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Definition: The commutator aba^(-1)b^(-1) is an element associated to a pair a, b of elements of a group.
Source: Algebra, second edition by Michael Artin
Example: Two elements a and b of a group commute, ab=ba, if and only aba^(-1)=b, and this is true if and only if aba^(-1)b^(-1)=1.