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For a group action, an orbit is in bijection with a coset space G/Stab(x)
Orbit–Stabilizer Theorem. Let be a group acting on a set via a group action. For , let be the orbit of and let be the stabilizer of . Then the map
is a bijection. In particular, if is finite then
This theorem converts problems about orbits into problems about cosets and index. It is the main input for the class equation and many counting arguments.
A group is a set together with a binary operation such that:
Equivalently, a group is a monoid in which every element is invertible. Much of group theory studies subgroups and structure-preserving maps called group homomorphisms.
Let a group action of a group on a set be given, written . For , the orbit of is the subset
Let a group action of a group on a set be given. For , the stabilizer of is
The stabilizer is always a subgroup of .
A left coset of a subgroup in a group is a subset of of the form
for some . A right coset is a subset of the form
Left cosets are the equivalence classes of the equivalence relation on defined by iff (equivalently, ). In particular, the set of left cosets forms a partition of , and the number of distinct left cosets is the index . If is a normal subgroup, then for all , and the set of cosets is the underlying set of the quotient group .
Let be a subgroup of a group . The index of in , denoted , is the cardinality of the set of left cosets of in :
(Equivalently, it is the number of distinct right cosets.)
For finite , Lagrange's theorem implies , so in particular divides . Small index has strong consequences: iff , and if then is normal, i.e. a normal subgroup.
Class Equation. Let be a finite group. For , let the conjugacy class be
and let the centralizer be
Then . If denotes the center of and are representatives of the distinct conjugacy classes contained in , then