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<h1>3 Examples</h1>

<H3>Sections</H3>
<oL>
<li> <A HREF="CHAP003.htm#SECT001">Constructing some nilpotent matrix groups</a>
<li> <A HREF="CHAP003.htm#SECT002">Testing nilpotency and other functions</a>
<li> <A HREF="CHAP003.htm#SECT003">Using the library of primitive nilpotent groups</a>
</ol>

<a name = "I0"></a>

In this chapter we give some examples of computing with the Package
Nilmat.


<h2><a name="SECT001">3.1 Constructing some nilpotent matrix groups</a></h2>

<pre>
gap> g1 := MaximalAbsolutelyIrreducibleNilpotentMatGroup(52,3,3);
<matrix group with 7 generators>
</pre>

The group <code>g1</code> is a subgroup of GL(52,33) generated by 7 matrices.

<pre>
gap> g2 := MaximalAbsolutelyIrreducibleNilpotentMatGroup(180,11,2);
<matrix group with 41 generators>
</pre>

The group <code>g2</code> is a subgroup of GL(180,112) generated by 41 matrices.

<pre>
gap> MaximalAbsolutelyIrreducibleNilpotentMatGroup(210,2,10);
fail
</pre>

In this third example, absolutely irreducible nilpotent subgroups of
GL(210,210) do not exist, because the degree of the matrices
and the field size are both even.

<pre>
gap> g3 := MonomialNilpotentMatGroup(450);
<matrix group with 24 generators>
</pre>

Here <code>g3</code> is a monomial nilpotent subgroup of GL(450,Q).

<pre>
gap> g4 := ReducibleNilpotentReducibleMatGroup(3,180,11,2);
<matrix group with 82 generators>
</pre>

Here <tt>g</tt><tt>4</tt> < GL(540,112) is the Kronecker product of a
unipotent subgroup of GL(3,112) and the group <code>g2</code>.

<pre>
gap> g5 := ReducibleNilpotentMatGroup(7,36);
<matrix group with 72 generators>
</pre>

Here <tt>g</tt><tt>5</tt> < GL(252, Q) is a reducible nilpotent group constructed
as the Kronecker product of a unipotent subgroup of GL(7,Q) with
<code>MonomialNilpotentMatGroup(36)</code>.


<h2><a name="SECT002">3.2 Testing nilpotency and other functions</a></h2>

We now illustrate use of the functions
<code>IsNilpotentMatGroup</code>,
<code>SylowSubgroupsOfNilpotentFFMatGroup</code>,
<code>IsFiniteNilpotentMatGroup</code>,
<code>SizeOfNilpotentMatGroup</code>, and
<code>IsCompletelyReducibleNilpotentMatGroup</code>.

<pre>
gap> IsNilpotentMatGroup(GL(200,Integers));
false

gap> IsNilpotentMatGroup(GL(150,11^3));
false

gap> g6 := MaximalAbsolutelyIrreducibleNilpotentMatGroup(127,2,7);
<matrix group with 3 generators>
gap> IsNilpotentMatGroup(g6);
true

gap> g7 := MonomialNilpotentMatGroup(350);
<matrix group with 6 generators>
gap> IsNilpotentMatGroup(g7);
true
gap> IsFiniteNilpotentMatGroup(g7);
true

gap> g8 := ReducibleNilpotentMatGroup(6,35);
<matrix group with 5 generators>
gap> IsNilpotentMatGroup(g8);
true
gap> IsFiniteNilpotentMatGroup(g8);
false

gap> g9 := ReducibleNilpotentMatGroup(2,36,5,2);
<matrix group with 21 generators>
gap> SylowSubgroupsOfNilpotentFFMatGroup(g9);
[ <matrix group with 5 generators>, <matrix group with 6 generators>,
 <matrix group with 1 generators> ]
gap> IsCompletelyReducibleNilpotentMatGroup(g9);
false

gap> g10 := MaximalAbsolutelyIrreducibleNilpotentMatGroup(24,5,2);
<matrix group with 17 generators>
gap> SizeOfNilpotentMatGroup(g10);
173946175488
gap> IsCompletelyReducibleNilpotentMatGroup(g10);
true

gap> g11 := MonomialNilpotentMatGroup(96);
<matrix group with 31 generators>
gap> SizeOfNilpotentMatGroup(g11);
6442450944
gap> IsCompletelyReducibleNilpotentMatGroup(g11);
true
</pre>


<h2><a name="SECT003">3.3 Using the library of primitive nilpotent groups</a></h2>

This section gives examples of applying the functions from the
Nilmat library of primitive nilpotent subgroups of GL(n,q).

<pre>
gap> L0 := NilpotentPrimitiveMatGroups(2,3,1);
[ Group([ [ [ 0*Z(3), Z(3)^0 ], [ Z(3)^0, Z(3)^0 ] ] ]),
 Group([ [ [ Z(3)^0, 0*Z(3) ], [ 0*Z(3), Z(3)^0 ] ],
 [ [ Z(3), Z(3)^0 ], [ Z(3), Z(3) ] ],
 [ [ Z(3)^0, 0*Z(3) ], [ 0*Z(3), Z(3) ] ] ]),
 Group([ [ [ Z(3)^0, 0*Z(3) ], [ 0*Z(3), Z(3)^0 ] ],
 [ [ 0*Z(3), Z(3)^0 ], [ Z(3), 0*Z(3) ] ],
 [ [ Z(3), Z(3) ], [ Z(3), Z(3)^0 ] ] ]) ]
gap> SizesOfNilpotentPrimitiveMatGroups(2,3,1);
[ 8, 16, 8 ]
gap> List(L0,Size);
[ 8, 16, 8 ]

gap> L1 := NilpotentPrimitiveMatGroups(2,2,10);;
gap> Length(L1);
40
gap> Size(L1[38]);
209715
gap> s := SizesOfNilpotentPrimitiveMatGroups(2,2,10);
[ 5, 15, 25, 41, 55, 75, 123, 155, 165, 205, 275, 451, 465, 615, 775, 825,
 1025, 1271, 1353, 1705, 2255, 2325, 3075, 3813, 5115, 6355, 6765, 8525,
 11275, 13981, 19065, 25575, 31775, 33825, 41943, 69905, 95325, 209715,
 349525, 1048575 ]

gap> L2 := NilpotentPrimitiveMatGroups(55,3,1);;
gap> Length(L2);
114

gap> L3 := NilpotentPrimitiveMatGroups(6,3,3);;
gap> Length(L3);
110

gap> L4 := NilpotentPrimitiveMatGroups(22,11,1);;
gap> Length(L4);
1002
</pre>

The lists <code>L1</code> and <code>L2</code> contain only abelian groups, while <code>L3</code> and
<code>L4</code> contain non-abelian nilpotent groups.

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