Most of this discussion is about cases where sage thinks it is acting on
a bigger set than the user wants it to.  I'll just point out bug 8963

  http://trac.sagemath.org/sage_trac/ticket/8963

where sage has the reverse problem:  it thinks that the set that is 
being acted on is smaller.  For example, it correctly computes that
the row stabilizer group of the tableau [[1,2],[3]] is a group with
two elements generated by the transposition (1,2), but it thinks that
this group is acting on the set {1, 2} rather than the set {1, 2, 3},
and this leads to difficulties.

That bug has a patch that fixes this particular problem, but enhancing
sage to know about the set that a permutation group acts on sounds like
a good idea.

Dan

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