Noun
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.
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. Source: Internet
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. Source: Internet
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. Source: Internet
But is not symmetric, since switching α and β yields (formally, this is termed a group action of the symmetric group of the roots). Source: Internet
Every group can, therefore, be embedded in the symmetric group on its own elements, Sym(G). Source: Internet
For the remainder of this article, "symmetric group" will mean a symmetric group on a finite set. Source: Internet