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In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions.
The symmetric group \( S_n\) is the group of permutations on \(n\) objects. Usually the objects are labeled \( \{1,2,\ldots,n\},\) and elements of \(S_n \) are given by bijective functions \( \sigma \colon \{1,2,\ldots,n\} \to \{1,2,\ldots,n\}.\)
Aug 22, 2024 · The symmetric group S_n of degree n is the group of all permutations on n symbols. S_n is therefore a permutation group of order n! and contains as subgroups every group of order n. The nth symmetric group is represented in the Wolfram Language as SymmetricGroup [n].
The members of $X$ represent the corners of a triangle, canonically numbered clockwise for the identity triangle. Each of the $f_i$ transform that triangle by flipping, rotation, or identity -- these are all symmetries. Thus, "symmetry group." $\endgroup$ –
\(S_n\) with compositions forms a group; this group is called a Symmetric group. \(S_n\) is a finite group of order \(n!\) and are permutation groups consisting of all possible permutations of n objects.
Notes on the symmetric group. 1 Computations in the symmetric group. Recall that, given a set X, the set SX of all bijections from X to itself (or, more brie y, permutations of X) is group under function composition.
In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient space which takes the object to itself, and which preserves all the relevant structure of the object.
