#9102: Docstring improvements for strong generating systems of permutation
groups
---------------------------+------------------------------------------------
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 nthiery):
* cc: mhansen (added)
Comment:
Replying to [comment:5 jasonbhill]:
> 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:7>
Sage <http://www.sagemath.org>
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