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In mathematics, an **isomorphism** (from the Ancient Greek: ἴσος *isos* "equal", and μορφή *morphe* "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that admits an inverse.^{[note 1]} Two mathematical objects are **isomorphic** if an isomorphism exists between them. An *automorphism* is an isomorphism whose source and target coincide. The interest of isomorphisms lies in the fact that two isomorphic objects cannot be distinguished by using only the properties used to define morphisms; thus isomorphic objects may be considered the same as long as one considers only these properties and their consequences.

For most algebraic structures, including groups and rings, a homomorphism is an isomorphism if and only if it is bijective.

In topology, where the morphisms are continuous functions, isomorphisms are also called *homeomorphisms* or *bicontinuous functions*. In mathematical analysis, where the morphisms are differentiable functions, isomorphisms are also called *diffeomorphisms*.

A **canonical isomorphism** is a canonical map that is an isomorphism. Two objects are said to be **canonically isomorphic** if there is a canonical isomorphism between them. For example, the canonical map from a finite-dimensional vector space *V* to its second dual space is a canonical isomorphism; on the other hand, *V* is isomorphic to its dual space but not canonically in general.

Isomorphisms are formalized using category theory. A morphism *f* : *X* → *Y* in a category is an isomorphism if it admits a two-sided inverse, meaning that there is another morphism *g* : *Y* → *X* in that category such that *gf* = 1_{X} and *fg* = 1_{Y} , where 1_{X} and 1_{Y} are the identity morphisms of *X* and *Y*, respectively.^{[1]}

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**Isomorphism - Wikipedia, the free encyclopedia**

s article is about mathematics. For other uses, see Isomorphism (disambiguation). The group of fifth roots of unity under multiplication is isomorphic to the group of rotations of the regular pentagon under composition. <span>In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that admits an inverse. [note 1] Two mathematical objects are isomorphic if an isomorphism exists between them. An automorphism is an isomorphism whose source and target coincide. The interest of isomorphisms lies in the fact that two isomorphic objects cannot be distinguished by using only the properties used to define morphisms; thus isomorphic objects may be considered the same as long as one considers only these properties and their consequences. For most algebraic structures, including groups and rings, a homomorphism is an isomorphism if and only if it is bijective. In topology, where the morphisms are continuous functions, isomorphisms are also called homeomorphisms or bicontinuous functions. In mathematical analysis, where the morphisms are differentiable functions, isomorphisms are also called diffeomorphisms. A canonical isomorphism is a canonical map that is an isomorphism. Two objects are said to be canonically isomorphic if there is a canonical isomorphism between them. For example, the canonical map from a finite-dimensional vector space V to its second dual space is a canonical isomorphism; on the other hand, V is isomorphic to its dual space but not canonically in general. Isomorphisms are formalized using category theory. A morphism f : X → Y in a category is an isomorphism if it admits a two-sided inverse, meaning that there is another morphism g : Y → X in that category such that gf = 1 X and fg = 1 Y , where 1 X and 1 Y are the identity morphisms of X and Y, respectively. [1] Contents 1 Examples 1.1 Logarithm and exponential 1.2 Integers modulo 6 1.3 Relation-preserving isomorphism 2 Isomorphism vs. bijective morphism 3 Applications 4 Relation with equ

s article is about mathematics. For other uses, see Isomorphism (disambiguation). The group of fifth roots of unity under multiplication is isomorphic to the group of rotations of the regular pentagon under composition. <span>In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that admits an inverse. [note 1] Two mathematical objects are isomorphic if an isomorphism exists between them. An automorphism is an isomorphism whose source and target coincide. The interest of isomorphisms lies in the fact that two isomorphic objects cannot be distinguished by using only the properties used to define morphisms; thus isomorphic objects may be considered the same as long as one considers only these properties and their consequences. For most algebraic structures, including groups and rings, a homomorphism is an isomorphism if and only if it is bijective. In topology, where the morphisms are continuous functions, isomorphisms are also called homeomorphisms or bicontinuous functions. In mathematical analysis, where the morphisms are differentiable functions, isomorphisms are also called diffeomorphisms. A canonical isomorphism is a canonical map that is an isomorphism. Two objects are said to be canonically isomorphic if there is a canonical isomorphism between them. For example, the canonical map from a finite-dimensional vector space V to its second dual space is a canonical isomorphism; on the other hand, V is isomorphic to its dual space but not canonically in general. Isomorphisms are formalized using category theory. A morphism f : X → Y in a category is an isomorphism if it admits a two-sided inverse, meaning that there is another morphism g : Y → X in that category such that gf = 1 X and fg = 1 Y , where 1 X and 1 Y are the identity morphisms of X and Y, respectively. [1] Contents 1 Examples 1.1 Logarithm and exponential 1.2 Integers modulo 6 1.3 Relation-preserving isomorphism 2 Isomorphism vs. bijective morphism 3 Applications 4 Relation with equ

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