[xgap] 4 Subgroup Lattices - Examples

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# 4 Subgroup Lattices - Examples

### Sections

1. The Subgroup Lattice of the Dihedral Group of Order 8
2. A Partial Subgroup Lattice of the Symmetric Group on 6 Points
3. A Partial Subgroup Lattice of the Cavicchioli Group
4. A Partial Subgroup Lattice of the Trefoil Knot Group
5. A Partial Subgroup Lattice of a Finitely Presented Group
6. A Partial Subgroup Lattice of a Space Group

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.

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