Editing Group action (section)

=== Fixed points and stabilizer subgroups ===
Given {{math|''g''}} in {{math|''G''}} and {{math|''x''}} in {{math|''X''}} with {{math|1=''g''&sdot;''x'' = ''x''}}, it is said that "{{math|''x''}} is a fixed point of {{math|''g''}}" or that "{{math|''g''}} fixes {{math|''x''}}". For every {{math|''x''}} in {{math|''X''}}, the '''{{visible anchor|stabilizer subgroup}}''' of {{math|''G''}} with respect to {{math|''x''}} (also called the '''isotropy group''' or '''little group'''<ref name="Procesi">{{cite book|last1=Procesi|first1=Claudio|title=Lie Groups: An Approach through Invariants and Representations|date=2007|publisher=Springer Science & Business Media|isbn=9780387289298|page=5|url=https://books.google.com/books?id=Sl8OAGYRz_AC&q=%22little+group%22+action&pg=PA5|access-date=23 February 2017|language=en}}</ref>) is the set of all elements in {{math|''G''}} that fix {{math|''x''}}:
<math display=block>G_x = \{g \in G : g{\cdot}x = x\}.</math>
This is a [[subgroup]] of {{math|''G''}}, though typically not a normal one. The action of {{math|''G''}} on {{math|''X''}} is [[Group action (mathematics)#Notable properties of actions|free]] if and only if all stabilizers are trivial. The kernel {{math|''N''}} of the homomorphism with the symmetric group, {{math|''G'' → Sym(''X'')}}, is given by the [[Intersection (set theory)|intersection]] of the stabilizers {{math|''G''<sub>''x''</sub>}} for all {{math|''x''}} in {{math|''X''}}. If {{math|''N''}} is trivial, the action is said to be faithful (or effective).

Let {{math|''x''}} and {{math|''y''}} be two elements in {{math|''X''}}, and let {{math|''g''}} be a group element such that {{math|1=''y'' = ''g''&sdot;''x''}}. Then the two stabilizer groups {{math|''G''<sub>''x''</sub>}} and {{math|''G''<sub>''y''</sub>}} are related by {{math|1=''G''<sub>''y''</sub> = ''gG''<sub>''x''</sub>''g''<sup>−1</sup>}}. Proof: by definition, {{math|''h'' ∈ ''G''<sub>''y''</sub>}} if and only if {{math|1=''h''&sdot;(''g''&sdot;''x'') = ''g''&sdot;''x''}}. Applying {{math|''g''<sup>−1</sup>}} to both sides of this equality yields {{math|1=(''g''<sup>−1</sup>''hg'')&sdot;''x'' = ''x''}}; that is, {{math|''g''<sup>−1</sup>''hg'' ∈ ''G''<sub>''x''</sub>}}. An opposite inclusion follows similarly by taking {{math|''h'' ∈ ''G''<sub>''x''</sub>}} and {{math|1=''x'' = ''g''<sup>−1</sup>&sdot;''y''}}.

The above says that the stabilizers of elements in the same orbit are [[Conjugacy class|conjugate]] to each other. Thus, to each orbit, we can associate a [[conjugacy class]] of a subgroup of {{math|''G''}} (that is, the set of all conjugates of the subgroup). Let {{math|(''H'')}} denote the conjugacy class of {{math|''H''}}. Then the orbit {{math|''O''}} has type {{math|(''H'')}} if the stabilizer {{math|''G''<sub>''x''</sub>}} of some/any {{math|''x''}} in {{math|''O''}} belongs to {{math|(''H'')}}. A maximal orbit type is often called a [[principal orbit type]].