActionHomomorphism) does what we need. It creates a homomorphism defined on a8 whose image is a new group that acts on [ 1 .. 56 ] in the same way that a8 acts 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 5.2, 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 (Reference: 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 (Reference: Blocks) to find a block system. The (optional) third argument in the following example tells Blocks (Reference: Blocks) that we want a block system where 1 and 3 lie in one block.
gap> blocks := Blocks( a8_56, [1..56], [1,3] );;
The result is a list of sets, such that a8_56 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 OnSets (Reference: 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 (Reference: OnSets) as third argument to a function like Stabilizer (Reference: 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 (Reference: 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> seed:=First(AllBlocks(a8_56),x->Length(x)=2);; gap> blocks := Blocks( a8_56, [1..56], seed);; gap> u28_56 := Stabilizer( a8_56, seed, 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 acts primitively on the 28 blocks.
gap> IsPrimitive( a8_56, blocks, OnSets ); true
Stabilizer (Reference: 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 acts regularly on the 60 triples of [ 4 .. 8 ] which contain no element twice. We construct the list of these 60 triples with the function Orbit (Reference: Orbit) (using OnTuples (Reference: OnTuples) as the natural generalization of OnPairs (Reference: OnPairs)) and then pass it as action domain to the function IsRegular (Reference: 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 acts on [ 1 .. 336 ] in the same way that a8 acts 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 5.2 that there is another standard action of permutations, namely the conjugation. E.g., since no other action is specified in the following example, OrbitLength (Reference: OrbitLength) simply acts via OnPoints (Reference: OnPoints), and because perm_1 ^ perm_2 is defined as the conjugation of perm_2 on perm_1, 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 \(g\) in any finite group \(G\) can be computed as OrbitLength( \(G\), \(g\) ), the command Size( ConjugacyClass( \(G\), \(g\) ) ) 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 (Reference: 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> orb:=Set(orb);; gap> blocks := Blocks( a8, orb );; Length( blocks ); 15 gap> Set(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)(5,7)(6,8), we must use an element of a8 that maps (1,2)(3,4)(5,6)(7,8) to (1,3)(2,4)(5,7)(6,8). The function RepresentativeAction (Reference: 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 (Reference: RepresentativeAction) returns fail.
gap> rep := RepresentativeAction( a8, (1,2)(3,4)(5,6)(7,8), > (1,3)(2,4)(5,7)(6,8) ); (2,3)(6,7) 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)(6,8) to u105.
gap> u15b := ClosureGroup( u105, (2,3)(6,8) );; Index( a8, u15b ); 15 gap> RepresentativeAction( a8, u15, u15b ); fail
RepresentativeAction (Reference: 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.
Summary. 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., Cycle (Reference: Cycle) and Permutation (Reference: Permutation). These are fully described in Chapter Reference: Group Actions.
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 \(G \rightarrow H\) 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 \(i\)th entry is the image of \(\textit{gens}[i]\) 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> Size( Image( hom, DerivedSubgroup(s4) ) ); 3
gap> PreImage( hom, (1,2,3) ); Error, <map> must be an inj. and surj. mapping called from <function "PreImage">( <arguments> ) called from read-eval loop at line 4 of *stdin* 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 \(S_4\) onto \(S_3\) is well known from elementary group theory. Images of elements and subgroups under hom can be calculated with the function Image (Reference: Image). But since the mapping hom is not bijective, we cannot use the function PreImage (Reference: PreImage) for preimages of elements (they can have several preimages). Instead, we have to use PreImagesRepresentative (Reference: 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 (Reference: 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 below.
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 (Reference: 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 (Reference: GroupHomomorphismByImagesNC) can be used; the NC stands for no check
.
But note that horrible things can happen if GroupHomomorphismByImagesNC (Reference: 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 (Reference: 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 (Reference: GroupHomomorphismByImages), which returns fail if the requested homomorphism does not exist, there is the function GroupGeneralMappingByImages (Reference: GroupGeneralMappingByImages), which returns a general mapping (that is, a possibly multi-valued mapping) that can be tested with IsGroupHomomorphism (Reference: 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 (Reference: GroupHomomorphismByImages) with left and right sides exchanged, in which case it is of course merely a GroupGeneralMappingByImages (Reference: GroupGeneralMappingByImages).
gap> rev := GroupGeneralMappingByImages( s3, s4, > [(1,2),(2,3)], GeneratorsOfGroup(s4) );;
Now \(hom\) maps \(a\) to \(b\) if and only if \(rev\) maps \(b\) to \(a\), for \(a \in\) s4 and \(b \in\) s3. Since every such \(b\) has four preimages under hom, it now has four images under rev. Just as the four preimages form a coset of the kernel \(V_4 \leq \)s4 of hom, they also form a coset of the cokernel \(V_4 \leq \)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 (Reference: One) returns the identity element of a group. 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 Reference: Group Homomorphisms.
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
Summary. 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).
For some types of groups, the best method for calculations in it is to use instead 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 (Reference: StabilizerOfExternalSet) in GAP (because conjugacy classes are represented as external sets, see Section Reference: Conjugacy Classes).
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 (Reference: 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([ (1,4,2,3), (1,5,4)(2,6,3), (1,2)(3,4), (3,4)(5,6) ]) gap> IsomorphismGroups( niceaut, SymmetricGroup( 4 ) ); [ (1,4,2,3), (1,5,4)(2,6,3), (1,2)(3,4), (3,4)(5,6) ] -> [ (1,4,2,3), (1,4,2), (1,2)(3,4), (1,3)(2,4) ]
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 (Reference: 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 (Reference: NiceMonomorphism), because a newly-constructed group will not yet have a NiceMonomorphism (Reference: 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([ f1*f2, f2^2*f3, f4, f3 ])
The star * denotes composition of mappings from the left to the right, as we have seen in Section 5.2 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.
Prescribing a known map as the nice monomorphism of a group is possible only if this map knows that it is injective. In the above example, this is the case because the map is created as the composition of two maps (a nice monomorphism and an isomorphism) which know that they are injective. In general, the injectivity flag of a map map gets set if one tests the property by calling IsInjective( map ); if one is sure that map is injective and wants to avoid the potentially expensive test then one can set the flag by explicitly calling SetIsInjective( map, true ).
Summary. In this section we have seen how calculations in groups can be carried out in isomorphic images in nicer groups. We have seen that GAP pursues this technique automatically for certain classes of groups, e.g., for matrix groups that are known to be finite.
Groups and the functions for groups are treated in Chapter Reference: Groups. There are several chapters dealing with groups in specific representations, for example Chapter Reference: Permutation Groups on permutation groups, Reference: Polycyclic Groups on polycyclic (including finite solvable) groups, Reference: Matrix Groups on matrix groups and Reference: Finitely Presented Groups on finitely presented groups. Chapter Reference: Group Actions deals with group actions. Group homomorphisms are the subject of Chapter Reference: Group Homomorphisms.
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