Sophie: gap-system-4.4.12-5mdv2010.0 i586

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<html><head><title>[xgap] 5.3 GraphicSubgroupLattice, Labelling of Levels</title></head>
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<h1>5.3 GraphicSubgroupLattice, Labelling of Levels</h1><p>
<p>
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. 
<p>
All vertices for subgroups of a certain finite index are
placed in exactly one common level. But what about infinite indices?
<p>
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 <strong>and</strong> the index is either not known
(to <font face="Gill Sans,Helvetica,Arial">GAP</font>) or <code>infinity</code>. The level parameter is the size but with a
negative sign. If the index is <code>infinity</code> and the size is
not known, the third type applies: ``infinity''. In each ``infinity''
level only one vertex can exist.
<p>
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. 
<p>
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.
<p>
For every graphic subgroup lattice the levels are partially ordered by
the following rules:
<p>
<ul>
<li> A ``finite index'' level is greater than an ``infinity'' level.
<p>
<li> An ``infinity'' is greater than a ``finite size'' level.
<p>
<li> The ``finite index'' levels are totally ordered by descending indices.
<p>
<li> The ``finite size'' levels are totally ordered by ascending sizes. 
<p>
<li> For space groups the ``infinity'' levels are partially ordered by
     the Hirsch lengths of the subgroups.
</ul>
<p>
Note that in general different ``infinity'' levels are not comparable in this
partial order!
<p>
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.
<p>
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.
<p>
<p>
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<P>
<address>xgap manual<br>Mai 2003
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