automorphism
in mathematics, a correspondence that associates to every element in a set a unique element of the set (perhaps itself) and for which there is a companion correspondence, known as its inverse, such that one followed by the other produces the identity correspondence (I); i.e., the correspondence that associates every element with itself. In symbols, if f is the original correspondence and g is its inverse, then g(f(a))=I(a)=a=I(a)=f(g(a)) for every a in the set. Furthermore, operations such as addition and multiplication must be preserved; for example, f(a+b)=f(a)+f(b) and f(ab)=f(a)f(b) for every a and b in the set
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