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In abstract algebra, the symmetric group S''n'' on a finite set of ''n'' symbols is the group whose elements are all the permutation operations that can be performed on ''n'' distinct symbols, and whose group operation is the composition of such permutation operations, which are defined as bijective functions from the set of symbols to itself.〔Jacobson (2009), p. 31.〕 Since there are ''n''! (''n'' factorial) possible permutation operations that can be performed on a tuple composed of ''n'' symbols, it follows that the order (the number of elements) of the symmetric group S''n'' is ''n''!. Although symmetric groups can be defined on infinite sets as well, this article discusses only the finite symmetric groups: their applications, their elements, their conjugacy classes, a finite presentation, their subgroups, their automorphism groups, and their representation theory. For the remainder of this article, "symmetric group" will mean a symmetric group on a finite set. The symmetric group is important to diverse areas of mathematics such as Galois theory, invariant theory, the representation theory of Lie groups, and combinatorics. Cayley's theorem states that every group ''G'' is isomorphic to a subgroup of the symmetric group on ''G''. == Definition and first properties == The symmetric group on a finite set ''X'' is the group whose elements are all bijective functions from ''X'' to ''X'' and whose group operation is that of function composition.〔 For finite sets, "permutations" and "bijective functions" refer to the same operation, namely rearrangement. The symmetric group of degree ''n'' is the symmetric group on the set The symmetric group on a set ''X'' is denoted in various ways including S''X'', 𝔖''X'', Σ''X'', ''X''! and Sym(''X'').〔 If ''X'' is the set then the symmetric group on ''X'' is also denoted S''n'',〔 𝔖''n'', Σ''n'', and Sym(''n''). Symmetric groups on infinite sets behave quite differently from symmetric groups on finite sets, and are discussed in , , and . This article concentrates on the finite symmetric groups. The symmetric group on a set of ''n'' elements has order ''n''! 〔Jacobson (2009), p. 32. Theorem 1.1.〕 It is abelian if and only if . For and (the empty set and the singleton set) the symmetric group is trivial (note that this agrees with ), and in these cases the alternating group equals the symmetric group, rather than being an index two subgroup. The group S''n'' is solvable if and only if . This is an essential part of the proof of the Abel–Ruffini theorem that shows that for every there are polynomials of degree ''n'' which are not solvable by radicals, i.e., the solutions cannot be expressed by performing a finite number of operations of addition, subtraction, multiplication, division and root extraction on the polynomial's coefficients. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Symmetric group」の詳細全文を読む スポンサード リンク
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