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In abstract algebra, an isomorphism can be thought of as a function that preserves sets and the relations among their elements. Formally an isomorphism between two groups is a bijective (one-to-one and onto) map φ that preserves the group operation. That is, for x and y elements of a group G, φ(xy) = φ(x)φ(y).

As an example, consider the group Z of integers under ordinary addition, and the function φ(x) = 2x which maps elements of Z to elements of the even integers 2Z, since φ(ab) = 2(ab) = 2a2b = φ(a)φ(b), this is an example of an isomorphism that is also an automorphism (an isomorphism from a set of elements onto itself).