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# 5.3 GraphicSubgroupLattice, Labelling of Levels

labellevelsintro We intend to represent subgroups of the same index at the same height of the graphic lattice. For this purpose we introduce ``levels'' as horizontal slices of a graphic subgroup lattice.

All vertices for subgroups of a certain finite index are placed in exactly one common level. But what about infinite indices?

Every level has a ``level parameter''. There are three types of levels: ``finite index'', ``finite size'', and ``infinity''. ``finite index'' means, that the index of the subgroups in this level is a certain, finite natural number, the level parameter is exactly this number. ``finite size'' means, that the size of the subgroups in this level is finite and the index is either not known (to GAP) or `infinity`. The level parameter is the size but with a negative sign. If the index is `infinity` and the size is not known, the third type applies: ``infinity''. In each ``infinity'' level only one vertex can exist.

This means that ``finite index'' takes precedence over ``finite size'' and ``infinity'', and ``finite size'' takes precedence over ``infinity'' if XGAP is in doubt which level to choose for a new vertex.

If the group in question is a space group provided by the CRYST share package, levels of type ``infinity'' are also labelled by the Hirsch length of the subgroup, which is the number of infinite cyclic factors in any given subnormal series. Note that this is only implemented for space groups as of this writing, although the Hirsch length is defined for a wider range of groups.

For every graphic subgroup lattice the levels are partially ordered by the following rules:

- A ``finite index'' level is greater than an ``infinity'' level.

- An ``infinity'' is greater than a ``finite size'' level.

- The ``finite index'' levels are totally ordered by descending indices.

- The ``finite size'' levels are totally ordered by ascending sizes.

- For space groups the ``infinity'' levels are partially ordered by the Hirsch lengths of the subgroups.

Note that in general different ``infinity'' levels are not comparable in this partial order!

On the screen, the levels are of course always totally ordered with respect to their height. XGAP ensures that this total ordering is always compatible with the abovementioned partial ordering. This means, that the user can permute ``infinity'' levels, as long as this does not violate the partial order by the Hirsch lengths and the principle, that a vertex ``containing'' another one is drawn higher on the screen.

Note that the level a vertex belongs to will be changed, when new information about the subgroup is available. This happens if the user chooses any entry in the information menu and new information is available thereafter.

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