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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.
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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.
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 \( \sigma \colon \{1,2,\ldots,n\} \to \{1,2,\ldots,n\}.\)
\(S_n\) 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.
The symmetric group on a finite set is the group whose elements are all bijective functions from to and whose group operation is that of function composition. For finite sets, "permutations" and "bijective functions" refer to the same operation, namely rearrangement.
Group theory is the mathematical study of symmetry, and explores general ways of studying it in many distinct settings. Group theory ties together many of the diverse topics we have already explored – including sets, cardinality, number theory, isomorphism, and modu-lar arithmetic – illustrating the deep unity of contemporary mathematics.
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Oct 15, 2021 · Definition: Symmetric Group. When \(A=\{1,2,\ldots, n\}\) (\(n\in \mathbb{Z}^+\)), we call \(S_A\) the symmetric group on \(n\) letters and denote it by \(S_n\text{.}\) (We can, in fact, define \(S_0\text{,}\) the set of all permutations on the empty set.
Jun 7, 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].
