The Subgroups
menu will be pulled down if you place the pointer inside
the Subgroups
button and press the left mouse button. Keep the button
down and choose an entry by moving the pointer on top of this entry.
Release the mouse button to select an entry.
Note that you can also get the Subgroups
menu as a popup menu by
clicking with the right mouse button into the graphic sheet of the subgroup
lattice, but not on a vertex.
The result of a computation from any of the following entries is colored green, if your screen supports color. In most cases there will also be short information message in the GAP window about the result.
Note that some of the menu entries make it necessary to compute presentations of subgroups using a modified Todd-Coxeter algorithm. This can be very time consuming and in some cases even impossible, if the index is too high.
In the following descriptions, we use ``vertices'' as abbreviation for ``subgroups associated with vertices''.
Abelian Prime Quotient
pops up a dialog box asking for a prime p. It then computes and
displays the largest elementary abelian p quotient of the selected
vertex. If no presentation for the subgroup associated to the vertex is
known a presentation is first computed using a modified Todd-Coxeter
algorithm. It then calls PrimeQuotient
to compute the largest
elementary abelian quotient. Abelian PrimeQuotient
requires exactly one
selected vertex.
All Overgroups
computes and displays all overgroups of the selected vertex. It first
computes the permutation action of the whole group on the cosets of the
subgroup associated with the selected vertex and then searches for all
block systems. If the subgroup of the selected vertex is normal, then
everything is calculated within the (finite) factor group in a better
representation. All Overgroups
requires exactly one selected vertex.
Closure
computes and displays the common closure of the selected vertices. Requires at least one selected vertex. See also ClosureGroup in the GAP reference manual.
Compare Subgroups
A non-empty set of vertices must be selected to choose this menu entry. All subgroups belonging to these vertices are compared pairwise, and the inclusion information is displayed in the lattice. It may happen that two or more vertices are merged if GAP notices, that the subgroups are equal.
Conjugacy Class
computes and displays the conjugacy class of the selected vertex.
Conjugacy Class
requires exactly one selected vertex.
Cores
computes and displays the cores of the selected vertices. Cores
requires at least one selected vertex.
Derived Subgroups
computes and displays the derived subgroups of the selected vertices.
If applied to a proper subgroup of the whole group it will only
display those derived subgroups whose index is finite. Derived
Subgroups
requires at least one selected vertex.
Epimorphisms (GQuotients)
pops up another menu. Requires exactly one selected vertex.
Sym(n) Alt(n) PSL(d,q) Library User Defined
Click on any of these entries to try to find a quotient isomorphic to the
symmetric group (Sym(n)
), the alternating group (Alt(n)
), the projective
special linear group (PSL(d,q)
), a group in a library supplied with
XGAP (this will pop up a file selector), or a user defined group stored
in the variable IMAGE_GROUP
. After supplying additional parameters, for
example, the degree of the symmetric group or the dimension and field of
PSL using dialog boxes, the corresponding entry will change, for example
to something like
Sym(3) 3 found
After one or more quotients were found click display to display them.
Note that in XGAP4 in fact the kernel of the epimorphism is marked whereas in XGAP3 this was not the case, even though the XGAP3 manual stated this.
In fact in XGAP3 a stabilizer of a permutation action on an orbit was put into the lattice.
In case that the image of the epimorphism is a permutation group you can get this functionality by clicking on display point stabilizer instead of display.
Intermediate Subgroups
computes and displays all intermediate subgroups between two selected groups. Requires exactly two selected vertices. See also IntermediateSubgroups in the GAP reference manual.
Intersection
computes and displays the common intersection of the selected vertices. Requires at least one selected vertex. See also Intersection in the GAP reference manual.
Intersections
computes and displays the pairwise intersections of the selected
vertices. Intersections
requires at least two selected vertices.
Low Index Subgroups
pops up a dialog box asking for index limit k. It will then do a low
index subgroup search for subgroups of index at most k of the selected
vertex using LowIndexSubgroupsFpGroup
. If no presentation for the
subgroup associated to the vertex is known a presentation is first
computed using a modified Todd-Coxeter algorithm. Low Index Subgroups
requires exactly one selected vertex.
Normalizers
computes and displays the normalizers of the selected vertices.
Normalizers
requires at least one selected vertex.
Prime Quotient
pops up a dialog box asking for a prime p and another dialog box asking
for a class c. It then computes and displays the largest p-quotient
of class c of the selected vertex. If no presentation for the subgroup
associated to the vertex is known a presentation is first computed using
a modified Todd-Coxeter algorithm. It then calls PrimeQuotient
.
Prime Quotient
requires exactly one selected vertex.
Test Conjugacy
walks through all levels and tests for all pairs of classes, that contain a
selected vertex, whether the groups in the classes are conjugates. If so,
the classes are merged. After these calculations Rearrange Classes
is
called. Note that conjugacy calculations can take lots of time for finitely
presented groups!
SelectedGroups to GAP
If the user selects this menu entry, the subgroups belonging to the
selected vertices are put into a list which is stored into the variable
last
. This is equivalent to the statement SelectedGroups(sheet);;
if
sheet
contains the graphic sheet object. If XGAP logging is on, then
the normal GAP logging via LogTo
is also directed to the XGAP log
file.
InsertVertices from GAP
If the user selects this menu entry, the value of the variable last
is
used to insert new vertices into the graphic sheet. If last
is equal to
one subgroup, it is inserted via InsertVertex
. If last
is a list of
subgroups, InsertVertex
is called for all those subgroups. There is no
error issued if one of the entries of last
is no subgroup. If XGAP
logging is on, then the normal GAP logging via LogTo
is switched off!
The idea of this is to switch the logging temporarily from XGAP logging
to normal GAP logging between two clicks to ``SelectedGroups to GAP''
and ``InsertVertices from GAP'' respectively.
Start Logging
After clicking on this menu entry the user is prompted for a filename. From this point on all commands issued via mouse clicks in the subgroup menu are logged into that file, such that one can afterwards see ``what happened'' in the XGAP session. The information displayed is the same as in the info displays in the GAP window.
Stop Logging
A click onto this menu entry stops the XGAP logging.
These menu entries represent only a small selection of the functions of
GAP which the authors of XGAP considered most frequently used. You
can calculate other subgroups from
the GAP command window. See sections gapxgap and
xgapgap for examples how to transfer information from the graphical
lattice of XGAP to GAP (via SelectedGroups
, see
GraphicSubgroupLattice, Selecting Vertices) and vice versa (via
SelectGroups
, see GraphicSubgroupLattice, Selecting Vertices, and
InsertVertex
, see GraphicSubgroupLattice, Inserting Vertices).
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xgap manual