Permutations in Group Theory
In group theory, the term permutation of a set means a bijective map, or bijection, from that set onto itself. The set of all permutations of any given set S forms a group, with composition of maps as product and the identity as neutral element. This is the symmetric group of S. Up to isomorphism, this symmetric group only depends on the cardinality of the set, so the nature of elements of S is irrelevant for the structure of the group. Symmetric groups have been studied most in the case of a finite sets, in which case one can assume without loss of generality that S={1,2,...,n} for some natural number n, which defines the symmetric group of degree n, written Sn.
Any subgroup of a symmetric group is called a permutation group. In fact by Cayley's theorem any group is isomorphic to some permutation group, and every finite group to a subgroup of some finite symmetric group. However, permutation groups have more structure than abstract groups, allowing for instance to define the cycle type of an element of a permutation group; different realizations of a group as a permutation group need not be equivalent for this additional structure. For instance S3 is naturally a permutation group, in which any transposition has cycle type (2,1), but the proof of Cayley's theorem realizes S3 as a subgroup of S6 (namely the permutations of the 6 elements of S3 itself), in which permutation group transpositions get cycle type (2,2,2). So in spite of Cayley's theorem, the study of permutation groups differs from the study of abstract groups.
Read more about this topic: Permutation
Famous quotes containing the words permutations, group and/or theory:
“The new shopping malls make possible the synthesis of all consumer activities, not least of which are shopping, flirting with objects, idle wandering, and all the permutations of these.”
—Jean Baudrillard (b. 1929)
“He hung out of the window a long while looking up and down the street. The world’s second metropolis. In the brick houses and the dingy lamplight and the voices of a group of boys kidding and quarreling on the steps of a house opposite, in the regular firm tread of a policeman, he felt a marching like soldiers, like a sidewheeler going up the Hudson under the Palisades, like an election parade, through long streets towards something tall white full of colonnades and stately. Metropolis.”
—John Dos Passos (1896–1970)
“Everything to which we concede existence is a posit from the standpoint of a description of the theory-building process, and simultaneously real from the standpoint of the theory that is being built. Nor let us look down on the standpoint of the theory as make-believe; for we can never do better than occupy the standpoint of some theory or other, the best we can muster at the time.”
—Willard Van Orman Quine (b. 1908)