3 p-power-poly-pcp-groups

returns the p-power-poly-pcp-groups described by the consistent pp-presentation with generators g₁, …, g_n, t₁, …t_d, c₁, …, c_m, for some non-negative integers n, d and m, and relations of the form

gip=rel[i,i],  tiexpo = rel[n+i,n+i],  ciexpo_vec[i] = rel[n+d+i,n+d+i],  gigj = rel[j,i],  tigj = rel[j,n+i],  titj = rel[n+j,n+i]·

The input consists of the following:

rel

is the list of relations, where each relation is presented by a list consisting of tuples; the first entry i of a tuple is the index of the generator (if i ≤ n, then it represents generator g_i, if n < i ≤ 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 ith 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 p,

cc

if the p-power-poly-pcp-groups represent an infinite coclass sequence of p-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 p-power-poly-pcp-groups.

rec

is a record of the form rec( rel, expo, n, d, m, prime, cc, expo_vec, name ).

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.

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 ] >

constructs an element in p-power-poly-pcp-groups, where G is a p-power-poly-pcp-group (thus representing an infinite coclass sequence through a pp-presentation) with generators g₁, …, g_n, t₁, …, t_d, c₁, …, c_m and word is a list of tuples, where the first entry i in the tuple gives the index of the generator (if i ≤ n, then it represents generator g_i, if n < i ≤ 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.

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)

3.3 Operations and functions for p-power-poly-pcp-group elements

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

collects the p-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 COLLECT_PPOWERPOLY_PCP.

3.4 Operations and functions for p-power-poly-pcp-groups

For p-power-poly-pcp-groups:

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

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

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

takes the pp-presentation given by the record ParPres as in PPPPcpGroups or the p-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 p-group.

takes pp-presentation given by the record ParPres as in PPPPcpGroups or the p-power-poly-pcp-groups G and takes n, a non-negative integer, as the parameter to obtain a pcp-presentation for the corresponding finite p-group, for further information we refer to the polycyclic package.

prints the p-power-poly-pcp-groups G defined by ParPres in the file file as a record that could be used as input to PPPPcpGroups to create p-power-poly-pcp-groups.

appends the pp-presentation of the p-power-poly-pcp-groups G defined by ParPres to the file file as a record that could be used as input to PPPPcpGroups to create p-power-poly-pcp-groups.

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.

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.

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

3.5 Info classes for the p-power-poly-pcp-groups

The following info classes are available:

is an InfoClass with the following levels.

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.

the default value is 1.

is an InfoClass with the following levels.

level 1

displays some information during collecting;

the default value is 0.

3.6 Global variables for the p-power-poly-pcp-groups

The following global variables are available with default value:

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

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