I'm just teaching algebra, and I was about to recommend to my students to
make some experiments with groups and permutation groups in Sage. Unless
I'm overlooking something, some of the functionality and in particular the
documentation is a mess.
Let's start with something like
sage: a = SymmetricGroup(3)('(1,2)')
sage: b = SymmetricGroup(3)('(2,3)')
For someone who is used to Gap and Magma, the first surprise is that
sage: 1^a
produces a NotImplementedError
I don't see in the documents how to apply a permutation group element to an
element. Well,
sage: a*b
(1,3,2)
indicates that Sage uses the right action, as do Gap and Magma. After some
experimenting one finds out that a group element a applied to x is a(x). In
my opinion, this quite irritating, for this produces things like
sage: (a*b)(1) == a(b(1))
False
Are there any plans (or possibilities) to fix such a misconception? (I
know, everyone is invited to help improve and extend Sage, and at some
point I hope to do so ...)
As to the docs, it seems to me that neither the thematic tutorial, nor the
permutation group pages in the Reference Manual and Constructions Manual
explain or give examples about how to apply permutations.
The thematic tutorial `Group Theory and Sage' looks particularly odd: It
starts with number theory which has nothing to do with what I would expect
under this heading. The second part then contains no material at all about
how to create and work with self-made permutation groups.
Considering the great possibilities which Sage offers, it is a pity that
especially for finite groups the docs appear unorganized and miss important
topics.
-- Peter Mueller
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