<html><head><title>[Nilmat] 3 Examples</title></head> <body text="#000000" bgcolor="#ffffff"> [<a href = "chapters.htm">Up</a>] [<a href ="CHAP002.htm">Previous</a>] [<a href ="CHAP004.htm">Next</a>] [<a href = "theindex.htm">Index</a>] <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> [<a href = "chapters.htm">Up</a>] [<a href ="CHAP002.htm">Previous</a>] [<a href ="CHAP004.htm">Next</a>] [<a href = "theindex.htm">Index</a>] <P> <address>Nilmat manual<br>June 2007 </address></body></html>