Editing Group action (section)

=== Invariant subsets ===
If {{math|''Y''}} is a [[subset]] of {{math|''X''}}, then {{math|''G''&sdot;''Y''}} denotes the set {{math|{{mset|''g''&sdot;''y'' : ''g'' ∈ ''G'' and ''y'' ∈ ''Y''}}}}. The subset {{math|''Y''}} is said to be ''invariant under ''{{math|''G''}} if {{math|1=''G''&sdot;''Y'' = ''Y''}} (which is equivalent {{math|''G''&sdot;''Y'' ⊆ ''Y''}}). In that case, {{math|''G''}} also operates on {{math|''Y''}} by [[Restriction (mathematics)|restricting]] the action to {{math|''Y''}}. The subset {{math|''Y''}} is called ''fixed under ''{{math|''G''}} if {{math|1=''g''&sdot;''y'' = ''y''}} for all {{math|''g''}} in {{math|''G''}} and all {{math|''y''}} in {{math|''Y''}}. Every subset that is fixed under {{math|''G''}} is also invariant under {{math|''G''}}, but not conversely.

Every orbit is an invariant subset of {{math|''X''}} on which {{math|''G''}} acts [[Group action (mathematics)#Notable properties of actions|transitively]]. Conversely, any invariant subset of {{math|''X''}} is a union of orbits. The action of {{math|''G''}} on {{math|''X''}} is ''transitive'' if and only if all elements are equivalent, meaning that there is only one orbit.

A {{math|''G''}}''-invariant'' element of {{math|''X''}} is {{math|''x'' ∈ ''X''}} such that {{math|1=''g''&sdot;''x'' = ''x''}} for all {{math|''g'' ∈ ''G''}}. The set of all such {{math|''x''}} is denoted {{math|''X''<sup>''G''</sup>}} and called the {{math|''G''}}''-invariants'' of {{math|''X''}}. When {{math|''X''}} is a [[G-module|{{math|''G''}}-module]], {{math|''X''<sup>''G''</sup>}} is the zeroth [[Group cohomology|cohomology]] group of {{math|''G''}} with coefficients in {{math|''X''}}, and the higher cohomology groups are the [[derived functor]]s of the [[functor]] of {{math|''G''}}-invariants.