<html><head><title>[xgap] 4.6 A Partial Subgroup Lattice of a Space Group</title></head>
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<h1>4.6 A Partial Subgroup Lattice of a Space Group</h1><p>
<p>
This section describes an investigation of an (infinite) space group
provided by the catalogue CRYSTCAT of crystallographic groups and
using the share package CRYST. Here you will see that subgroups of
finite index and of finite size can occur at the same time in a
graphic subgroup lattice of an infinite group. Note that you have to
install these share packages to try this example.
<p>
Load the share packages, define the group and open the subgroup
lattice window:
<p>
<pre>
gap> RequirePackage("crystcat");;
gap> RequirePackage("cryst");;
gap> g := SpaceGroupBBNWZ(4,6,3,1,2);
SpaceGroupOnRightBBNWZ( 4, 6, 3, 1, 2 )
gap> s := GraphicSubgroupLattice(g);
<graphic subgroup lattice "GraphicSubgroupLattice of SpaceGroupOnRightBBNWZ( 4, \
6, 3, 1, 2 )">
</pre>
<p>
This fetches the space group of dimension <var>4</var>, associated crystal
system number <var>6</var>, <var><font face="helvetica,arial">Q</font></var>-class 3, <var><font face="helvetica,arial">Z</font></var>-class <var>1</var>, and space group type
<var>2</var> (see the CRYSTCAT documentation for an explanation of this).
<p>
Now we calculate some maximal subgroups with finite index and choose three of
them:
<p>
<pre>
gap> m := MaximalSubgroupClassReps(g,rec(latticeequal := true));;
gap> mm := m{[1..3]};
[ <matrix group with 7 generators>, <matrix group with 7 generators>,
<matrix group with 7 generators> ]
</pre>
<p>
Again refer to the CRYST share package documentation for an
explanation of these commands. Insert this list of three subgroups
into the lattice by selecting <code>InsertSubgroups from GAP</code> from the
<code>Subgroups</code> menu.
<p>
Next calculate subgroups of infinite index but with finite size:
<p>
<pre>
gap> w := WyckoffPositions(g);;
gap> ww := w{[1..3]};
[ < Wyckoff position, point group 11, translation := [ 0, 1/2, 0, 0 ],
basis := [ ] >
, < Wyckoff position, point group 11, translation := [ 0, 1/2, 0, 1/2 ],
basis := [ ] >
, < Wyckoff position, point group 11, translation := [ 0, 1/2, 1/2, 0 ],
basis := [ ] >
]
gap> www := List(ww,WyckoffStabilizer);
[ <matrix group with 3 generators>, <matrix group with 3 generators>,
<matrix group with 3 generators> ]
</pre>
<p>
Insert these subgroups into the lattice by selecting
<code>InsertSubgroups from GAP</code> from the <code>Subgroups</code> menu. They will be
inserted in the level for groups of size 8.
<p>
Now you can compute the intersection of a subgroup with finite index
and a subgroup with finite size, select for example vertex <var>2</var> and vertex
<var>5</var> and choose <code>Intersection</code> from the <code>Subgroups</code> menu. You get a new
vertex representing a subgroup of size <var>4</var>.
<p>
If you now calculate the centralizers of the fifteen latticeequal maximal
subgroups from above, you get among them four non-trivial cyclic subgroups:
<p>
<pre>
gap> c := List(m,x->Centralizer(g,x));
[ <matrix group with 1 generators>, Group([ ]), Group([ ]), Group([ ]),
<matrix group with 1 generators>, Group([ ]), Group([ ]),
<matrix group with 1 generators>, Group([ ]),
<matrix group with 1 generators>, Group([ ]), Group([ ]), Group([ ]),
Group([ ]), Group([ ]) ]
</pre>
<p>
Insert these into the graphic sheet by selecting <code>Insert Vertices from GAP</code>
from the <code>Subgroups</code> menu. You
will get four different new vertices representing groups with infinite
index <strong>and</strong> infinite size. Each such vertex is placed into a level on
its own, which is marked by <code>[ H1, </code><var>n</var><code>]</code> where <var>n</var> is replaced
with subsequent natural numbers (see section <a href="C005S003.htm">levelsintro</a> for details
about levels). ``H1'' means Hirsch length 1, that is, each subnormal series
of the group contains one and only one infinite cycle. In fact, since
these are subgroups of space groups, it indicates that the translation
subgroup is of dimension 1.
<p>
(Note that by calculating the point groups of these centralizers you can
in fact see, that the infinite cyclic groups consist of translations only.)
<p>
Take two of these centralizers and calculate the closure by selecting
<code>Closure</code> from the <code>Subgroups</code> menu. You will get a new subgroup
of Hirsch length 2, which is also placed on a level of its own. Next
select three of them, and calculate the closure. What do you observe?
Also, select all four of them and calculate the closure. This time
you get a subgroup of index 16, hence its level is marked by this finite
index rather than a Hirsch number (which would be 4 here). Note that
the finite index is used rather than the Hirsch length for this placement.
<p>
Finally, check, whether the centralizers are normal in the whole space
group by clicking on the vertices with the right mouse button and
choosing <code>IsNormal</code> in the ``Information'' window, which springs up.
Now form the closures of each of them with each of the size 8 point
stabilizers. You will get some other subgroups of Hirsch length 1.
Both the centralizers and the point stabilizers are abelian. Is this also
true for the closures?
<p>
This is now the end of our partial investigation of the (partial)
subgroup lattice of <var>G</var>, close the graphic sheet by selecting <code>close
graphic sheet</code> from the <code>Sheet</code> menu.
<p>
<p>
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<P>
<address>xgap manual<br>Mai 2003
</address></body></html>
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