Properties of Permutations A permutation is a function that assigns to each element of a set a unique element of another set. In other words, it is a biject...
Properties of Permutations
A permutation is a function that assigns to each element of a set a unique element of another set. In other words, it is a bijection between two sets.
Closure:
The set of permutations of a set with n elements is closed under the operations of composition and inversion. This means that if A and B are sets with n elements, and f: A -> B and g: B -> A are permutations, then the permutation h: A -> A defined by h(x) = g(x) for all x in A is also a permutation.
Symmetry:
If f: A -> B is a permutation, then the permutation f^-1: B -> A is also a permutation. This means that f is bijective and its inverse is also a permutation.
Transitivity:
If f: A -> B and g: B -> C are permutations, then the composition of f and g, fo g: A -> C, is also a permutation.
Identity element:
The identity permutation, id: A -> A, which leaves each element unchanged, is a permutation that is the identity element for the composition of permutations.
Cyclic groups:
A permutation group is a group under the operation of composition, in which the identity element is a generator. A permutation group G is cyclic if and only if it has a finite order.
Alternating groups:
An alternating group is a group in which every element can be expressed as the composition of a finite number of permutations