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<html><head><title>[aclib] 4 Example computations with almost crystallographic groups</title></head>
<body text="#000000" bgcolor="#ffffff">
[<a href = "chapters.htm">Up</a>] [<a href ="CHAP003.htm">Previous</a>] [<a href = "theindex.htm">Index</a>]
<h1>4 Example computations with almost crystallographic groups</h1><p>
<P>
<H3>Sections</H3>
<oL>
<li> <A HREF="CHAP004.htm#SECT001">Example computations I</a>
<li> <A HREF="CHAP004.htm#SECT002">Example computations II</a>
<li> <A HREF="CHAP004.htm#SECT003">Example computations III</a>
</ol><p>
<p>
<p>
<h2><a name="SECT001">4.1 Example computations I</a></h2>
<p><p>
Using the functions available for pcp groups in the share package
<font face="Gill Sans,Helvetica,Arial">polycyclic</font> it is now easy to redo some of the calculations of
<a href="biblio.htm#KD"><cite>KD</cite></a>. As a first example we check whether the groups indicated
as torsion free in <a href="biblio.htm#KD"><cite>KD</cite></a> are also determined as torsion free
ones by <font face="Gill Sans,Helvetica,Arial">GAP</font>. In <a href="biblio.htm#KD"><cite>KD</cite></a> these almost Bieberbach groups are listed as
``AB-groups''. So for type ``013'' these are the groups with parameters
<var>(k,0,1,0,1,0)</var> where <var>k</var> is an even integer. Let's look at some examples
in <font face="Gill Sans,Helvetica,Arial">GAP</font>:
<p>
<pre>
gap> G:=AlmostCrystallographicPcpDim4("013",[8,0,1,0,1,0]);
Pcp-group with orders [ 2, 2, 0, 0, 0, 0 ]
gap> IsTorsionFree(G);
true
gap> G:=AlmostCrystallographicPcpDim4("013",[9,0,1,0,1,0]);
Pcp-group with orders [ 2, 2, 0, 0, 0, 0 ]
gap> IsTorsionFree(G);
false
</pre>
<p>
Further, there is also some cohomology information in the tables
of <a href="biblio.htm#KD"><cite>KD</cite></a>. In fact, the groups in this library were obtained
as extensions <var>E</var> of the form
<p>
<p><var>
1 rightarrow<font face="helvetica,arial">Z</font>rightarrowE rightarrowQ rightarrow1
<p></var>
<p>
where, in the 4-dimensional case <var>Q = E/langled rangle</var>. The
cohomology information for the particular example above shows that
the groups determined by a parameter set <var>(k<sub>1</sub>,k<sub>2</sub>,k<sub>3</sub>,k<sub>4</sub>,k<sub>4</sub>,k<sub>6</sub>)</var>
are equivalent as extensions to the groups determined by the parameters
<var>(k<sub>1</sub>, k<sub>2</sub> bmod2, k<sub>3</sub> bmod2, k<sub>4</sub> bmod2, k<sub>5</sub> bmod2, 0)</var>. This is
also visible in finding torsion:
<p>
<pre>
gap> G:=AlmostCrystallographicPcpDim4("013",[10,0,2,0,1,0]);
Pcp-group with orders [ 2, 2, 0, 0, 0, 0 ]
gap> IsTorsionFree(G);
false
gap> G:=AlmostCrystallographicPcpDim4("013",[10,0,3,0,1,9]);
Pcp-group with orders [ 2, 2, 0, 0, 0, 0 ]
gap> IsTorsionFree(G);
true
</pre>
<p>
<p>
<h2><a name="SECT002">4.2 Example computations II</a></h2>
<p><p>
The computation of cohomology groups played an important role in the
classification of the almost Bieberbach groups in <a href="biblio.htm#KD"><cite>KD</cite></a>. Using
<font face="Gill Sans,Helvetica,Arial">GAP</font>, it is now possible to check these computations. As an example we
consider the 4-dimensional almost crystallographic groups of type 85 on
page 202 of <a href="biblio.htm#KD"><cite>KD</cite></a>. This group <var>E</var> has 6 generators. In the table, one
also finds the information
<p>
<p><var>
H<sup>2</sup>(Q,<font face="helvetica,arial">Z</font>) = <font face="helvetica,arial">Z</font>oplus(<font face="helvetica,arial">Z</font><sub>2</sub>)<sup>2</sup> oplus<font face="helvetica,arial">Z</font><sub>4</sub>
<p></var>
<p>
for <var>Q=E/langled rangle</var> as above. Moreover, the <var>Q</var>--module <var><font face="helvetica,arial">Z</font></var> is
in fact the group <var>langled rangle</var>, where the <var>Q</var>-action comes from
conjugation inside <var>E</var>. In the case of groups of type 85, <var><font face="helvetica,arial">Z</font></var> is a
trivial <var>Q</var>-module. The following example demonstrates how to (re)compute
this two-cohomology group <var>H<sup>2</sup>(Q,<font face="helvetica,arial">Z</font>)</var>.
<p>
<pre>
gap> G:=AlmostCrystallographicPcpGroup(4, "085", false);
Pcp group with orders [ 2, 4, 0, 0, 0, 0 ]
gap> GroupGeneratedByd:=Subgroup(G, [G.6] );
Pcp group with orders [ 0 ]
gap> Q:=G/GroupGeneratedByd;
Pcp group with orders [ 2, 4, 0, 0, 0 ]
gap> action:=List( Pcp(Q), x -> [[1]] );
[ [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ]
gap> C:=CRRecordByMats( Q, action);;
gap> TwoCohomologyCR( C ).factor.rels;
[ 2, 2, 4, 0 ]
</pre>
<p>
This last line gives us the abelian invariants of the second
cohomology group <var>H<sup>2</sup>(Q,<font face="helvetica,arial">Z</font>)</var>. So we should read this line as
<p>
<p><var>
H<sup>2</sup>(Q,<font face="helvetica,arial">Z</font>) = <font face="helvetica,arial">Z</font><sub>2</sub> oplus<font face="helvetica,arial">Z</font><sub>2</sub> oplus<font face="helvetica,arial">Z</font><sub>4</sub> oplus<font face="helvetica,arial">Z</font>
<p></var>
<p>
which indeed coincides with the information in <a href="biblio.htm#KD"><cite>KD</cite></a>.
<p>
<p>
<h2><a name="SECT003">4.3 Example computations III</a></h2>
<p><p>
As another application of the capabilities of the combination of
<code>aclib</code> and <font face="Gill Sans,Helvetica,Arial">polycyclic</font> we check some computations of <a href="biblio.htm#DM"><cite>DM</cite></a>.
<p>
Section 5 of the paper <a href="biblio.htm#DM"><cite>DM</cite></a> is completely devoted to an example
of the computation of the <var>P</var>-localization of a virtually nilpotent group,
where <var>P</var> is a set of primes. Although it is not our intention to
develop the theory of <var>P</var>-localization of groups at this place, let us
summarize some of the main results concerning this topic here.
<p>
For a set of primes <var>P</var>, we say that <var>n inP</var> if and only if <var>n</var> is
a product of primes in <var>P</var>. A group <var>G</var> is said to be <var>P</var>-local if and
only if the map <var>mu<sub>n</sub>:GrightarrowG: g mapstog<sup>n</sup></var> is bijective for
all <var>n inP'</var>, where <var>P'</var> is the set of all primes not in <var>P</var>. The
<var>P</var>-localization of a group <var>G</var>, is a <var>P</var>-local group <var>G<sub>P</sub></var> together
with a morphism <var>alpha:G rightarrowG<sub>P</sub></var> which satisfy the following
universal property: For each <var>P</var>-local group <var>L</var> and any morphism
<var>varphi: G rightarrowL</var>, there exists a unique morphism <var>psi:G<sub>P</sub>
rightarrowL</var>, such that <var>psicircalpha= varphi</var>.
<p>
This concept of localization is well developed for finite groups and
for nilpotent groups. For a finite group <var>G</var>, the <var>P</var>-localization is
the largest quotient of <var>G</var>, having no elements with an order belonging to
<var>P'</var> (the morphism <var>alpha</var>, mentioned above is the natural projection).
<p>
In <a href="biblio.htm#DM"><cite>DM</cite></a> a contribution is made towards the localization of virtually
nilpotent groups. The theory developed in the paper is then illustrated
in the last section of the paper by means of the computation of the
<var>P</var>-localization of an almost crystallographic group. For their example
the authors have chosen an almost crystallographic group <var>G</var> of dimension 3
and type 17. For the set of parameters <var>(k<sub>1</sub>,k<sub>2</sub>,k<sub>3</sub>,k<sub>4</sub>)</var> they have
considered all cases of the form <var>(k<sub>1</sub>,k<sub>2</sub>,k<sub>3</sub>,k<sub>4</sub>)=(2,0,0,k<sub>4</sub>)</var>.
<p>
Here we will check their computations in two cases <var>k<sub>4</sub>=0</var> and <var>k<sub>4</sub>=1</var>
using the set of primes <var>P={2}</var>. The holonomy group of these almost
crystallographic groups <var>G</var> is the dihedral group <var>calD<sub>6</sub></var> of order
12. Thus there is a short exact sequence of the form
<p><var> 1 rightarrowFitt(G) rightarrowG
rightarrowcalD<sub>6</sub> rightarrow1. <p></var>
<p>
As a first step in their computation, Descheemaeker and Malfait determine
the group <var>I<sub>P'</sub>calD<sub>6</sub></var>, which is the unique subgroup of order 3 in
<var>calD<sub>6</sub></var>. One of the main objects in <a href="biblio.htm#DM"><cite>DM</cite></a> is the group <var>K=p<sup>-1</sup>
(I<sub>P'</sub>calD<sub>6</sub>)</var>, where <var>p</var> is the natural projection of <var>G</var> onto its
holonomy group. It is known that the <var>P</var>-localization of <var>G</var> coincides
with the <var>P</var>-localization of <var>G/gamma<sub>3</sub>(K)</var>, where <var>gamma<sub>3</sub>(K)</var> is the
third term in the lower central series of <var>K</var>. As <var>G/gamma<sub>3</sub>(K)</var> is
finite in this example, we exactly know what this <var>P</var>-localization is.
Let us now show, how GAP can be used to compute this <var>P</var>-localization in
two cases:
<p>
<p>
First case: The parameters are <var>(k<sub>1</sub>,k<sub>2</sub>,k<sub>3</sub>,k<sub>4</sub>)=(2,0,0,0)</var>
<pre>
gap> G := AlmostCrystallographicPcpGroup(3, 17, [2,0,0,0] );
Pcp group with orders [ 2, 6, 0, 0, 0 ]
gap> projection := NaturalHomomorphismOnHolonomyGroup( G );
[ g1, g2, g3, g4, g5 ] -> [ g1, g2, identity, identity, identity ]
gap> F := HolonomyGroup( G );
Pcp group with orders [ 2, 6 ]
gap> IPprimeD6 := Subgroup( F , [F.2^2] );
Pcp group with orders [ 3 ]
gap> K := PreImage( projection, IPprimeD6 );
Pcp group with orders [ 3, 0, 0, 0 ]
gap> PrintPcpPresentation( K );
pcp presentation on generators [ g2^2, g3, g4, g5 ]
g2^2 ^ 3 = identity
g3 ^ g2^2 = g3^-1*g4^-1
g3 ^ g2^2^-1 = g4*g5^-2
g4 ^ g2^2 = g3*g5^2
g4 ^ g2^2^-1 = g3^-1*g4^-1*g5^2
g4 ^ g3 = g4*g5^2
g4 ^ g3^-1 = g4*g5^-2
gap> Gamma3K := CommutatorSubgroup( K, CommutatorSubgroup( K, K ));
Pcp group with orders [ 0, 0, 0 ]
gap> quotient := G/Gamma3K;
Pcp group with orders [ 2, 6, 3, 3, 2 ]
gap> S := SylowSubgroup( quotient, 3);
Pcp group with orders [ 3, 3, 3 ]
gap> N := NormalClosure( quotient, S);
Pcp group with orders [ 3, 3, 3 ]
gap> localization := quotient/N;
Pcp group with orders [ 2, 2, 2 ]
gap> PrintPcpPresentation( localization );
pcp presentation on generators [ g1, g2, g3 ]
g1 ^ 2 = identity
g2 ^ 2 = identity
g3 ^ 2 = identity
</pre>
This shows that <var>G<sub>P</sub>cong<font face="helvetica,arial">Z</font><sub>2</sub><sup>3</sup></var>.
<p>
<p>
Second case: The parameters are <var>(k<sub>1</sub>,k<sub>2</sub>,k<sub>3</sub>,k<sub>4</sub>)=(2,0,0,1)</var>
<pre>
gap> G := AlmostCrystallographicPcpGroup(3, 17, [2,0,0,1]);;
gap> projection := NaturalHomomorphismOnHolonomyGroup( G );;
gap> F := HolonomyGroup( G );;
gap> IPprimeD6 := Subgroup( F , [F.2^2] );;
gap> K := PreImage( projection, IPprimeD6 );;
gap> Gamma3K := CommutatorSubgroup( K, CommutatorSubgroup( K, K ));;
gap> quotient := G/Gamma3K;;
gap> S := SylowSubgroup( quotient, 3);;
gap> N := NormalClosure( quotient, S);;
gap> localization := quotient/N;
Pcp group with orders [ 2, 2, 2 ]
gap> PrintPcpPresentation( localization );
pcp presentation on generators [ g1, g2, g3 ]
g1 ^ 2 = identity
g2 ^ 2 = g3
g3 ^ 2 = identity
g2 ^ g1 = g2*g3
g2 ^ g1^-1 = g2*g3
</pre>
<p>
In this case, we see that <var>G<sub>P</sub>=calD<sub>4</sub></var>.
<p>
<p>
The reader can check that these results coincide with those obtained in
<a href="biblio.htm#DM"><cite>DM</cite></a>. Note also that we used a somewhat different scheme to compute
this localization than the one used in <a href="biblio.htm#DM"><cite>DM</cite></a>. We invite the reader to
check the same computations, tracing exactly the steps made in <a href="biblio.htm#DM"><cite>DM</cite></a>.
<p>
<p>
[<a href = "chapters.htm">Up</a>] [<a href ="CHAP003.htm">Previous</a>] [<a href = "theindex.htm">Index</a>]
<P>
<address>aclib manual<br>August 2025
</address></body></html>
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