[xgap] 5.12 GraphicSubgroupLattice for FpGroups, Subgroups Menu

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# 5.12 GraphicSubgroupLattice for FpGroups, Subgroups Menu

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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