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A map f between groups A and B is a homomorphism of A into B if
f(a1 * a2) = f(a1) * f(a2) for all a1, a2 in A.
where the *s are the respective group operations.
(from Greek homoios morphe, "similar form"), a special correspondence between the members (elements) of two algebraic systems, such as two groups, two rings, or two fields. Two homomorphic systems have the same basic structure, and, while their elements and operations may appear entirely different, results on one system often apply as well to the other system. Thus, if a new system can be shown to be homomorphic to a known system, certain known features of one can be applied to the other, thereby simplifying the analysis of the new system