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In group theory, Cayley's theorem, named in honour of Arthur Cayley, states that every group ''G'' is isomorphic to a subgroup of the symmetric group acting on ''G''.〔Jacobson (2009), p. 38.〕 This can be understood as an example of the group action of ''G'' on the elements of ''G''.〔Jacobson (2009), p. 72, ex. 1.〕 A permutation of a set ''G'' is any bijective function taking ''G'' onto ''G''; and the set of all such functions forms a group under function composition, called ''the symmetric group on'' ''G'', and written as Sym(''G'').〔Jacobson (2009), p. 31.〕 Cayley's theorem puts all groups on the same footing, by considering any group (including infinite groups such as (''R'',+)) as a permutation group of some underlying set. Thus, theorems that are true for subgroups of permutation groups are true for groups in general. Nevertheless, Alperin and Bell note that "in general the fact that finite groups are imbedded in symmetric groups has not influenced the methods used to study finite groups". The regular action used in the standard proof of Cayley's theorem does not produce the representation of ''G'' in a ''minimal''-order permutation group. For example, , itself already a symmetric group of order 6, would be represented by the regular action as a subgroup of (a group of order 720). The problem of finding an embedding of a group in a minimal-order symmetric group is rather more difficult. == History == While it seems elementary enough, it should be noted that at the time, the modern definitions didn't exist, and when Cayley introduced what are now called ''groups'' it wasn't immediately clear that this was equivalent to the previously known groups, which are now called ''permutation groups''. Cayley's theorem unifies the two. Although Burnside attributes the theorem to Jordan, Eric Nummela nonetheless argues that the standard name—"Cayley's Theorem"—is in fact appropriate. Cayley, in his original 1854 paper, showed that the correspondence in the theorem is one-to-one, but he failed to explicitly show it was a homomorphism (and thus an isomorphism). However, Nummela notes that Cayley made this result known to the mathematical community at the time, thus predating Jordan by 16 years or so. 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「Cayley's theorem」の詳細全文を読む スポンサード リンク
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