XGAP provides a graphical interface to the lattice or a partial lattice of subgroups of groups. For finitely presented groups it gives you easy access for example to the low index, prime quotient and Reidemeister-Schreier algorithms in order to build a partial lattice interactively. For other types of groups it provides easy access to many of the group functions (for example, the normalizer, normal subgroups, and Sylow subgroups).
This chapter explains how to use this interface by way of examples. Chapter Subgroup Lattices - Systematic Description gives details about the various options and menus available. These two chapters will not describe how to write your own programs using the graphic extensions supplied by XGAP, see chapters Graphic Sheets - Basic graphic operations to Graphic Graphs for details.
It is assumed that you have already started XGAP. On most systems you do this by typing
user@host:~> xgap
on the command line. Ask your system administrator if this does not work. This command will create a new window, the so called GAP window, in which GAP is awaiting your input. Depending on the window system and window manager you use, placing a new window on your screen might be done automatically or might require you to use the mouse to choose a position for the window and pressing the left mouse button to place the window.
The small arrow or cross you see on your screen is called a pointer. Although the device used to move this pointer can be anything, a mouse, a track ball, a glide-pad, or even something as exotic as a rat, we will use the term mouse to refer to this pointer device.
In case that some computation takes longer than expected, for instance
the low index and the prime quotient can be quite time consuming, you
can always interrupt a computation by making the GAP window active
and pressing CTRL-C or selecting Interrupt in the Run menu.
Again, making a window active is system and window manager dependent.
In most cases you either have to move the pointer inside the GAP
window or you have to click on the title bar of the GAP window.
Note that for each of the following examples there is a small GAP
script in the examples subdirectory of the XGAP home directory
which contains the necessary commands. However we consider it better
for learning XGAP in a first time session if you type the commands
by hand as suggested in the next few sections of this manual.
This section gives you an example on how to use the function
GraphicSubgroupLattice (see GraphicSubgroupLattice for details), which
will display the Hasse diagram of the subgroup lattice of a given group.
Using the dihedral group of size 8 as example the following will show you
most features of the GraphicSubgroupLattice program. This exercise is
best carried out in front of XGAP, trying the various commands yourself.
First you have to define a group in GAP, this example uses a dihedral group defined as polycyclic group.
gap> d8 := DihedralGroup(8); <pc group of size 8 with 3 generators> gap> SetName(d8,"d8");
Now you ask for a graphical display by
gap> s := GraphicSubgroupLattice(d8); <graphic subgroup lattice "GraphicSubgroupLattice of d8">
XGAP will open a window containing a new graphic sheet, a menu bar (menus are described below) above the graphic sheet and a title. On most systems the title will be either below the graphic sheet or above the menu bar. The dimension of the graphic sheet is fixed, changing the size of the window will not change the size of the graphic sheet, see GraphicSubgroupLattice, Poset Menu how to resize the graphic sheet. It is possible that the graphic sheet is larger (depending on the lattice it might be much larger) than the window. In this case the window will contain so called scrollbars which allow you to select the portion of the graphic sheet which will be displayed.
XGAP first shows only the whole group (which is already selected) and the trivial subgroup, connected by a line indicating inclusion.
ConjugacyClassesSubgroups computes and returns the conjugacy classes of
subgroups, so summing up the sizes of the classes tells you how many
elements the lattice has.
gap> Sum( List( ConjugacyClassesSubgroups(d8), Size ) );
10
10 is small enough to use AllSubgroups without painting the screen
black. After you have clicked this menu entry in the Subgroups menu you
see the complete Hasse diagram in the graphic sheet.
The following initial remarks can be made about the graphical representation of the subgroup lattice:
The initial placement of the vertices chosen by GraphicSubgroupLattice
might not be optimal or you might want to choose a different one in order
to exploit certain features of the diagram. It is therefore possible to
move the vertices around using the mouse.
The mouse together with the left mouse button can be used to move and select vertices. A selected vertex is represented by a thicker circle, colored red if your screen supports color. For example, in order to move vertex 4 use the mouse to place the pointer inside the circle around 4 and press the left mouse button. Keep the mouse button pressed and start moving the mouse. The vertex will now follow the pointer. Because of the height restrictions given by the size it is not possible to move 4 above 6 or below 1. It must always stay within its level. If you release the left mouse button vertex 4 will stay at its current position and the rest of the conjugacy class (in this example 5) will be moved to this new position.
In order to select vertex G place the pointer inside the diamond around G, press the left mouse button and release it immediately. Do not move the mouse while you hold down the left mouse button. Vertex G now has a slightly thicker boundary and is red if you have a color screen. There are two different ways to select more than one vertex, see A Partial Subgroup Lattice of the Symmetric Group on 6 Points or GraphicSubgroupLattice, Selecting Vertices.
On the top of the window, above the graphic sheet, you can see a list
of menu names: Sheet, Poset, and Subgroups. In order to
open any of these pull down menus place the pointer inside the
button containing the menu name and press the left mouse button. Keep
the button pressed. A pull down menu will be shown and by moving the
pointer down you can choose a menu entry. By choosing an entry and
then releasing the mouse button the entry is selected, the
corresponding function is executed and the pull down menu is closed.
If you release the mouse button while the pointer is outside the pull
down menu the menu is closed without selecting any entry. Note that
this behaviour is different from that of some other graphical user
interfaces such as for example Windows.
Now select Change Labels from the Poset menu. If this entry
is not available you have failed to select vertex G. After
selecting Change Labels a small dialog box is opened asking for a
label. Type in D8 and press the return key or click on OK. The
label of vertex G will now be changed to ``D8''. Note that in the
X Window System you have to move the pointer on the text field if you
want to edit the label.
In order to find out which vertex represents the centre of D8,
first select vertex D8 and then the menu entry Centres from the
Subgroups menu. In case of a color screen, vertex D8 will be
selected and colored red, and vertex 2 will be colored green. The
color green indicates that vertex 2 is the result of a computation.
There will also be a message in the GAP window saying that vertex
2 represents the centre of the group belonging to vertex D8.
#I Centres (D8) --> (2)
Most of the menu entries in Subgroups should be self-explanatory, for
details and the difference between Closure and Closures see
GraphicSubgroupLattice, Subgroups Menu.
If you have selected some vertices (in the example D8 is now
selected), and you want to investigate the subgroups corresponding to
these vertices further in GAP, the function SelectedGroups will
return a list of these subgroups (note that you can also achieve
calling this function by selecting SelectedGroups to GAP in the
Subgroups menu):
labelgapxgap
gap> SelectedGroups(s); [ d8 ]
On the other hand, the functions supplied via the Subgroups menu are by
far not all functions applicable to groups. In order to show results of a
computation in GAP in the diagram, you can use SelectGroups. The
function SelectGroups allows you to mark any set of subgroups of D8 in
the diagram.
For instance, you can compute the lower central series of this (nilpotent) group in GAP.
gap> l := LowerCentralSeries(d8); [ d8, Group([ f3 ]), Group([ <identity> of ... ]) ] gap> SelectGroups(s,l);
This lower central series corresponds to the vertices D8, 2 and
1 which will now be selected. If, as it is not the case in this
example, the subgroups are not yet depicted in the lattice, a warning
appears in the GAP command window. You have to use InsertVertex
to insert a new vertex into the lattice (note that you can also
achieve this by selecting InsertVertices from GAP in the
Subgroups menu, see section InsertVertex for the complete
description of this function or section xgapgap for an example).
To summarize the above: the function SelectedGroups can be used to
transfer information from the diagram to GAP, the functions
SelectGroups and InsertVertex can be used to transfer information
from GAP to the diagram.
In order to finish this example, close the window by selecting close
graphic sheet from the Sheet menu. This will close the window
containing the Hasse diagram of D8.
In this example you have learned, how to display the Hasse diagram of the
subgroup lattice of a group using GraphicSubgroupLattice, how to use the
mouse to move and select vertices, how to select a menu entry and how to
transfer information between the Hasse diagram and GAP using SelectGroups
and SelectedGroups.
In order to learn more about the menus Sheet and Poset, which were only
mentioned very briefly, see GraphicSubgroupLattice, Sheet Menu, and
GraphicSubgroupLattice, Poset Menu.
This section investigates the subgroup lattice of S6.
gap> s6 := SymmetricGroup(6); Sym( [ 1 .. 6 ] ) gap> SetName(s6,"S6"); gap> cc := ConjugacyClassesSubgroups(s6);; gap> Sum(List(cc,Size)); 1455
As there are 1455 subgroups, displaying the whole lattice of subgroups would not be helpful because there are simply too many. Therefore this example builds only a partial subgroup lattice. We assume that you are familiar with the general ideas, mouse actions and menus, which were discussed in The Subgroup Lattice of the Dihedral Group of Order 8.
We again start to build a partial lattice, by using
GraphicSubgroupLattice (see GraphicSubgroupLattice). After you have
entered
gap> s := GraphicSubgroupLattice(s6); <graphic subgroup lattice "GraphicSubgroupLattice of S6">
XGAP will open a window containing a new graphic sheet with two connected vertices labeled 1 and G. Vertex 1 represents the trivial subgroup and vertex G the group S6. Vertex G is already selected, so it will be red if your screen supports color.
XGAP can automatically write a protocol of all the subsequent
actions you perform via mouse clicks. This is convenient because in
comparison to normal GAP sessions you do not have the script of
typed commands. You can activate this feature by selecting Start
Logging from the Subgroups menu. XGAP prompts you for a filename
via a file selector box. See loggingfacility for details about this
feature.
In order to find all subgroups of size 60, we cannot not use the
Subgroups menu directly, so go back into the GAP window and extract
the conjugacy classes of cc whose representatives have size 60.
gap> c60 := Filtered(cc,x->Size(Representative(x))=60);; gap> s60 := List(c60,Representative); [ Group([ (1,2)(3,4), (1,3,5) ]), Group([ (1,2)(3,4), (1,2,3)(4,5,6) ]) ]
labelxgapgap
We now use the function InsertVertex (see InsertVertex) to add
these two subgroups to your partial lattice.
gap> for g in s60 do InsertVertex(s,g); od;
Note that we could have achieved this result with the entry
InsertVertices from GAP in the Subgroups menu, see
InsertVertex. The Hasse diagram now contains four vertices. The
new vertices are not selected automatically. You can do this as
mentioned above by clicking with the left mouse button on them.
Selecting Conjugate Subgroups from the Subgroups menu adds the
complete conjugacy classes. Please do this first for vertex 2 and
then for vertex 3 such that the numbering of vertices in the
following description is correct!
In order to find out what type of subgroups we are looking at, use another kind of menu not discussed so far, namely the ``Information'' menu. Place the pointer inside vertex 3, press the right mouse button and release it immediately. This will pop up a new window, containing some text describing vertex 3 (as mentioned above, depending on the window system and window manager, placing this window on the screen might require some interaction with the mouse).
Size 60 Index 12 IsAbelian unknown IsCentral unknown IsCyclic unknown IsNilpotent false IsNormal false IsPerfect true IsSimple unknown IsSolvable false Isomorphism unknown
Note that GAP does not yet automatically draw the conclusion that a nonsolvable subgroup is also not abelian, cyclic or central. Place the pointer on top of the entry ``Isomorphism'' and press the left mouse button. After a while this entry is changed to
Isomorphism [ 60, 5 ]
telling you that the subgroup represented by vertex 3 is isomorphic to
the alternating group on five symbols. The notation [ 60, 5 ] comes out
of the small groups library and is the only information about the
isomorphism type we can get from GAP4 up to now. Select close to close
the ``Information'' menu. Repeat this with vertex 2, you will see that the
subgroup of vertex 2 is also isomorphic to A5, however these two A5
inside S6 are not conjugate in S6. The ``Information'' menu is described
in detail in GraphicSubgroupLattice, Information Menu.
Now we want to compute the normalizers of the elements of the conjugacy
class containing the subgroup of vertex 3. You could either select
vertex 3 and then Normalizers and repeat this process for the vertices
9 to 13, or you can first select the vertices 3, 9 to 13 and then
select Normalizers. But how to select more than one vertex? If you
first select 3 and then 9, vertex 3 will get deselected as soon as
9 gets selected. However, if you select vertex 3, place the pointer
inside vertex 9, hold down the SHIFT key on your keyboard and then
select vertex 9 using the left mouse button, vertex 9 will be selected
in addition to vertex 3. Another method to select more than one vertex
is to use the rubber band to catch vertices inside a rectangle. Place the
pointer left and a bit higher than vertex 3 outside any other vertex.
Press the left mouse button and hold it down. Now, using the mouse, move
the pointer right and slightly below vertex 13. You see a rectangle, one
corner at your start position and the other following the pointer. If
vertices 3 and 9 to 13 are all inside this rectangle, release the
mouse button. Now these vertices are selected. Select Normalizers from
the Subgroups menu to compute and display the normalizers.
Now select vertex 3 and 4 and compute the intersection. The
intersection is of size 10. Select this intersection and use
SelectedGroups to get a GAP record describing the subgroup.
gap> l := SelectedGroups(s); [ Group([ (2,3)(4,6), (1,2)(3,4) ]) ] gap> u := l[1]; Group([ (2,3)(4,6), (1,2)(3,4) ])
In order to find out which subgroups of the complete lattice lie above
the subgroup u you can use Intermediate Subgroups. You select the
whole group in addition to u and choose Intermediate Subgroups in the
Subgroups menu. You get 6 groups, some of them are already in the
lattice, the others are added.
There is another feature we have not seen yet. Close the current graphic sheet and start again with a fresh one.
gap> Close(s); gap> s := GraphicSubgroupLattice(s6); <graphic subgroup lattice "GraphicSubgroupLattice of S6">
In order to compute a Sylow 2 subgroup select Sylow Subgroup from the
Subgroups menu. A small dialog box will pop up asking for a prime,
type in 2 and press return or click on OK. Now select this new
vertex 2 representing the Sylow 2 subgroup and compute its normal
subgroups. This is rather slow because the function checks for each new
vertex if the corresponding subgroup is conjugate to an old one of the
same size.
This is now the end of our partial investigation of the (partial)
subgroup lattice of S6, close the graphic sheet(s) using close
graphic sheet of the Sheet menu. If you started the logging facility
of XGAP as described above you now have a file (probably called
xgap.log if you did not change the default) describing the actions
we performed.
This section investigates the following finitely presented group C2,
which was first investigated by Alberto Cavicchioli in Cav86:
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In this example we will show a way to prove a finitely presented group to be infinite, and to find some big nonabelian factor groups of it.
The following GAP commands define C2.
gap> f := FreeGroup( "a", "b" ); a := f.1;; b := f.2;; <free group on the generators [ a, b ]> gap> c2 := f / [ a*b*a^-2*b*a/b, (b^-1*a^3*b^-1*a^-3)^2*a ]; <fp group on the generators [ a, b ]> gap> SetName(c2,"c2");
We again assume that you are familiar with the general ideas, mouse actions and menus, which were discussed in The Subgroup Lattice of the Dihedral Group of Order 8 and A Partial Subgroup Lattice of the Symmetric Group on 6 Points.
In order to build a partial lattice of a finitely presented group, you
again use the function GraphicSubgroupLattice. But if the first argument
to GraphicSubgroupLattice is a finitely presented group the available menus
are different from the example in the previous section. After you have
entered
gap> s := GraphicSubgroupLattice(c2); <graphic subgroup lattice "GraphicSubgroupLattice of c2">
XGAP will open a window containing a new graphic sheet. Compared to the interactive lattice of a permutation group as described in the previous section, there are the following differences:
-- There is only one vertex instead of two. This vertex labeled G is the whole group C2. There is no vertex for the trivial subgroup (yet).
-- If you pull down the Subgroups menu, you will see that this menu is
now very different. It gives you access to various algorithms for
finitely presented groups but most of the entries from the last two
examples are missing because most of the GAP functions behind these
entries are not applicable to (infinite) finitely presented groups.
This example will show you how to prove that C2 is infinite. First look at the abelian invariants in order to see what the commutator factor group is. In order to compute the abelian invariants pop up the ``Information'' menu. This is done in exactly the same manner as in the previous section. Place the pointer inside vertex G, press the right mouse button and release it immediately. This ``Information'' menu is described in detail in GraphicSubgroupLattice for FpGroups, Information Menu.
Index 1 IsNormal true IsFpGroup unknown Abelian Invariants unknown Coset Table unknown IsomorphismFpGroup unknown Factor Group unknown
This tells you what XGAP already knows about the group associated with vertex G. In order to compute the abelian invariants click onto this line. After a while this entry will change to
Abelian Invariants perfect
telling you that C2 is perfect. So none of the Subgroups menu
entries Abelian Prime Quotient, All Overgroups, Conjugacy Class,
Cores, Derived Subgroups, Intersection, Intersections,
Normalizers or Prime Quotient will compute any new subgroups.
In order to avoid accidents the menu entries Abelian Prime Quotient,
All Overgroups, Epimorphisms (GQuotients), Conjugacy Class,
Low Index Subgroups, and Prime Quotient from the Subgroups menu are
only selectable if exactly one vertex is selected because the
functions behind these entries are in general quite time and space
consuming.
Close the ``Information'' window and select Low Index Subgroups from the
Subgroups menu. A small dialog box will pop up asking for a limit on
the index. Type in 12 and press return or click on OK. In general
it is hard to say what kind of index limit will still work, for some
groups even 5 might be too much while for others 20 works fine, see
also LowIndexSubgroupsFpGroup.
GAP computes 10 subgroups of index 11 and 8 subgroups of index 12. If you now start to check the abelian invariants of the index 12 subgroups you will find out that all subgroups represented by vertices 3 to 10 have a finite commutator factor group except the subgroup belonging to vertex 4 which has an infinite abelian quotient. Therefore the group C2 itself is infinite.
Now we want to investigate C2 a little further using GAP. Select
vertices 3, 4, and 5 and switch to the GAP window. Use
SelectedGroups to get the subgroups associated with these vertices.
gap> u := SelectedGroups( s );
[ Group([ a, b*a^2*b^-2, b*a*b^2*a^-1*b^-1*a^-1*b^-1, b^4*a^-2*b^-2,
b^2*a^3*b^-1*a^-1*b^-2 ]),
Group([ a, b^2*a*b^-1*a^-1*b^-1, b^3*a^-1*b^-1, b*a*b*a^3*b^-1 ]),
Group([ a, b^2*a*b^-1*a^-1*b^-1, b*a^3*b^-2, b^4*a^-1*b^-3,
b*a*b^3*a^-1*b^-1 ]) ]
FactorCosetAction computes for each of these subgroups ui the
operation of C2 on its cosets. It returns the result as a homomorphism
of C2 onto a permutation group. The operation on ui is therefore a
permutation representation of the factor group
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DisplayCompositionSeries we can identify these factor groups.
gap> p := List( u, x -> FactorCosetAction( c2, x ) );;
gap> l := List( p, Image );;
gap> for x in l do DisplayCompositionSeries(x); Print("\n"); od;
G (2 gens, size 95040)
| M(12)
1 (0 gens, size 1)
G (2 gens, size 660)
| A(1,11) = L(2,11) ~ B(1,11) = O(3,11) ~ C(1,11) = S(2,11) ~ 2A(1,11) = U(2,11)
1 (0 gens, size 1)
G (2 gens, size 239500800)
| A(12)
1 (0 gens, size 1)
(This display can look a little different according to the GAP version you use.)
So C2 contains the Mathieu group M12, the alternating group on
12 symbols and PSL(2,11) as factor groups. Therefore it would
have been possible to find vertex 4 using
Epimorphisms (GQuotients) instead of Low Index Subgroups.
Close the graphic
sheet by selecting the menu entry close graphic sheet from the Sheet
menu and start with a fresh one.
gap> s := GraphicSubgroupLattice(c2); <graphic subgroup lattice "GraphicSubgroupLattice of c2">
Select Epimorphisms (GQuotients) from the Subgroups menu. This
pops up a menu similar to the ``Information'' menu (see
GraphicSubgroupLattice for FpGroups, Subgroups Menu).
Sym(n) Alt(n) PSL(d,q) Library User Defined
Select PSL(d,q), which pops up a dialog box asking for a dimension.
Enter 2 and click on OK. Then a second dialog box pops up asking
for a field size. Enter 11 and click on OK. After a short time
of computation the display in the Epimorphisms (GQuotients) menu
changes and shows
PSL(2,11) 1 found
telling you, that GAP has found 1 epimorphism (up to inner automorphisms of PSL(2,11)) from C2 onto PSL(2,11). Click on display point stabilizer to create a new vertex representing a subgroup u such that the factor group of C2 / Core(u) is isomorphic to PSL(2,11). You could have clicked on display to create a new vertex representing the kernel of the epimorphism.
This is now the end of our partial investigation of the (partial)
subgroup lattice of C2, you have seen that C2 is infinite and
contains M12, Alt(12), and PSL(2,11) as factor groups. Close
the graphic sheet by selecting close graphic sheet from the Sheet
menu.
This section investigates the following finitely presented group, the
trefoil knot group K3.
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This examples shows some limitations of the methods available, in particular if infinite factors occur.
gap> f := FreeGroup( "a", "b" ); <free group on the generators [ a, b ]> gap> k3 := f / [ f.1*f.2*f.1 / (f.2*f.1*f.2) ]; <fp group on the generators [ a, b ]> gap> s := GraphicSubgroupLattice(k3); <graphic subgroup lattice "GraphicSubgroupLattice">
If you compute the Abelian invariants of K3 you will see that the
commutator factor group is isomorphic to the infinite cyclic group.
If you try to compute the derived subgroups it works! Just click on
Derived Subgroups in the Subgroups menu. A vertex appears in a
level marked with [ infinity, 1 ]. However, there are not too many
things you can do with such infinite index subgroups up to now, as we
will illustrate below:
First produce some more subgroups by Low Index Subgroups (for
example with index limit 5). If you now try to compare one of the new
subgroups with the derived subgroup, this is possible. If you however
try to calculate the intersection of one of the finite-index subgroups
with the derived subgroups, GAP will run into an error:
Error the coset enumeration has defined more than 256000 cosets: type 'return;' if you want to continue with a new limit of 512000 cosets, type 'quit;' if you want to quit the coset enumeration, type 'maxlimit := 0; return;' in order to continue without a limit, ... (a few lines follow)
This can happen if the coset enumeration algorithm tries to enumerate the cosets of a subgroup with infinite index. This situation can also occur with other operations.
You can leave this break loop by entering the command quit; or by
clicking Leave Break Loop in the Run menu of the main XGAP
window.
Earlier you have computed the subgroups of index at most 5. There
is one normal subgroup of index 2 belonging to vertex 6 and one of
index 4 belonging to vertex 8. There is no line between those
two vertices. Select both and click on Compare Subgroups in the
Subgroups menu. A line appears and the line between vertices 8 and
G vanishes. The reason for this is, that the
LowIndexSubgroupsFpGroup call did not deliver the complete inclusion
info. This can always happen for finitely presented groups in XGAP.
In this case you have to compare the subgroups manually by
Compare Subgroups. Note that this can mean large computations, especially if
the indices are huge.
Now select vertex 10 and choose Cores from the Subgroups menu.
You will get a new vertex 12 for an index 24 subgroup. Select the
vertices 12 and G and choose Intermediate Subgroups from the
Subgroups menu. You will get lots of new vertices. Note that some
of them are duplicates of those which were already in the lattice.
This is because comparison of subgroups can be quite expensive and is
therefore not performed automatically in the case of finitely
presented groups.
Select all vertices with a rubber band (click into the top left corner
of the sheet, hold down the mouse and move the pointer to the lower
right corner, then release the mouse button), and choose
Compare Subgroups from the Subgroups menu. A few vertices will disappear
and you get some messages in the GAP window about merging of
vertices.
The display is also not fully correct with respect to conjugacy
classes. IntermediateSubgroups does not return the complete
information about conjugacy of subgroups. Because also conjugacy tests
can be very expensive, they are also not performed automatically for
finitely presented groups. Select Test Conjugacy from the
Subgroups menu to trigger this test manually (note that all
vertices are still selected!). The vertices belonging to conjugate
subgroups are arranged together and if you move those containing the
normal subgroup of index 24 above this one you recognize the
subgroup lattice of the symmetric group on 4 points above that
normal subgroup.
This is now the end of our partial investigation of the (partial)
subgroup lattice of K3, close the graphic sheet by selecting close
graphic sheet from the Sheet menu.
This section describes the investigation of the following finitely presented
group:
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Define the group and open the subgroup lattice window:
gap> f := FreeGroup(2); <free group on the generators [ f1, f2 ]> gap> g := f/[f.1^6]; <fp group on the generators [ f1, f2 ]> gap> s := GraphicSubgroupLattice(g); <graphic subgroup lattice "GraphicSubgroupLattice">
First compute prime quotients by Prime Quotient in the Subgroups
menu. You are asked for a prime and a class. Enter 2 and 7
respectively. You get lots of output in the GAP command window and
seven new vertices. Some of the corresponding subgroups have huge
indices. Note that these groups are only represented as kernels of
epimorphisms within GAP. So explicit calculation of a coset table
or a presentation could take very long or be absolutely impossible!
Now compute epimorphisms onto the symmetric group on 3 points by
Epimorphisms (GQuotients) in the Subgroups menu, but use a polycyclic
presentation as follows (the reason for this will be explained below):
gap> IdGroup(SymmetricGroup(3)); [ 6, 1 ] gap> s3 := SmallGroup(6,1); <pc group with 2 generators> gap> IMAGE_GROUP := s3;;
This first determines the identification number of the symmetric group
on 3 points within the small groups library, and then fetches this
group as a polycyclic group. For groups of size less than 1000 this
is often a good way to get a polycyclic presentation. Note that
SymmetricGroup(3) leads to a permutation group. The last statement
stores the group in a variable which can be used by XGAP.
Select vertex G, then click on Epimorphisms (GQuotient) in the
Subgroups menu and select User defined in the window that pops up.
This will always use the group stored in the global variable
IMAGE_GROUP. GAP finds three epimorphisms. Display the three
kernels by selecting display in the epimorphisms window.
Note that the new vertex 9 will be drawn on the line between vertices 2 and 3 because there is not yet a vertex in the level corresponding to index 6. You can move it aside by dragging it with the mouse to some better position within its level.
Now select vertices 8 and 11 and calculate the intersection of the two subgroups of indices 137438953472 and 6 respectively. GAP can calculate this intersection by calculating the subdirect product of the image groups of the epimorphisms (the index of the subgroup belonging to vertex 12 in G is 412316860416 which is three times of the index of the subgroup belonging to vertex 11). Note that this subdirect product can only be calculated because the two image groups are polycyclic groups. This is the reason why we needed S3 as polycyclic group earlier.
This is now the end of our partial investigation of the (partial)
subgroup lattice of G, close the graphic sheet by selecting close
graphic sheet from the Sheet menu.
This section describes an investigation of an (infinite) space group provided by the catalogue CRYSTCAT of crystallographic groups and using the package CRYST. Here you will see that subgroups of finite index and of finite size can occur at the same time in a graphic subgroup lattice of an infinite group. Note that you have to install these packages to try this example.
Load the packages, define the group and open the subgroup lattice window:
gap> LoadPackage("crystcat");;
gap> LoadPackage("cryst");;
gap> g := SpaceGroupBBNWZ(4,6,3,1,2);
SpaceGroupOnRightBBNWZ( 4, 6, 3, 1, 2 )
gap> s := GraphicSubgroupLattice(g);
<graphic subgroup lattice "GraphicSubgroupLattice of SpaceGroupOnRightBBNWZ( 4, \
6, 3, 1, 2 )">
This fetches the space group of dimension 4, associated crystal system number 6, Q-class 3, Z-class 1, and space group type 2 (see the CRYSTCAT documentation for an explanation of this).
Now we calculate some maximal subgroups with finite index and choose three of them:
gap> m := MaximalSubgroupClassReps(g,rec(latticeequal := true));;
gap> mm := m{[1..3]};
[ <matrix group with 7 generators>, <matrix group with 7 generators>,
<matrix group with 7 generators> ]
Again refer to the CRYST package documentation for an
explanation of these commands. Insert this list of three subgroups
into the lattice by selecting InsertSubgroups from GAP from the
Subgroups menu.
Next calculate subgroups of infinite index but with finite size:
gap> w := WyckoffPositions(g);;
gap> ww := w{[1..3]};
[ < Wyckoff position, point group 11, translation := [ 0, 1/2, 0, 0 ],
basis := [ ] >
, < Wyckoff position, point group 11, translation := [ 0, 1/2, 0, 1/2 ],
basis := [ ] >
, < Wyckoff position, point group 11, translation := [ 0, 1/2, 1/2, 0 ],
basis := [ ] >
]
gap> www := List(ww,WyckoffStabilizer);
[ <matrix group with 3 generators>, <matrix group with 3 generators>,
<matrix group with 3 generators> ]
Insert these subgroups into the lattice by selecting
InsertSubgroups from GAP from the Subgroups menu. They will be
inserted in the level for groups of size 8.
Now you can compute the intersection of a subgroup with finite index
and a subgroup with finite size, select for example vertex 2 and vertex
5 and choose Intersection from the Subgroups menu. You get a new
vertex representing a subgroup of size 4.
If you now calculate the centralizers of the fifteen latticeequal maximal subgroups from above, you get among them four non-trivial cyclic subgroups:
gap> c := List(m,x->Centralizer(g,x)); [ <matrix group with 1 generators>, Group([ ]), Group([ ]), Group([ ]), <matrix group with 1 generators>, Group([ ]), Group([ ]), <matrix group with 1 generators>, Group([ ]), <matrix group with 1 generators>, Group([ ]), Group([ ]), Group([ ]), Group([ ]), Group([ ]) ]
Insert these into the graphic sheet by selecting Insert Vertices from GAP
from the Subgroups menu. You
will get four different new vertices representing groups with infinite
index and infinite size. Each such vertex is placed into a level on
its own, which is marked by [ H1, n] where n is replaced
with subsequent natural numbers (see section levelsintro for details
about levels). ``H1'' means Hirsch length 1, that is, each subnormal series
of the group contains one and only one infinite cycle. In fact, since
these are subgroups of space groups, it indicates that the translation
subgroup is of dimension 1.
(Note that by calculating the point groups of these centralizers you can in fact see, that the infinite cyclic groups consist of translations only.)
Take two of these centralizers and calculate the closure by selecting
Closure from the Subgroups menu. You will get a new subgroup
of Hirsch length 2, which is also placed on a level of its own. Next
select three of them, and calculate the closure. What do you observe?
Also, select all four of them and calculate the closure. This time
you get a subgroup of index 16, hence its level is marked by this finite
index rather than a Hirsch number (which would be 4 here). Note that
the finite index is used rather than the Hirsch length for this placement.
Finally, check, whether the centralizers are normal in the whole space
group by clicking on the vertices with the right mouse button and
choosing IsNormal in the ``Information'' window, which springs up.
Now form the closures of each of them with each of the size 8 point
stabilizers. You will get some other subgroups of Hirsch length 1.
Both the centralizers and the point stabilizers are abelian. Is this also
true for the closures?
This is now the end of our partial investigation of the (partial)
subgroup lattice of G, close the graphic sheet by selecting close
graphic sheet from the Sheet menu.
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XGAP manual