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Cyclic permutation
A cyclic permutation, also called a cycle, is a permutation of a finite set that moves its elements in a single closed loop; a cycle on all n elements is an n-cycle.
Frobenius group
A Frobenius group is a transitive permutation group in which no non-identity element fixes more than one point, studied by Georg Frobenius in 1900 and 1901 using his new theory of group characters.
Parity of a permutation
The parity of a permutation, also called its sign or signature, classifies a permutation as even or odd according to the number of transpositions in any decomposition.
Permutation group
A permutation group is a group whose elements are permutations of a set, with composition as the operation; by Cayley's theorem, every group is isomorphic to one.
Permutation representation
A permutation representation is a way of realizing a group's elements as permutations of a set, equivalently a group action of the group on that set.
Primitive permutation group
A primitive permutation group is a transitive permutation group preserving no nontrivial block system, equivalently one whose point stabilizers are maximal subgroups; these groups are the basic building blocks of all permutation groups.
Symmetric group
The symmetric group on a set is the group of all permutations of the set under composition; on n elements it is written Sn and has n! elements.