Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In abstract algebra, an isomorphism (Greek: isos "equal", and morphe "shape") is a bijective map f such that both f and its inverse f –1 are homomorphisms, i.e., structure-preserving mappings. In the more general setting of category theory, an isomorphism is a morphism f: X -> Y in a category for which there exists an "inverse" f –1: Y -> X, with the property that both f –1f = idX and f f –1 = idY. Informally, an isomorphism is a kind of mapping between objects, which shows a relationship between two properties or operations. If there exists an isomorphism between two structures, we call the two structures isomorphic. In a certain sense, isomorphic structures are structurally identical, if you choose to ignore finer-grained differences that may arise from how they are defined.

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