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A permutation group is a subgroup of a symmetric group; that is, its elements are permutations of a given set. It is thus a subset of a symmetric group that is closed under composition of permutations, contains the identity permutation, and contains the inverse permutation of each of its elements. [2]
While primitive permutation groups are transitive, not all transitive permutation groups are primitive. The simplest example is the Klein four-group acting on the vertices of a square, which preserves the partition into diagonals. On the other hand, if a permutation group preserves only trivial partitions, it is transitive, except in the case ...
Pages in category "Permutation groups" The following 33 pages are in this category, out of 33 total. This list may not reflect recent changes. ...
A Zassenhaus group is a permutation group G on a finite set X with the following three properties: G is doubly transitive. Non-trivial elements of G fix at most two points. G has no regular normal subgroup. ("Regular" means that non-trivial elements do not fix any points of X; compare free action.)
The symmetric group on a finite set is the group whose elements are all bijective functions from to and whose group operation is that of function composition. [1] For finite sets, "permutations" and "bijective functions" refer to the same operation, namely rearrangement.
Class groups of algebraic number fields were among the earliest examples of factor groups, of much interest in number theory. If a group G is a permutation group on a set X, the factor group G/H is no longer acting on X; but the idea of an abstract group permits one not to worry about this discrepancy.
Considering the symmetric group S n of all permutations of the set {1, ..., n}, we can conclude that the map sgn: S n → {−1, 1} that assigns to every permutation its signature is a group homomorphism. [2] Furthermore, we see that the even permutations form a subgroup of S n. [1] This is the alternating group on n letters, denoted by A n. [3]
It is a classical result of Jordan that the symmetric and alternating groups (of degree k and k + 2 respectively), and M 12 and M 11 are the only sharply k-transitive permutation groups for k at least 4. Important examples of multiply transitive groups are the 2-transitive groups and the Zassenhaus groups.