Group structure and the axiom of choice information
Ernst Zermelo in 1904 proved the wellordering theorem using what was to become known as the axiom of choice.
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 X there exists a binary operation • such that (X, •) is a group.[1]
The axiom of choice is true.
^A cancellative binary operation suffices, i.e. such that (X, •) is a cancellative magma. See below.
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