#9102: Docstring improvements for strong generating systems of permutation
groups
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Reporter: rbeezer | Owner: AlexGhitza
Type: enhancement | Status: needs_work
Priority: blocker | Milestone: sage-4.4.4
Component: algebra | Keywords:
Author: Rob Beezer | Upstream: N/A
Reviewer: | Merged:
Work_issues: |
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Changes (by jasonbhill):
* status: needs_review => needs_work
Comment:
While looking at the amssymb issue, I came across a problem elsewhere in
what has been added to the docstring.
"``base_of_group`` is a list of the positions on which ``self`` acts,"
This is not correct. The base of a permutation group G is defined as a
list L of points in the domain of the action such that no element of G
fixes all points of L.
http://en.wikipedia.org/wiki/Base_(group_theory)
Computationally, this is equivalent to requiring no strong generator to
fix all points of the base, and hence the size of a base corresponds
directly to the length of the stabilizer chain given by the Schreier-Sims
algorithm. (In fact, this algorithm yields a base and the strong
generating system relative to that specific base.)
The assertion that the base is not unique is correct, but the specific
base used must correspond to the stabilizer chain constructed from a
strong generating set. Otherwise, the information of the strong generating
set is of less computational value.
For instance, a symmetric group of degree n has base of size n-1. So, S_5
on points [1..5] has bases [1,2,4,5], [1,2,3,4], etc., depending on the
SGS. At the same time, any cyclic group has a base of size 1.
You may of course artificially inflate the base and it will still follow
the definition, but then anyone using the strong generating set for
calculations (MANY invariants of a group follow from this) must
recalculate the base and this will often result in a different SGS.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/9102#comment:5>
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