1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
|
<Chapter Label="Libraries and examples of pcp-groups">
<Heading>Libraries and examples of pcp-groups</Heading>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Libraries of various types of polycyclic groups">
<Heading>Libraries of various types of polycyclic groups</Heading>
There are the following generic pcp-groups available.
<ManSection>
<Func Name="AbelianPcpGroup" Arg="n[, rels]"/>
<Func Name="AbelianPcpGroup" Arg="rels" Label="rels only"/>
<Description>
constructs the abelian group on <A>n</A> generators such that
generator <M>i</M> has order <M>rels[i]</M>. If this order is infinite,
then <M>rels[i]</M> should be either unbound or 0 or infinity.
If <A>n</A> is not provided then the length of <A>rels</A> is used.
If <A>rels</A> is omitted then all generators will have infinite order.
</Description>
</ManSection>
<ManSection>
<Func Name="DihedralPcpGroup" Arg="n"/>
<Description>
constructs the dihedral group of order <A>n</A>. If <A>n</A> is an odd
integer, then 'fail' is returned. If <A>n</A> is zero or not an
integer, then the infinite dihedral group is returned.
</Description>
</ManSection>
<ManSection>
<Func Name="UnitriangularPcpGroup" Arg="n, c"/>
<Description>
returns a pcp-group isomorphic to the group of upper triangular
in <M>GL(n, R)</M> where <M>R = &ZZ;</M> if <M>c = 0</M> and <M>R = \mathbb{F}_p</M> if <M>c = p</M>.
The natural unitriangular matrix representation of the returned
pcp-group <M>G</M> can be obtained as <M>G!.isomorphism</M>.
</Description>
</ManSection>
<ManSection>
<Func Name="SubgroupUnitriangularPcpGroup" Arg="mats"/>
<Description>
<A>mats</A> should be a list of upper unitriangular <M>n \times n</M>
matrices over <M>&ZZ;</M> or over <M>\mathbb{F}_p</M>. This function returns the
subgroup of the corresponding 'UnitriangularPcpGroup' generated
by the matrices in <A>mats</A>.
</Description>
</ManSection>
<ManSection>
<Func Name="InfiniteMetacyclicPcpGroup" Arg="n, m, r"/>
<Description>
Infinite metacyclic groups are classified in <Cite Key="B-K00"/>. Every
infinite metacyclic group <M>G</M> is isomorphic to a finitely presented
group <M>G(m,n,r)</M> with two generators <M>a</M> and <M>b</M> and relations of the
form <M>a^m = b^n = 1</M> and <M>[a,b] = a^{1-r}</M>, where (differing from the
conventions used by GAP) we have <M>[a,b] = a b a^-1 b^-1</M>, and <M>m,n,r</M> are three
non-negative integers with <M>mn=0</M> and <M>r</M> relatively prime to <M>m</M>.
If <M>r \equiv -1</M> mod <M>m</M> then <M>n</M> is even, and if <M>r \equiv 1</M> mod
<M>m</M> then <M>m=0</M>. Also <M>m</M> and <M>n</M> must not be <M>1</M>.
<P/>
Moreover, <M>G(m,n,r)\cong G(m',n',s)</M> if and only if <M>m=m'</M>, <M>n=n'</M>,
and either <M>r \equiv s</M> or <M>r \equiv s^{-1}</M> mod <M>m</M>.
<P/>
This function returns the metacyclic group with parameters <A>n</A>,
<A>m</A> and <A>r</A> as a pcp-group with the pc-presentation <M>\langle
x,y | x^n, y^m, y^x = y^r\rangle</M>. This presentation is easily
transformed into the one above via the mapping <M>x \mapsto b^{-1},
y \mapsto a</M>.
</Description>
</ManSection>
<ManSection>
<Func Name="HeisenbergPcpGroup" Arg="n"/>
<Description>
returns the Heisenberg group on <M>2<A>n</A>+1</M> generators as pcp-group.
This gives a group of Hirsch length <M>2<A>n</A>+1</M>.
</Description>
</ManSection>
<ManSection>
<Func Name="MaximalOrderByUnitsPcpGroup" Arg="f"/>
<Description>
takes as input a normed, irreducible polynomial over the integers.
Thus <A>f</A> defines a field extension <A>F</A> over the rationals. This
function returns the split extension of the maximal order <A>O</A> of <A>F</A>
by the unit group <A>U</A> of <A>O</A>, where <A>U</A> acts by right multiplication
on <A>O</A>.
</Description>
</ManSection>
<ManSection>
<Func Name="BurdeGrunewaldPcpGroup" Arg="s, t"/>
<Description>
returns a nilpotent group of Hirsch length 11 which has been
constructed by Burde und Grunewald. If <A>s</A> is not 0, then this
group has no faithful 12-dimensional linear representation.
</Description>
</ManSection>
</Section>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Some asorted example groups">
<Heading>Some assorted example groups</Heading>
The functions in this section provide some more example groups to play
with. They come with no further description and their investigation is
left to the interested user.
<ManSection>
<Func Name="ExampleOfMetabelianPcpGroup" Arg="a, k"/>
<Description>
returns an example of a metabelian group. The input parameters must
be two positive integers greater than 1.
</Description>
</ManSection>
<ManSection>
<Func Name="ExamplesOfSomePcpGroups" Arg="n"/>
<Description>
this function takes values <A>n</A> in 1 up to 16 and returns for each
input an example of a pcp-group. The groups in this example list
have been used as test groups for the functions in this package.
</Description>
</ManSection>
</Section>
</Chapter>
|
