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The orbit-stabilizer relationship (also known as the orbit-stabilizer theorem) is very clear for finite groups. Is there an equivalent relation for continuous groups? Also, is there a similar notion for monoid (or semigroup) actions on vector spaces?

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    $\begingroup$ You need a condition on the action. For example, an action of the integers on the circle by irrational rotations has trivial stabilizers, but the orbit of a point is not discrete. $\endgroup$
    – S. Carnahan ♦
    Commented Jun 13, 2014 at 0:14
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    $\begingroup$ Orbit-stabilizer doesn't really work nicely for monoids and semigroups because of irreversibility. $\endgroup$
    – Benjamin Steinberg
    Commented Jun 13, 2014 at 1:29
  • $\begingroup$ Any reference reading on this? Somehow, I couldn't spot anything concrete via google; but I suspect quite a lot must have been studied (may be much less than the case of groups). $\endgroup$
    – Arnab
    Commented Jun 13, 2014 at 16:28

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