<html><head><title>[Nilmat] 3 Examples</title></head>
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<h1>3 Examples</h1><p>
<P>
<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><p>
<p>
<a name = "I0"></a>
In this chapter we give some examples of computing with the Package
<font face="Gill Sans,Helvetica,Arial">Nilmat</font>.
<p>
<p>
<h2><a name="SECT001">3.1 Constructing some nilpotent matrix groups</a></h2>
<p><p>
<pre>
gap> g1 := MaximalAbsolutelyIrreducibleNilpotentMatGroup(52,3,3);
<matrix group with 7 generators>
</pre>
<p>
The group <code>g1</code> is a subgroup of <var>GL(52,3<sup>3</sup>)</var> generated by 7 matrices.
<p>
<pre>
gap> g2 := MaximalAbsolutelyIrreducibleNilpotentMatGroup(180,11,2);
<matrix group with 41 generators>
</pre>
<p>
The group <code>g2</code> is a subgroup of <var>GL(180,11<sup>2</sup>)</var> generated by 41 matrices.
<p>
<pre>
gap> MaximalAbsolutelyIrreducibleNilpotentMatGroup(210,2,10);
fail
</pre>
<p>
In this third example, absolutely irreducible nilpotent subgroups of
<var>GL(210,2<sup>10</sup>)</var> do not exist, because the degree of the matrices
and the field size are both even.
<p>
<pre>
gap> g3 := MonomialNilpotentMatGroup(450);
<matrix group with 24 generators>
</pre>
<p>
Here <code>g3</code> is a monomial nilpotent subgroup of <var>GL(450,<font face="helvetica,arial">Q</font>)</var>.
<p>
<pre>
gap> g4 := ReducibleNilpotentReducibleMatGroup(3,180,11,2);
<matrix group with 82 generators>
</pre>
<p>
Here <var><code>g4</code> < GL(540,11<sup>2</sup>)</var> is the Kronecker product of a
unipotent subgroup of <var>GL(3,11<sup>2</sup>)</var> and the group <code>g2</code>.
<p>
<pre>
gap> g5 := ReducibleNilpotentMatGroup(7,36);
<matrix group with 72 generators>
</pre>
<p>
Here <var><code>g5</code> < GL(252, <font face="helvetica,arial">Q</font>)</var> is a reducible nilpotent group constructed
as the Kronecker product of a unipotent subgroup of <var>GL(7,<font face="helvetica,arial">Q</font>)</var> with
<code>MonomialNilpotentMatGroup(36)</code>.
<p>
<p>
<h2><a name="SECT002">3.2 Testing nilpotency and other functions</a></h2>
<p><p>
We now illustrate use of the functions
<code>IsNilpotentMatGroup</code>,
<code>SylowSubgroupsOfNilpotentFFMatGroup</code>,
<code>IsFiniteNilpotentMatGroup</code>,
<code>SizeOfNilpotentMatGroup</code>, and
<code>IsCompletelyReducibleNilpotentMatGroup</code>.
<p>
<pre>
gap> IsNilpotentMatGroup(GL(200,Rationals));
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>
<p>
<p>
<h2><a name="SECT003">3.3 Using the library of primitive nilpotent groups</a></h2>
<p><p>
This section gives examples of applying the functions from the
<font face="Gill Sans,Helvetica,Arial">Nilmat</font> library of primitive nilpotent subgroups of <var>GL(n,q)</var>.
<p>
<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, 8, 16 ]
gap> List(L0,Size);
[ 8, 8, 16 ]
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(L3);
1002
</pre>
<p>
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.
<p>
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<P>
<address>Nilmat manual<br>June 2007
</address></body></html>
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