Quelle ppowerpolypcpgroup.tex Sprache: Latech

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%W ppowerpolypcpgroup.tex GAP documentation Dörte Feichtenschlager
%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Chapter{p-power-poly-pcp-groups}

Eick and Leedham-Green \cite{ELG08} defined for a prime and a fixed
coclass <r> infinite coclass sequences. These sequences consist of finite
-groups of coclass <r>. For each infinite coclass sequence there exists a
consistent pp-presentation (see
Section~"Background on (polycyclic) parametrised presentations")
such that if we choose a natural number for the parameter and possibly reduce
the exponents modulo the relative orders, we obtain a consistent polycyclic
presentation for a group in the sequence; and for each group in the sequence
there exists a natural number such that using this as a value for the
parameter, we obtain a polycyclic presentation for the group.

We use these consistent pp-presentations to compute parametrised
groups, which we call -power-poly-pcp-groups. Furthermore, methods for
these are presented. Without specifying the parameter we compute certain
properties and using the -power-poly-pcp-groups we do this for all groups
they represent at once.

The -power-poly-pcp-groups have a consistent pp-presentation with
generators $g_1, \ldots, g_n, t_1, \ldots t_d$ and $c_1, \ldots, c_m$, for some
non-negative integers <n>, <d> and <m>, and relations of the form, where
$rel[i,j]$ stores the right hand sides of the relations (see
Section~"Background on (polycyclic) parametrised presentations" for more
information on pp-presentations),

%display{nontext}
$$
\eqalign{
&\, g_i^p=rel[i,i],\cr
&\, t_i^{expo} = rel[n+i,n+i],\cr
&\, c_i^{expo\_vec[i]} = rel[n+d+i,n+d+i],\cr
&\, g_i^{g_j} = rel[j,i], \cr
&\, t_i^{g_j} = rel[j,n+i], \cr
&\, t_i^{t_j} = rel[n+j,n+i],
}
$$
%display{text}
% g_i^p = rel[i,i],
% t_i^{expo} = rel[n+i,n+i],
% c_i^{expo\_vec[i]} = rel[n+d+i,n+d+i],
% g_i^{g_j} = rel[j,i],
% t_i^{g_j} = rel[j,n+i],
% t_i^{t_j} = rel[n+j,n+i],
%enddisplay
where the $t_i$'s commute modulo $\langle c_1,\ldots, c_m\rangle$ and the
$c_i$'s are central. So (see Section~"Obtaining p-power-poly-pcp-groups")
are the right hand sides of the relations, where some depend on the parameter.
The relative orders <expo> and <expo\_vec[i]> of the generators $t_j$ and
$c_i$ depend on the parameter.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Example}

In this section we present the well-known example of quaternion groups
$Q_{2^{x+3}}$. They have a pp-presentation of the following form:

%display{nontext}
$$
\eqalign{
\{ g_1,g_2,t_1 \mid &g_1^{2} = t_1^{2^x},\, g_2^{g_1} = g_2
t_1^{-1+2^{x+1}},\cr
&\, g_2^{2} = t_1,\, t_1^{g_1} = t_1^{-1+2^{x+1}},\cr
&\, t_1^{2^{x+1}} = 1 \}.
}
$$
%display{text}
% { g_1,g_2,t_1|g_1^2 = t_1^{2^x}, g_2^{g_1} = g_2t_1^{-1+2^{x+1}},
% g_2^{2} = t_1, t_1^{g_1} = t_1^{-1+2^{x+1}},
% t_1^{2^{x+1}} = 1 }.
%enddisplay

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Obtaining p-power-poly-pcp-groups}

To obtain -power-poly-pcp-groups:

\>PPPPcpGroups( <rel>, <n>, <d>, <m>, <expo>, <expo_vec>, <prime>, <cc>, <name> ) F
\>PPPPcpGroups( <rec> ) F

returns the p-power-poly-pcp-groups described by the consistent
pp-presentation with generators $g_1, \ldots, g_n$, $t_1, \ldots t_d$,
$c_1, \ldots, c_m$, for some non-negative integers <n>, <d> and <m>, and
relations of the form

%display{nontext}
$$
\eqalign{
&\, g_i^p=rel[i,i],\cr
&\, t_i^{expo} = rel[n+i,n+i],\cr
&\, c_i^{expo\_vec[i]} = rel[n+d+i,n+d+i],\cr
&\, g_i^{g_j} = rel[j,i], \cr
&\, t_i^{g_j} = rel[j,n+i], \cr
&\, t_i^{t_j} = rel[n+j,n+i].
}
$$
%display{text}
% g_i^p = rel[i,i],
% t_i^{expo} = rel[n+i,n+i],
% c_i^{expo\_vec[i]} = rel[n+d+i,n+d+i],
% g_i^{g_j} = rel[j,i],
% t_i^{g_j} = rel[j,n+i],
% t_i^{t_j} = rel[n+j,n+i].
%enddisplay

The input consists of the following:

%display{nonhtml}
\beginitems
`<rel>' & is the list of the right hand sides of the relations, where each
relation is presented by a list consisting of tuples; the first entry of
a tuple is the index of the generator (if $i \le n$, then it represents
generator $g_i$, if $n \< i \le d$, then it represents generator $t_{i-n}$
and otherwise it represents generator $c_{i-n-d}$) and the second entry of
the tuple is the corresponding exponent.
Note that the exponents of the $g_i$'s are saved as integers and all other
exponents as lists, representing elements depending on the parameter.

`<n>' & is the number of generators $g_i$,

`<d>' & is the number of generators $t_i$,

`<m>' & is the number of generators $c_i$,

`<expo>' & is the relative order of all generators $t_i$; note that is
a list that represents an element depending on the parameter,

`<expo_vec>' & is the list of relative orders, where the th entry of the
list gives the relative order of the generator $c_i$; note that each
relative order is a list that represents an element depending on the
parameter,

`<prime>' & is the underlying prime
,

`<cc>' & if the
-power-poly-pcp-groups represent an infinite coclass
sequence of -groups of coclass <r>, then <cc> = <r>. If they represent
Schur extensions of groups in an infinite coclass sequence, then <cc> is
the coclass of the groups in this infinite coclass sequence.

`<name>' & a string to name the
-power-poly-pcp-groups.

`<rec>' & is a record of the form
<rec( rel, expo, n, d, m, prime, cc, expo_vec, name )>.
\enditems
%display{nontext}
%\beginitems
%`<rel>' & is the list of relations, where each relation is presented by a
%list consisting of tuples; the first entry of a tuple is the index of the
%generator (if $i \le n$, then it represents generator $g_i$, if $n \< i \le d$,
%then it represents generator $t_{i-n}$ and otherwise it represents generator
%$c_{i-n-d}$) and the second entry of the tuple is the corresponding exponent.
%Note that the exponents of the $g_i$'s are saved as integers and all other
%exponents as lists, representing elements depending on the parameter.
%`<n>' & is the number of generators $g_i$,
%`<d>' & is the number of generators $t_i$,
%`<m>' & is the number of generators $c_i$,
%`<expo>' & is the relative order of all generators $t_i$; note that expo is
%given as a list to represent an element depending on the parameter,
%`<expo_vec>' & is the list of relative orders, where the th entry of the
%list gives the relative order of the generator $c_i$; note that each
%relative order is given as a list to represent an element depending on the
%parameter,
%`<prime>' & is the underlying prime ,
%`<cc>' & if the -power-poly-pcp-groups represent an infinite coclass
%sequence of -groups of coclass <r>, then <cc> = <r>. If they represent
%Schur extensions of groups in an infinite coclass sequence, then <cc> is
%the coclass of the groups in this infinite coclass sequence.
%`<name>' & a string to name the -power-poly-pcp-groups.
%`<rec>' & is a record of the form
%<rec( rel, expo, n, d, m, prime, cc, expo_vec, name )>.
%\enditems
%enddisplay

The pp-presentation is described at the beginning of Chapter
"p-power-poly-pcp-group". Note that the consistency of the presentation is
checked and that the presentation has to be consistent.

\beginexample
gap> ParPresGlobalVar_2_1[1];
rec(
 rel := [ [ [ [ 1, 0 ] ] ], [ [ [ 2, 1 ], [ 3, -1+2*2^x ] ], [ [ 3, 1 ] ] ],
 [ [ [ 3, -1+2*2^x ] ], [ [ 3, 1 ] ], [ [ 3, 0 ] ] ] ], expo := 2*2^x,
 n := 2, d := 1, m := 0, prime := 2, cc := 1, expo_vec := [ ], name := "D" )
gap> G := PPPPcpGroups( ParPresGlobalVar_2_1[1] );
< P-Power-Poly-pcp-groups with 3 generators of relative orders [ 2,2,2*2^x ] >
\endexample

\>PPPPcpGroupsElement( <G>, <word> ) F

constructs an element in -power-poly-pcp-groups, where <G> is a
-power-poly-pcp-group (thus representing an infinite coclass sequence
through a pp-presentation) with generators $g_1, \ldots, g_n, t_1,
\ldots, t_d, c_1, \ldots, c_m$ and <word> is a list of tuples, where the first
entry in the tuple gives the index of the generator (if $i \le n$, then
it represents generator $g_i$, if $n \< i \le d$, then it represents generator
$t_{i-n}$ and otherwise it represents generator $c_{i-n-d}$) and the second
entry of the tuple is the corresponding exponent. Note that the exponents
of the $g_i$'s must be integers, while all other exponents can be integers
or lists, representing an element depending on the parameter.

\beginexample
gap> G := PPPPcpGroups( ParPresGlobalVar_2_1[3] );
< P-Power-Poly-pcp-groups with 3 generators of relative orders [ 2,2,2*2^x ] >
gap> g1 := PPPPcpGroupsElement( G , [[1,1]] );
g1
gap> g := PPPPcpGroupsElement( G , [[1,1],[2,1],[3,1]] );
g1*g2*t1
gap> h := PPPPcpGroupsElement( G , [[1,1],[2,1],[3,G!.expo-1]] );
g1*g2*t1^(-1+2*2^x)
\endexample

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Operations and functions for p-power-poly-pcp-group elements}

The typical operations for group elements can be carried out for
-power-poly-pcp-group elements, like `*', `/', Inverse, One, equality and
ShallowCopy.

\>CollectPPPPcp( <a> ) F

collects the -power-poly-pcp-group element <a> so that after reducing to
integers for every specific value for the parameter <x>, the element is
collected in the polycyclic group, represented by <x> in the underlying
pp-presentation.

Note that the global
variable `COLLECT_PPOWERPOLY_PCP' determines whether every element will be
collected immediately, when created, or not, see
%display{tex}
{\tt COLLECT_PPOWERPOLY_PCP},
%enddisplay
"COLLECT_PPOWERPOLY_PCP".

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Operations and functions for p-power-poly-pcp-groups}

For -power-poly-pcp-groups:

\> GeneratorsOfGroup( <G> )

returns a set of generators for the -power-poly-pcp-groups <G>.

\> One( <G> )

obtains the identity element of the -power-poly-pcp-groups <G>.

\>IsConsistentPPPPcp( <G> ) F
\>IsConsistentPPPPcp( <ParPres> ) F

checks if the underlying pp-presentation of the
-power-poly-pcp-groups <G> is consistent or if the pp-presenta-tion
<ParPres> is consistent.

\>GetPcGroupPPowerPoly( <ParPres>, <n> ) F
\>GetPcGroupPPowerPoly( <G>, <n> ) F

takes the pp-presentation given by the record <ParPres> as in
%display{tex}
{\tt PPPPcpGroups},
%enddisplay
"PPPPcpGroups" or the -power-poly-pcp-groups <G> and takes <n>, a
non-negative integer, as a value for the parameter to obtain a
pc-presentation for the corresponding finite -group.

\>GetPcpGroupPPowerPoly( <ParPres>, <n> ) F
\>GetPcpGroupPPowerPoly( <G>, <n> ) F

takes pp-presentation given by the record <ParPres> as in
%display{tex}
{\tt PPPPcpGroups},
%enddisplay
"PPPPcpGroups" or the -power-poly-pcp-groups <G> and takes <n>, a
non-negative integer, as the parameter to obtain a pcp-presentation for the
corresponding finite -group, for further information we refer to the
polycyclic package.

\>GAPInputPPPPcpGroups( <file>, <G> ) F
\>GAPInputPPPPcpGroups( <file>, <ParPres> ) F

prints the -power-poly-pcp-groups <G> defined by <ParPres> in the file
<file> as a record that could be used as input to
%display{tex}
{\tt PPPPcpGroups},
%enddisplay
"PPPPcpGroups" to create -power-poly-pcp-groups.

\>GAPInputPPPPcpGroupsAppend( <file>, <G> ) F
\>GAPInputPPPPcpGroupsAppend( <file>, <ParPres> ) F

appends the pp-presentation of the -power-poly-pcp-groups <G> defined by
<ParPres> to the file <file> as a record that could be used as input to
%display{tex}
{\tt PPPPcpGroups},
%enddisplay
"PPPPcpGroups" to create -power-poly-pcp-groups.

\>LatexInputPPPPcpGroups( <file>, <G> ) F
\>LatexInputPPPPcpGroups( <file>, <ParPres> ) F

prints the pp-presentation of <G> as given by <ParPres> in latex-code to the
file <file>. Note that only non-trivial relations are printed.

\>LatexInputPPPPcpGroupsAppend( <file>, <G> ) F
\>LatexInputPPPPcpGroupsAppend( <file>, <ParPres> ) F

appends the pp-presentation of <G> as given by <ParPres> in latex-code to the
file <file>. Note that only non-trivial relations are appended.

\> LatexInputPPPPcpGroupsAllAppend( <file>, <G> ) F
\> LatexInputPPPPcpGroupsAllAppend( <file>, <ParPres> ) F

appends the pp-presentation of <G> as given by <ParPres> in latex-code to the
file <file>. Note that all relations are appended.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Info classes for the p-power-poly-pcp-groups}

The following info classes are available:

\>`InfoConsistencyPPPPcp' V

is an InfoClass with the following levels.

%display{nonhtml}
\beginitems
`level 1' & displays the first consistency relation that fails during the consistency check;

`level 2' & displays which family of consistency relations have been checked during a consistency check.
\enditems
%display{nontext}
%\beginitems
%`level 1' & displays the first consistency relation that fails during the consistency check;
%`level 2' & displays which family of consistency relations have been checked during a consistency check.
%\enditems
%enddisplay

the default value is 1.

\>`InfoCollectingPPPPcp' V

is an InfoClass with the following levels.

\beginitems
`level 1' & displays some information during collecting;
\enditems

the default value is 0.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Global variables for the p-power-poly-pcp-groups}

The following global variables are available with default value:

\>`COLLECT_PPOWERPOLY_PCP' V

is a global variable determining if every -power-poly-pcp-group
element is collected, when created, the default value is true.

Messung V0.5

¤ Dauer der Verarbeitung: 0.5 Sekunden ¤

*© Formatika GbR, Deutschland

Wurzel

Suchen

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung und die Messung sind noch experimentell.