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<a id="X8171DAF2833FF728" name="X8171DAF2833FF728"></a>
<div class="ChapSects"><a href="chap5.html#X8171DAF2833FF728">5 Groups and Homomorphisms</a>
<div class="ContSect"> <a href="chap5.html#X85ED46007CED6191">5.1 Permutation groups</a>

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<div class="ContSect"> <a href="chap5.html#X8493BB7D7ECB5AB2">5.2 Actions of Groups</a>

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<div class="ContSect"> <a href="chap5.html#X8742F6A083ADE513">5.3 Subgroups as Stabilizers</a>

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<div class="ContSect"> <a href="chap5.html#X8131EE9182385680">5.4 Group Homomorphisms by Images</a>

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<div class="ContSect"> <a href="chap5.html#X7FFD731684606BC6">5.5 Nice Monomorphisms</a>

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<div class="ContSect"> <a href="chap5.html#X84579FB48474079C">5.6 Further Information about Groups and Homomorphisms</a>

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<h3>5 Groups and Homomorphisms</h3>

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 <code class="func">Group</code> (Reference: Group), <code class="func">Size</code> (Reference: Size) or <code class="func">SylowSubgroup</code> (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.

<a id="X85ED46007CED6191" name="X85ED46007CED6191"></a>

<h4>5.1 Permutation groups</h4>

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:

<div class="example"><pre>
gap> s8 := Group( (1,2), (1,2,3,4,5,6,7,8) );
Group([ (1,2), (1,2,3,4,5,6,7,8) ])
</pre></div>

We formed the group generated by the permutations <code class="code">(1,2)</code> and <code class="code">(1,2,3,4,5,6,7,8)</code>, which is well known to be the symmetric group S_8 on eight points, and assigned it to the identifier <code class="code">s8</code>. 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 <code class="code">s8</code>, 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.

<div class="example"><pre>
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 ] )
</pre></div>

Once information about a group like <code class="code">s8</code> or <code class="code">a8</code> 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, <code class="code">a8</code> stores its order and that it is nonabelian and perfect, and <code class="code">s8</code> stores its derived subgroup <code class="code">a8</code>. Had we computed <code class="code">a8</code> as <code class="code">CommutatorSubgroup( s8, s8 )</code>, 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 <code class="func">CommutatorSubgroup</code> (Reference: CommutatorSubgroup) can compute the commutator subgroup of two arbitrary subgroups.) The situation is a bit different for Sylow p-subgroups: The function <code class="func">SylowSubgroup</code> (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 <code class="code">ComputedSylowSubgroups</code>, but we won't dwell on the details here.

<div class="example"><pre>
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 ]
</pre></div>

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 <code class="code">a8</code>, 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.

<div class="example"><pre>
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
</pre></div>

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 <code class="code">a8</code> and <code class="code">elab</code>, 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 <code class="func">SetName</code> (Reference: Name). We do this with the name <code class="code">2^3</code> 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.

<div class="example"><pre>
gap> SetName( elab, "<group of type 2^3>" ); elab;
<group of type 2^3>
gap> norm := Normalizer( a8, elab );; Size( norm );
1344
</pre></div>

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

<div class="example"><pre>
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
</pre></div>

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 <code class="code">hom</code>). However, the action domain of this factor group has nothing to do with the action domain of <code class="code">norm</code>. (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 <code class="code">hom</code> is a coset modulo <code class="code">elab</code>. We use the function <code class="func">PreImages</code> (Reference: PreImages) here because <code class="code">hom</code> is not a bijection, so an element of the range can have several preimages or none at all.

<div class="example"><pre>
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))
</pre></div>

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.

<div class="example"><pre>
gap> rep:= Representative( coset );
(2,8,6,7,3,4,5)
gap> x * rep^-1 in ker;
true
</pre></div>

The factor group <code class="code">f</code> 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.)

<div class="example"><pre>
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)" );
</pre></div>

We give <code class="code">f</code> the name <code class="code">L_3(2)</code> 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 <code class="code">=</code> sign are different names for the same matrix group, e.g., <code class="code">A(2,2)</code> is the Lie type notation for the classical notation <code class="code">L(3,2)</code>. Other pairs of names are connected via <code class="code">~</code>, these then specify other classical groups that are isomorphic to that linear group (e.g., the symplectic group <code class="code">S(2,7)</code>, whose Lie type notation would be <code class="code">C(1,7)</code>).

The group <code class="code">norm</code> acts on the eight elements of its normal subgroup <code class="code">elab</code> by conjugation, yielding a representation of L_3(2) in <code class="code">s8</code> which leaves one point fixed (namely point <code class="code">1</code>). The image of this representation can be computed with the function <code class="func">Action</code> (Reference: Action homomorphisms); it is even contained in the group <code class="code">norm</code> and we can show that <code class="code">norm</code> is indeed a split extension of the elementary abelian group 2^3 with this image of L_3(2).

<div class="example"><pre>
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)" );
</pre></div>

By the way, you should not try the operator <code class="code"><</code> instead of the function <code class="func">IsSubgroup</code> (Reference: IsSubgroup). Something like

<div class="example"><pre>
gap> elab < a8;
false
</pre></div>

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

<a id="X8493BB7D7ECB5AB2" name="X8493BB7D7ECB5AB2"></a>

<h4>5.2 Actions of Groups</h4>

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

<div class="example"><pre>
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 ]
</pre></div>

Note the difference between <code class="func">Order</code> (Reference: Order) (which means the element order), <code class="func">Size</code> (Reference: Size) (which means the size of the conjugacy class) and <code class="func">Length</code> (Reference: Length) (which means the length of a list). We choose to let <code class="code">a8</code> operate on the class of length 112.

<div class="example"><pre>
gap> class := First( ccl, c -> Size(c) = 112 );;
gap> op := Action( a8, AsList( class ),OnPoints );;
</pre></div>

We use <code class="func">AsList</code> (Reference: AsList) here to convert the conjugacy class into a list of its elements whereas we wrote <code class="code">Action( norm, elab )</code> directly in the previous section. The reason is that the elementary abelian group <code class="code">elab</code> 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 <code class="func">First</code> (Reference: First), used to find the first element in a list which passes some test.

In this example, we have specified the action function <code class="func">OnPoints</code> (Reference: OnPoints) in this example, which is defined as <code class="code">OnPoints( </code>d<code class="code">, </code>g<code class="code"> ) = </code>d<code class="code"> ^ </code>g. This <q>caret</q> 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 <code class="func">OnRight</code> (Reference: OnRight), which means right multiplication, defined as d<code class="code"> * </code>g. (Group actions in GAP are always from the right.)

We now have a permutation representation <code class="code">op</code> 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 <code class="func">Blocks</code> (Reference: Blocks).

<div class="example"><pre>
gap> IsPrimitive( op, [ 1 .. 112 ] );
false
gap> blocks := Blocks( op, [ 1 .. 112 ] );;
</pre></div>

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

Now <code class="code">blocks</code> 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 <code class="code">op</code> act on these blocks setwise.

<div class="example"><pre>
gap> Length( blocks[1] ); Length( blocks );
2
56
gap> op2 := Action( op, blocks, OnSets );;
gap> IsPrimitive( op2, [ 1 .. 56 ] );
true
</pre></div>

Note that we give a third argument (the action function <code class="func">OnSets</code> (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 <code class="code">op</code> 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 <code class="code">*</code> operator, reading left-to-right.

<div class="example"><pre>
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) ])
</pre></div>

Alternatively, it is possible to create action homomorphisms immediately (without creating the action first) by giving the same set of arguments to <code class="func">ActionHomomorphism</code> (Reference: ActionHomomorphism).

<div class="example"><pre>
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)
</pre></div>

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 <code class="code">"surjective"</code> as an extra last argument. The <code class="code">Image</code> of the action homomorphism of course is the same group in either case.

<div class="example"><pre>
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
</pre></div>

Continuing the example, the normalizer of an element in the conjugacy class <code class="code">class</code> 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 <code class="code">a8</code> is found by the function <code class="func">RepresentativeAction</code> (Reference: RepresentativeAction).

<div class="example"><pre>
gap> sgp := Normalizer( a8, Subgroup(a8,[Representative(class)]) );;
gap> Size( sgp );
360
gap> RepresentativeAction( a8, sgp, preim );
(2,4,3)
</pre></div>

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.

<div class="example"><pre>
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
</pre></div>

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 <code class="func">Enumerator</code> (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:

<div class="example"><pre>
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
</pre></div>

For the action on cosets the object of interest is the <code class="func">RightTransversal</code> (Reference: RightTransversal) of a subgroup. Again, it does not write out actual elements and thus can be created even for subgroups of large index.

<div class="example"><pre>
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
</pre></div>

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 <code class="func">PositionCanonical</code> (Reference: PositionCanonical) instead of <code class="func">Position</code> (Reference: Position). In general, for elements contained in a list, <code class="func">PositionCanonical</code> (Reference: PositionCanonical) returns the same as <code class="code">Position</code>. If the element is not contained in the list (and for special lists, such as transversals), <code class="code">PositionCanonical</code> returns the list element representing the same objects, e.g. the transversal element representing the same coset.

<div class="example"><pre>
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
</pre></div>

Thus, acting on a <code class="code">RightTransversal</code> with the <code class="code">OnRight</code> 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.

<div class="example"><pre>
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) ])
</pre></div>

Summary. In this section we have learned how groups can operate on GAP objects such as integers and group elements. We have used <code class="func">ActionHomomorphism</code> (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.

<a id="X8742F6A083ADE513" name="X8742F6A083ADE513"></a>

<h4>5.3 Subgroups as Stabilizers</h4>

Action functions can also be used to construct subgroups. We will try to find several subgroups in <code class="code">a8</code> 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.

<div class="example"><pre>
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
</pre></div>

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 <code class="func">Tuples</code> (Reference: Tuples) we first generate a list of all pairs.

<div class="example"><pre>
gap> pairs := Tuples( [1..8], 2 );;
</pre></div>

Now we would like to have <code class="code">a8</code> operate on this domain. But we cannot use the default action <code class="func">OnPoints</code> (Reference: OnPoints) because powering a list by a permutation via the caret operator <code class="code">^</code> 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 <q>package</q> refers to the GAP functionality for group actions, not to a GAP package). In our example we can do this by simply passing <code class="func">OnPairs</code> (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 <code class="func">IsTransitive</code> (Reference: IsTransitive).

<div class="example"><pre>
gap> IsTransitive( a8, pairs, OnPairs );
false
</pre></div>

The action is of course not transitive, since the pairs <code class="code">[ 1, 1 ]</code> and <code class="code">[ 1, 2 ]</code> cannot lie in the same orbit. So we want to find out what the orbits are. The function <code class="func">Orbits</code> (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.

<div class="example"><pre>
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 ] ]
</pre></div>

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

<div class="example"><pre>
gap> u56 := Stabilizer( a8, orbs[2][1], OnPairs );; Index( a8, u56 );
56
</pre></div>

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 <code class="code">[ 1 .. 56 ]</code> instead of the list of pairs. The function <code class="func">ActionHomomorphism</code> (Reference: ActionHomomorphism) does what we need. It creates a homomorphism defined on <code class="code">a8</code> whose image is a new group that acts on <code class="code">[ 1 .. 56 ]</code> in the same way that <code class="code">a8</code> acts on the second orbit.

<div class="example"><pre>
gap> h56 := ActionHomomorphism( a8, orbs[2], OnPairs );;
gap> a8_56 := Image( h56 );;
</pre></div>

We would now like to know if the subgroup <code class="code">u56</code> of index 56 that we found is maximal or not. As we have used already in Section <a href="chap5.html#X8493BB7D7ECB5AB2">5.2</a>, a subgroup is maximal if and only if the action on the cosets of this subgroup is primitive.

<div class="example"><pre>
gap> IsPrimitive( a8_56, [1..56] );
false
</pre></div>

Remember that we can leave out the function if we mean <code class="func">OnPoints</code> (Reference: OnPoints) but that we have to specify the action domain for all action functions.

We see that <code class="code">a8_56</code> is not primitive. This means of course that the action of <code class="code">a8</code> on <code class="code">orb[2]</code> is not primitive, because those two actions are equivalent. So the stabilizer <code class="code">u56</code> is not maximal. Let us try to find its supergroups. We use the function <code class="func">Blocks</code> (Reference: Blocks) to find a block system. The (optional) third argument in the following example tells <code class="func">Blocks</code> (Reference: Blocks) that we want a block system where 1 and 3 lie in one block.

<div class="example"><pre>
gap> blocks := Blocks( a8_56, [1..56], [1,3] );;
</pre></div>

The result is a list of sets, such that <code class="code">a8_56</code> 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 <code class="func">OnSets</code> (Reference: OnSets) as third argument.

<div class="example"><pre>
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
</pre></div>

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

Here is a warning: If you specify <code class="func">OnSets</code> (Reference: OnSets) as third argument to a function like <code class="func">Stabilizer</code> (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 <code class="code">blocks[1]</code> came from the function <code class="func">Blocks</code> (Reference: Blocks), which returns a list of sets, so everything was OK.

Actually there is a third block system of <code class="code">a8_56</code> that gives rise to a third subgroup.

<div class="example"><pre>
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
</pre></div>

We know that the subgroup <code class="code">u28</code> of index 28 is maximal, because we know that <code class="code">a8</code> has no subgroups of index 2, 4, or 7. However we can also quickly verify this by checking that <code class="code">a8_56</code> acts primitively on the 28 blocks.

<div class="example"><pre>
gap> IsPrimitive( a8_56, blocks, OnSets );
true
</pre></div>

<code class="func">Stabilizer</code> (Reference: Stabilizer) is not only applicable to groups like <code class="code">a8</code> but also to their subgroups like <code class="code">u56</code>. So another method to find a new subgroup is to compute the stabilizer of another point in <code class="code">u56</code>. Note that <code class="code">u56</code> already leaves 1 and 2 fixed.

<div class="example"><pre>
gap> u336 := Stabilizer( u56, 3 );;
gap> Index( a8, u336 );
336
</pre></div>

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

<div class="example"><pre>
gap> IsRegular( u336, Orbit( u336, [4,5,6], OnTuples ), OnTuples );
true
</pre></div>

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

<div class="example"><pre>
gap> t := RightTransversal( a8, u336 );;
gap> a8_336 := Action( a8, t, OnRight );;
</pre></div>

To find subgroups above <code class="code">u336</code> we again look for nontrivial block systems.

<div class="example"><pre>
gap> blocks := Blocks( a8_336, [1..336] );; blocks[1];
[ 1, 43, 85 ]
</pre></div>

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

<div class="example"><pre>
gap> u112 := ClosureGroup( u336, t[43] );;
gap> Index( a8, u112 );
112
</pre></div>

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

<div class="example"><pre>
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
</pre></div>

We already mentioned in Section <a href="chap5.html#X8493BB7D7ECB5AB2">5.2</a> that there is another standard action of permutations, namely the conjugation. E.g., since no other action is specified in the following example, <code class="func">OrbitLength</code> (Reference: OrbitLength) simply acts via <code class="func">OnPoints</code> (Reference: OnPoints), and because <var class="Arg">perm_1</var><code class="code"> ^ </code><var class="Arg">perm_2</var> is defined as the conjugation of <var class="Arg">perm_2</var> on <var class="Arg">perm_1</var>, in fact we compute the length of the conjugacy class of <code class="code">(1,2)(3,4)(5,6)(7,8)</code>.

<div class="example"><pre>
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
</pre></div>

Note that although the length of a conjugacy class of any element g in any finite group G can be computed as <code class="code">OrbitLength( </code>G<code class="code">, </code>g<code class="code"> )</code>, the command <code class="code">Size( ConjugacyClass( </code>G<code class="code">, </code>g<code class="code"> ) )</code> is probably more efficient.

<div class="example"><pre>
gap> Size( ConjugacyClass( a8, (1,2)(3,4)(5,6)(7,8) ) );
105
</pre></div>

Of course the stabilizer <code class="code">u105</code> is in fact the centralizer of the element <code class="code">(1,2)(3,4)(5,6)(7,8)</code>. <code class="func">Stabilizer</code> (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 <code class="code">u105</code>.

<div class="example"><pre>
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) ]
</pre></div>

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

<div class="example"><pre>
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
</pre></div>

<code class="code">u15</code> is of course a maximal subgroup, because <code class="code">a8</code> has no subgroups of index 3 or 5. There is in fact another class of subgroups of index 15 above <code class="code">u105</code> that we get by adding <code class="code">(2,3)(6,8)</code> to <code class="code">u105</code>.

<div class="example"><pre>
gap> u15b := ClosureGroup( u105, (2,3)(6,8) );; Index( a8, u15b );
15
gap> RepresentativeAction( a8, u15, u15b );
fail
</pre></div>

<code class="func">RepresentativeAction</code> (Reference: RepresentativeAction) tells us that there is no element g in <code class="code">a8</code> such that <code class="code">u15 ^ </code>g<code class="code"> = u15b</code>. Because <code class="code">^</code> also denotes the conjugation of subgroups this tells us that <code class="code">u15</code> and <code class="code">u15b</code> 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., <code class="func">Cycle</code> (Reference: Cycle) and <code class="func">Permutation</code> (Reference: Permutation). These are fully described in Chapter Reference: Group Actions.

<a id="X8131EE9182385680" name="X8131EE9182385680"></a>

<h4>5.4 Group Homomorphisms by Images</h4>

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 <code class="code">GroupHomomorphismByImages( <var class="Arg">G</var>, <var class="Arg">H</var>, <var class="Arg">gens</var>, <var class="Arg">imgs</var> )</code> where <var class="Arg">gens</var> is a generating set for <var class="Arg">G</var> and <var class="Arg">imgs</var> is a list whose ith entry is the image of <var class="Arg">gens</var>[i] under the homomorphism.

<div class="example"><pre>
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
</pre></div>

<div class="example"><pre>
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;
</pre></div>

<div class="example"><pre>
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) ])
</pre></div>

This homomorphism from S_4 onto S_3 is well known from elementary group theory. Images of elements and subgroups under <code class="code">hom</code> can be calculated with the function <code class="func">Image</code> (Reference: Image). But since the mapping <code class="code">hom</code> is not bijective, we cannot use the function <code class="func">PreImage</code> (Reference: PreImage) for preimages of elements (they can have several preimages). Instead, we have to use <code class="func">PreImagesRepresentative</code> (Reference: PreImagesRepresentative), which returns one preimage if at least one exists (and would return <code class="keyw">fail</code> if none exists, which cannot occur for our surjective <code class="code">hom</code>). On the other hand, we can use <code class="func">PreImage</code> (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.

<div class="example"><pre>
gap> GroupHomomorphismByImages( s4, s3,
> GeneratorsOfGroup(s4), [(1,2,3),(2,3)] );
fail
</pre></div>

There is no such homomorphism, hence <code class="keyw">fail</code> is returned. But note that because of this, <code class="func">GroupHomomorphismByImages</code> (Reference: GroupHomomorphismByImages) must do some checks, and this was also done for the mapping <code class="code">hom</code> above. One can avoid these checks if one is sure that the desired homomorphism really exists. For that, the function <code class="func">GroupHomomorphismByImagesNC</code> (Reference: GroupHomomorphismByImagesNC) can be used; the <code class="code">NC</code> stands for <q>no check</q>.

But note that horrible things can happen if <code class="func">GroupHomomorphismByImagesNC</code> (Reference: GroupHomomorphismByImagesNC) is used when the input does not describe a homomorphism.

<div class="example"><pre>
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
</pre></div>

In other words, GAP claims that the kernel is the full <code class="code">s4</code>, yet <code class="code">hom2</code> 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 <code class="func">GroupHomomorphismByImagesNC</code> (Reference: GroupHomomorphismByImagesNC), GAP trusts you.

<div class="example"><pre>
gap> IsGroupHomomorphism( hom2 );
true
</pre></div>

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

Besides the safe command <code class="func">GroupHomomorphismByImages</code> (Reference: GroupHomomorphismByImages), which returns <code class="keyw">fail</code> if the requested homomorphism does not exist, there is the function <code class="func">GroupGeneralMappingByImages</code> (Reference: GroupGeneralMappingByImages), which returns a general mapping (that is, a possibly multi-valued mapping) that can be tested with <code class="func">IsGroupHomomorphism</code> (Reference: IsGroupHomomorphism).

<div class="example"><pre>
gap> hom2 := GroupGeneralMappingByImages( s4, s3,
> GeneratorsOfGroup(s4), [(1,2,3),(2,3)] );;
gap> IsGroupHomomorphism( hom2 );
false
</pre></div>

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 <q>reversal of arrows</q> in a homomorphism such as our original <code class="code">hom</code>. By this we mean the <code class="func">GroupHomomorphismByImages</code> (Reference: GroupHomomorphismByImages) with left and right sides exchanged, in which case it is of course merely a <code class="func">GroupGeneralMappingByImages</code> (Reference: GroupGeneralMappingByImages).

<div class="example"><pre>
gap> rev := GroupGeneralMappingByImages( s3, s4,
> [(1,2),(2,3)], GeneratorsOfGroup(s4) );;
</pre></div>

Now hom maps a to b if and only if rev maps b to a, for a ∈ <code class="code">s4</code> and b ∈ <code class="code">s3</code>. Since every such b has four preimages under <code class="code">hom</code>, it now has four images under <code class="code">rev</code>. Just as the four preimages form a coset of the kernel V_4 ≤<code class="code">s4</code> of <code class="code">hom</code>, they also form a coset of the cokernel V_4 ≤<code class="code">s4</code> of <code class="code">rev</code>. The cokernel itself is the set of all images of <code class="code">One( s3 )</code>. (It is a normal subgroup in the group of all images under <code class="code">rev</code>.) The operation <code class="func">One</code> (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 <code class="code">hom</code> as the cokernel of the reversed group general mapping (here <code class="code">rev</code>).

<div class="example"><pre>
gap> CoKernel( rev );
Group([ (1,4)(2,3), (1,3)(2,4) ])
</pre></div>

The reason why <code class="code">rev</code> is not a homomorphism is that it is not single-valued (because <code class="code">hom</code> 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 <code class="keyw">fail</code> if you say <code class="code">GroupHomomorphismByImages( <var class="Arg">G</var>, <var class="Arg">H</var>, <var class="Arg">gens</var>, <var class="Arg">imgs</var> )</code> where <var class="Arg">gens</var> does not generate <var class="Arg">G</var> (even if this would give a decent homomorphism on the subgroup generated by <var class="Arg">gens</var>). 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 <code class="code">hom2</code> nor its inverse general mapping (with arrows reversed) is a homomorphism.

<div class="example"><pre>
gap> CoKernel( hom2 ); Kernel( hom2 );
Group([ (2,3), (1,3) ])
Group([ (3,4), (2,3,4), (1,2,4) ])
gap> IsGroupHomomorphism( InverseGeneralMapping( hom2 ) );
false
</pre></div>

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).

<a id="X7FFD731684606BC6" name="X7FFD731684606BC6"></a>

<h4>5.5 Nice Monomorphisms</h4>

For some types of groups, the best method for calculations in it is to use instead an isomorphic group in a <q>better</q> representation (say, a permutation group). We call an injective homomorphism, that will give such an isomorphic image a <q>nice monomorphism</q>.

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:

<div class="example"><pre>
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) ])
</pre></div>

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 <code class="code">hom</code>. Lifting back a conjugacy class means finding the preimage of the representative and of the centralizer; the latter is called <code class="func">StabilizerOfExternalSet</code> (Reference: StabilizerOfExternalSet) in GAP (because conjugacy classes are represented as external sets, see Section Reference: Conjugacy Classes).

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Wiener Entwicklungsmethode

Haftungshinweis

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Bemerkung:

Die farbliche Syntaxdarstellung ist noch experimentell.