[Nilmat] 3 Examples

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# 3 Examples

### Sections

1. Constructing some nilpotent matrix groups
2. Testing nilpotency and other functions
3. Using the library of primitive nilpotent groups

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

## 3.1 Constructing some nilpotent matrix groups

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

The group `g1` is a subgroup of GL(52,33) generated by 7 matrices.

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

The group `g2` is a subgroup of GL(180,112) generated by 41 matrices.

    gap> MaximalAbsolutelyIrreducibleNilpotentMatGroup(210,2,10);
    fail

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.

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

Here `g3` is a monomial nilpotent subgroup of GL(450,Q).

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

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

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

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

## 3.2 Testing nilpotency and other functions

We now illustrate use of the functions `IsNilpotentMatGroup`, `SylowSubgroupsOfNilpotentFFMatGroup`, `IsFiniteNilpotentMatGroup`, `SizeOfNilpotentMatGroup`, and `IsCompletelyReducibleNilpotentMatGroup`.

    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

## 3.3 Using the library of primitive nilpotent groups

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

    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

The lists `L1` and `L2` contain only abelian groups, while `L3` and `L4` contain non-abelian nilpotent groups.

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