Editing Group action (section)

=== Transitivity properties ===
The action of {{math|''G''}} on {{math|''X''}} is called ''{{visible anchor|transitive}}'' if for any two points {{math|''x'', ''y'' ∈ ''X''}} there exists a {{math|''g'' ∈ ''G''}} so that {{math|1=''g'' &sdot; ''x'' = ''y''}}.

The action is ''{{visible anchor|simply transitive}}'' (or ''sharply transitive'', or ''{{visible anchor|regular}}'') if it is both transitive and free. This means that  given {{math|''x'', ''y'' ∈ ''X''}} there is exactly one {{math|''g'' ∈ ''G''}} such that {{math|1=''g'' &sdot; ''x'' = ''y''}}. If {{math|''X''}} is acted upon simply transitively by a group {{math|''G''}} then it is called a [[principal homogeneous space]] for {{math|''G''}} or a {{math|''G''}}-torsor.

For an integer {{math|''n'' ≥ 1}}, the action is {{visible anchor|n-transitive|text=''{{mvar|n}}-transitive''}} if {{math|''X''}} has at least {{math|''n''}} elements, and for any pair of {{math|''n''}}-tuples {{math|(''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>), (''y''<sub>1</sub>, ..., ''y''<sub>''n''</sub>) ∈ ''X''<sup>''n''</sup>}} with pairwise distinct entries (that is {{math|''x''<sub>''i''</sub> ≠ ''x''<sub>''j''</sub>}}, {{math|''y''<sub>''i''</sub> ≠ ''y''<sub>''j''</sub>}} when {{math|''i'' ≠ ''j''}}) there exists a {{math|''g'' ∈ ''G''}} such that {{math|1=''g''&sdot;''x''<sub>''i''</sub> = ''y''<sub>''i''</sub>}} for {{math|1=''i'' = 1, ..., ''n''}}. In other words, the action on the subset of {{math|''X''<sup>''n''</sup>}} of tuples without repeated entries is transitive. For {{math|1=''n'' = 2, 3}} this is often called double, respectively triple, transitivity. The class of [[2-transitive group]]s (that is, subgroups of a finite symmetric group whose action is 2-transitive) and more generally [[multiply transitive group]]s is well-studied in finite group theory.

An action is {{visible anchor|sharply n-transitive|text=''sharply {{mvar|n}}-transitive''}} when the action on tuples without repeated entries in {{math|''X''<sup>''n''</sup>}} is sharply transitive.

==== Examples ====
The action of the symmetric group of {{math|''X''}} is transitive, in fact {{math|''n''}}-transitive for any {{math|''n''}} up to the cardinality of {{math|''X''}}. If {{math|''X''}} has cardinality {{math|''n''}}, the action of the [[alternating group]] is {{math|(''n'' − 2)}}-transitive but not {{math|(''n'' − 1)}}-transitive.

The action of the [[general linear group]] of a vector space {{math|''V''}} on the set {{math|''V'' &setminus; {{mset|0}}}} of non-zero vectors is transitive, but not 2-transitive (similarly for the action of the [[special linear group]] if the dimension of {{math|''v''}} is at least 2). The action of the [[orthogonal group]] of a Euclidean space is not transitive on nonzero vectors but it is on the [[unit sphere]].