Editing Permutation group (section)

==Transitive actions==
The action of a group ''G'' on a set ''M'' is said to be ''transitive'' if, for every two elements ''s'', ''t'' of ''M'', there is some group element ''g'' such that ''g''(''s'') = ''t''.  Equivalently, the set ''M'' forms a single [[orbit (group theory)|orbit]] under the action of ''G''.<ref>{{harvnb|Artin|1991|p=177}}</ref>  Of the examples [[#Examples|above]], the group {e, (1 2), (3 4), (1 2)(3 4)} of permutations of {1, 2, 3, 4} is not transitive (no group element takes 1 to 3) but the group of symmetries of a square is transitive on the vertices.

=== Primitive actions ===
{{main|Primitive permutation group}}
A permutation group ''G'' acting transitively on a non-empty finite set ''M'' is ''imprimitive'' if there is some nontrivial [[set partition]] of ''M'' that is preserved by the action of ''G'', where "nontrivial" means that the partition isn't the partition into [[singleton set]]s nor the partition with only one part.  Otherwise, if ''G'' is transitive but does not preserve any nontrivial partition of ''M'', the group ''G'' is ''primitive''.

For example, the group of symmetries of a square is imprimitive on the vertices: if they are numbered 1, 2, 3, 4 in cyclic order, then the partition {{1, 3}, {2, 4}} into opposite pairs is preserved by every group element.  On the other hand, the full symmetric group on a set ''M'' is always primitive.