GroupStabilizerChain

GroupStabilizerChain[group]

returns a list of successive stabilizers in group of the points in a base of group.

Details and Options

  • A base of a group is a list of points of its domain of action such that the only element in the group fixing them all is the identity.
  • GroupStabilizerChain gives strong generators for a group, automatically choosing an appropriate base.
  • An explicit base can be specified by setting the GroupActionBase option.
  • For a base {b1,…,bn} the stabilizer chain is given as a list of elements of the form {b1,…,bi}->GroupStabilizer[group,{b1,…,bi}] with i=0,…,n. The first element is the stabilizer of {}, which is the complete group. The last element is the stabilizer of the base, which is the trivial group.
  • The list of generators of each stabilizer is a subset of the list of generators of the previous stabilizer in the list. Therefore they are strong generators for the respective groups.

Examples

open allclose all

Basic Examples  (1)

Stabilizer chain for a group:

These are strong generators for the group:

This is a base for the group:

Scope  (1)

Group orders in the chain of stabilizers. The first line is associated with the original group and the last line corresponds to the identity group:

Options  (1)

GroupActionBase  (1)

By default, GroupStabilizerChain chooses an appropriate base:

We can specify the initial points of a base:

Applications  (1)

GroupStabilizerChain is the function that gives access to the strong generators and their base. It is hence the starting point for further programming of tools for computation with permutation groups.

Those groups for which the length of the base is much shorter than the degree of the permutations can be efficiently manipulated using strong generators.

This is a permutation representation of the sporadic simple group ConwayGroupCo3[] acting on 276 points, but here a base with only 6 points is used:

Possible Issues  (1)

There is no guarantee that bases are minimal, so that they can have redundant points:

In this example point 4 is redundant because the stabilizer of {2,4} equals the stabilizer of {2}:

Wolfram Research (2010), GroupStabilizerChain, Wolfram Language function, https://reference.wolfram.com/language/ref/GroupStabilizerChain.html.

Text

Wolfram Research (2010), GroupStabilizerChain, Wolfram Language function, https://reference.wolfram.com/language/ref/GroupStabilizerChain.html.

CMS

Wolfram Language. 2010. "GroupStabilizerChain." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/GroupStabilizerChain.html.

APA

Wolfram Language. (2010). GroupStabilizerChain. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/GroupStabilizerChain.html

BibTeX

@misc{reference.wolfram_2024_groupstabilizerchain, author="Wolfram Research", title="{GroupStabilizerChain}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/GroupStabilizerChain.html}", note=[Accessed: 26-December-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_groupstabilizerchain, organization={Wolfram Research}, title={GroupStabilizerChain}, year={2010}, url={https://reference.wolfram.com/language/ref/GroupStabilizerChain.html}, note=[Accessed: 26-December-2024 ]}