Editing Isomorphism (section)

{{Short description|In mathematics, invertible homomorphism}}
{{About|mathematics}}
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| footer    = The [[Group (mathematics)|group]] of fifth [[roots of unity]] under multiplication is isomorphic to the group of rotations of the regular pentagon under composition.
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| image1    = One5Root.svg
| alt1      = Fifth roots of unity
| image2    = Regular polygon 5 annotated.svg
| alt2      = Rotations of a pentagon
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In [[mathematics]], an '''isomorphism''' is a structure-preserving [[Map (mathematics)|mapping]] or [[morphism]] between two [[Mathematical structure|structures]] of the same type that can be reversed by an [[inverse function|inverse mapping]]. Two mathematical structures are '''isomorphic''' if an isomorphism exists between them. The word is derived {{ety|grc|''{{linktext|ἴσος}}'' (isos)|equal||''{{linktext|μορφή}}'' (morphe)|form, shape}}.

The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may often be identified. In [[mathematical jargon]], one says that two objects are the same [[up to]] an isomorphism. A common example where isomorphic structures cannot be identified is when the structures are substructures of a larger one. For example, all subspaces of dimension one of a [[vector space]] are isomorphic and cannot be identified.

An [[automorphism]] is an isomorphism from a structure to itself. An isomorphism between two structures is a '''canonical isomorphism'''  (a [[canonical map]] that is an isomorphism) if there is only one isomorphism between the two structures (as is the case for solutions of a [[universal property]]), or if the isomorphism is much more natural (in some sense) than other isomorphisms. For example, for every [[prime number]] {{mvar|p}}, all [[Field (mathematics)|fields]] with {{mvar|p}} elements are canonically isomorphic, with a unique isomorphism. The [[isomorphism theorems]] provide canonical isomorphisms that are not unique.

The term {{em|isomorphism}} is mainly used for [[algebraic structure]]s and [[category (mathematics)|categories]]. In the case of algebraic structures, mappings are called [[homomorphism]]s, and a homomorphism is an isomorphism [[if and only if]] it is [[bijective]]. 

In various areas of mathematics, isomorphisms have received specialized names, depending on the type of structure under consideration. For example:
* An [[isometry]] is an isomorphism of [[metric space]]s.
* A [[homeomorphism]] is an isomorphism of [[topological space]]s.
* A [[diffeomorphism]] is an isomorphism of spaces equipped with a [[differential structure]], typically [[differentiable manifold]]s.
* A [[symplectomorphism]] is an isomorphism of [[symplectic manifold]]s.
* A [[permutation]] is an automorphism of a [[set (mathematics)|set]].
* In [[geometry]], isomorphisms and automorphisms are often called [[transformation (function)|transformations]], for example [[rigid transformation]]s, [[affine transformation]]s, [[projective transformation]]s.

[[Category theory]], which can be viewed as a formalization of the concept of mapping between structures, provides a language that may be used to unify the approach to these different aspects of the basic idea.