MeatAxe  2.4
mktree - Enumerate Group Elements

Command Line

mktree Options [-n] [-g NGen] Name 
Options
Standard options, see Standard Command Line Options
-n
Do not write the outpt file.
-g
Set the number of generators (default: 2).
Name
Name of the representation.

Input Files

Name.1, Name.2, ...
Generators.

Output Files

Name.elt
Element tree.

Description

This program enumerates all elements of a finitely generated matrix group. By default, the program assumes that the group has two generators, which are read from Name.1 and Name.2. A different number of generators can be specified with "-g".

Unless the "-n" option is used, the program writes the element tree to Name.elt. The element tree describes how the group elements can be calculated as products of generators. It is actually a matrix with two columns and one row for each group element. The i-th row of this matrix describes how the i-th element is calculated:

  • The row (-1,-1) represents the unit element. This row appears first in the output.
  • (0,k) means that this element is the k-th generator. Note that the generator number starts with 0, i.e., the first generator has k=0.
  • (s,k) means that the corresponding element is obtained by multiplying the s-th element from the right by the k-th generator. Both generator and element numbers start with 0. Thus, the (0,k) lines decribed above are actually a special case of the general (s,k) lines.

The following example shows the output for a group of the order 10 with two generators, a and b:

Line   Contents     Meaning
----   --------     ---------------
 1      -1   -1     identity matrix
 2      0    0      a
 3      0    1      b
 4      1    0      aa
 5      1    1      ab
 6      3    0      aaa
 7      3    1      aab
 8      5    0      aaaa
 9      5    1      aaab
10      7    1      aaaab

Implementation Details

The program holds all group elements in memory. This limits the application of the program to fairly small groups and representations of small degree.


MeatAxe 2.4 documentation, generated on Mon Jun 7 2021 11:42:24