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In mathematics a group is a set together with a binary operation on the set called multiplication that obeys the group axioms. The axiom of choice is an axiom of ZFC set theory which in one form states that every set can be wellordered. In ZF set theory, i.e. ZFC without the axiom of choice, the following statements are equivalent: * For every nonempty set there exists a binary operation such that is a group.〔A cancellative binary operation suffices, i.e. such that is a cancellative magma. See below.〕 * The axiom of choice is true. == A group structure implies the axiom of choice == In this section it is assumed that every set can be endowed with a group structure . Let be a set. Let be the Hartogs number of . This is the least cardinal number such that there is no injection from into . It exists without the assumption of the axiom of choice. Assume here for technical simplicity of proof that has no ordinal. Let denote multiplication in the group . For any there is an such that . Suppose not. Then there is an such that ''for all'' . But by elementary group theory, the are all different as α ranges over (i). Thus such a gives an injection from into . This is impossible since is a cardinal such that no injection into exists. Now define a map of into endowed with the lexicographical wellordering by sending to the least such that . By the above reasoning the map exists and is unique since least elements of subsets of wellordered sets are unique. It is, by elementary group theory, injective. Finally, define a wellordering on by if . It follows that every set can be wellordered and thus that the axiom of choice is true. For the crucial property expressed in (i) above to hold, and hence the whole proof, it is sufficient for to be a cancellative magma, e.g. a quasigroup. The cancellation property is enough to ensure that the are all different. 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「Group structure and the axiom of choice」の詳細全文を読む スポンサード リンク
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