[xgap] 4.6 A Partial Subgroup Lattice of a Space Group

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# 4.6 A Partial Subgroup Lattice of a Space Group

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.

Load the share packages, define the group and open the subgroup lattice window:

    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 )">

This fetches the space group of dimension 4, associated crystal system number 6, Q-class 3, Z-class 1, and space group type 2 (see the CRYSTCAT documentation for an explanation of this).

Now we calculate some maximal subgroups with finite index and choose three of them:

    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> ]

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 `InsertSubgroups from GAP` from the `Subgroups` menu.

Next calculate subgroups of infinite index but with finite size:

    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> ]

Insert these subgroups into the lattice by selecting `InsertSubgroups from GAP` from the `Subgroups` menu. They will be inserted in the level for groups of size 8.

Now you can compute the intersection of a subgroup with finite index and a subgroup with finite size, select for example vertex 2 and vertex 5 and choose `Intersection` from the `Subgroups` menu. You get a new vertex representing a subgroup of size 4.

If you now calculate the centralizers of the fifteen latticeequal maximal subgroups from above, you get among them four non-trivial cyclic subgroups:

    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([  ]) ]

Insert these into the graphic sheet by selecting `Insert Vertices from GAP` from the `Subgroups` menu. You will get four different new vertices representing groups with infinite index and infinite size. Each such vertex is placed into a level on its own, which is marked by `[ H1,` n`]` where n is replaced with subsequent natural numbers (see section levelsintro 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.

(Note that by calculating the point groups of these centralizers you can in fact see, that the infinite cyclic groups consist of translations only.)

Take two of these centralizers and calculate the closure by selecting `Closure` from the `Subgroups` 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.

Finally, check, whether the centralizers are normal in the whole space group by clicking on the vertices with the right mouse button and choosing `IsNormal` 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?

This is now the end of our partial investigation of the (partial) subgroup lattice of G, close the graphic sheet by selecting `close graphic sheet` from the `Sheet` menu.

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