symmetric group
Noun
Meaning
symmetric group (plural symmetric groups)
(mathematics) A group whose elements are precisely all of the bijections of some set with itself and whose operation is composition of those bijections.
Synonyms
Examples
Cayley's theorem states that every group G is isomorphic to a subgroup of the symmetric group on the elements of G, as a group acts on itself faithfully by (left or right) multiplication.
For example, the symmetric group of a finite set consists of all bijective transformations of that set; thus, applying any element of the permutation group to an element of the set will produce another element of the set.
Generalizations Parity can be generalized to Coxeter groups : one defines a length function which depends on a choice of generators (for the symmetric group, adjacent transpositions ), and then the function gives a generalized sign map.
But is not symmetric, since switching α and β yields (formally, this is termed a group action of the symmetric group of the roots).
Every group can, therefore, be embedded in the symmetric group on its own elements, Sym(G).
For the remainder of this article, "symmetric group" will mean a symmetric group on a finite set.