Orbit-stabilizer theorem
Revision as of 17:53, 12 September 2015 by AstrapiGnosis (talk | contribs)
The orbit-stabilizer theorem is a combinatorial result in group theory.
Let be a group acting on a set
. For any
, let
denote the stabilizer of
, and let
denote the orbit of
. The orbit-stabilizer theorem states that
Proof. Without loss of generality, let operate on
from the right. We note that if
are elements of
such that
, then
. Hence for any
, the set of elements
of
for which
constitute a unique left coset modulo
. Thus
The result then follows from Lagrange's Theorem.