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In mathematics, an **isomorphism** is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are **isomorphic** if an isomorphism exists between them. The word isomorphism is derived from the Ancient Greek: ἴσος *isos* "equal", and μορφή *morphe* "form" or "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 be identified. In mathematical jargon, one says that two objects are .

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 it 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, all fields with elements are canonically isomorphic, with a unique isomorphism. The isomorphism theorems provide canonical isomorphisms that are not unique.

The term is mainly used for algebraic structures. In this case, mappings are called homomorphisms, 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 spaces.
- A homeomorphism is an isomorphism of topological spaces.
- A diffeomorphism is an isomorphism of spaces equipped with a differential structure, typically differentiable manifolds.
- A permutation is an automorphism of a set.
- In geometry, isomorphisms and automorphisms are often called transformations, for example rigid transformations, affine transformations, projective transformations.

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.

Let

*\R*^{+}

*\R*

log*:**\R*^{+}*\to**\R*

log*(xy)*=log*x*+log*y*

*x,**y**\in**\R*^{+,}

*\exp**:**\R**\to**\R*^{+}

*\exp(x*+*y)*=*(\exp**x)(\exp**y)*

*x,**y**\in**\R,*

The identities

log*\exp**x*=*x*

*\exp*log*y*=*y*

log

*\exp*

log

log

The

log

Consider the group

*(\Z*_{6,}+*),*

*\left(\Z*_{2} x *\Z*_{3,}+*\right),*

These structures are isomorphic under addition, under the following scheme:$$\backslash begin(0,\; 0)\; \&\backslash mapsto\; 0\; \backslash \backslash (1,\; 1)\; \&\backslash mapsto\; 1\; \backslash \backslash (0,\; 2)\; \&\backslash mapsto\; 2\; \backslash \backslash (1,\; 0)\; \&\backslash mapsto\; 3\; \backslash \backslash (0,\; 1)\; \&\backslash mapsto\; 4\; \backslash \backslash (1,\; 2)\; \&\backslash mapsto\; 5\; \backslash \backslash \backslash end$$or in general

*(a,**b)**\mapsto**(*3*a*+4*b)**\mod*6*.*

For example,

*(*1*,*1*)*+*(*1*,*0*)*=*(*0*,*1*),*

1+3=4*.*

Even though these two groups "look" different in that the sets contain different elements, they are indeed **isomorphic**: their structures are exactly the same. More generally, the direct product of two cyclic groups

*\Z*_{m}

*\Z*_{n}

*(\Z*_{mn}*,*+*)*

If one object consists of a set *X* with a binary relation R and the other object consists of a set *Y* with a binary relation S then an isomorphism from *X* to *Y* is a bijective function

*f**:**X**\to**Y*

S is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partial order, total order, well-order, strict weak order, total preorder (weak order), an equivalence relation, or a relation with any other special properties, if and only if R is.

For example, R is an ordering ≤ and S an ordering

*\scriptstyle**\sqsubseteq,*

*f**:**X**\to**Y*

If

*X*=*Y,*

In algebra, isomorphisms are defined for all algebraic structures. Some are more specifically studied; for example:

- Linear isomorphisms between vector spaces; they are specified by invertible matrices.
- Group isomorphisms between groups; the classification of isomorphism classes of finite groups is an open problem.
- Ring isomorphism between rings.
- Field isomorphisms are the same as ring isomorphism between fields; their study, and more specifically the study of field automorphisms is an important part of Galois theory.

Just as the automorphisms of an algebraic structure form a group, the isomorphisms between two algebras sharing a common structure form a heap. Letting a particular isomorphism identify the two structures turns this heap into a group.

In mathematical analysis, the Laplace transform is an isomorphism mapping hard differential equations into easier algebraic equations.

In graph theory, an isomorphism between two graphs *G* and *H* is a bijective map *f* from the vertices of *G* to the vertices of *H* that preserves the "edge structure" in the sense that there is an edge from vertex *u* to vertex *v* in *G* if and only if there is an edge from

*f(u)*

*f(v)*

In mathematical analysis, an isomorphism between two Hilbert spaces is a bijection preserving addition, scalar multiplication, and inner product.

In early theories of logical atomism, the formal relationship between facts and true propositions was theorized by Bertrand Russell and Ludwig Wittgenstein to be isomorphic. An example of this line of thinking can be found in Russell's *Introduction to Mathematical Philosophy*.

In cybernetics, the good regulator or Conant–Ashby theorem is stated "Every good regulator of a system must be a model of that system". Whether regulated or self-regulating, an isomorphism is required between the regulator and processing parts of the system.

In category theory, given a category *C*, an isomorphism is a morphism

*f**:**a**\to**b*

*g**:**b**\to**a,*

*f**g*=1_{b}

*g**f*=1_{a.}

Two categories and are isomorphic if there exist functors

*F**:**C**\to**D*

*G**:**D**\to**C*

*FG*=1_{D}

*GF*=1_{C}

In a concrete category (that is, a category whose objects are sets (perhaps with extra structure) and whose morphisms are structure-preserving functions), such as the category of topological spaces or categories of algebraic objects (like the category of groups, the category of rings, and the category of modules), an isomorphism must be bijective on the underlying sets. In algebraic categories (specifically, categories of varieties in the sense of universal algebra), an isomorphism is the same as a homomorphism which is bijective on underlying sets. However, there are concrete categories in which bijective morphisms are not necessarily isomorphisms (such as the category of topological spaces).

See also: Equality (mathematics).

In certain areas of mathematics, notably category theory, it is valuable to distinguish between on the one hand and on the other. Equality is when two objects are exactly the same, and everything that is true about one object is true about the other, while an isomorphism implies everything that is true about a designated part of one object's structure is true about the other's. For example, the sets$$A\; =\; \backslash left\backslash \; \backslash quad\; \backslash text\; \backslash quad\; B\; =\; \backslash $$are ; they are merely different representations—the first an intensional one (in set builder notation), and the second extensional (by explicit enumeration)—of the same subset of the integers. By contrast, the sets

*\{A,**B,**C\}*

*\{*1*,*2*,*3*\}*

A*\mapsto*1*,*B*\mapsto*2*,*C*\mapsto*3*,*

A*\mapsto*3*,*B*\mapsto*2*,*C*\mapsto*1*,*

Sometimes the isomorphisms can seem obvious and compelling, but are still not equalities. As a simple example, the genealogical relationships among Joe, John, and Bobby Kennedy are, in a real sense, the same as those among the American football quarterbacks in the Manning family: Archie, Peyton, and Eli. The father-son pairings and the elder-brother-younger-brother pairings correspond perfectly. That similarity between the two family structures illustrates the origin of the word (Greek *iso*-, "same", and -*morph*, "form" or "shape"). But because the Kennedys are not the same people as the Mannings, the two genealogical structures are merely isomorphic and not equal.

*V*^{*}=*\left\{**\varphi**:**V**\to*K*\right\}*

K*.*

*\scriptstyle**V*l{\overset{\sim}{\to}}*V*^{*.}

*u,**v**\in**V,*

*V*^{**}=*\left\{**x**:**V*^{*}*\to*K*\right\}*

*v**\in**V*and*\varphi**\in**V*^{*,}

This leads to a third notion, that of a natural isomorphism: while

*V*

*V*^{**}

*\scriptstyle**V*l{\overset{\sim}{\to}}*V*^{**}*,*

However, there is a case where the distinction between natural isomorphism and equality is usually not made. That is for the objects that may be characterized by a universal property. In fact, there is a unique isomorphism, necessarily natural, between two objects sharing the same universal property. A typical example is the set of real numbers, which may be defined through infinite decimal expansion, infinite binary expansion, Cauchy sequences, Dedekind cuts and many other ways. Formally, these constructions define different objects which are all solutions with the same universal property. As these objects have exactly the same properties, one may forget the method of construction and consider them as equal. This is what everybody does when referring to " set of the real numbers". The same occurs with quotient spaces: they are commonly constructed as sets of equivalence classes. However, referring to a set of sets may be counterintuitive, and so quotient spaces are commonly considered as a pair of a set of undetermined objects, often called "points", and a surjective map onto this set.

If one wishes to distinguish between an arbitrary isomorphism (one that depends on a choice) and a natural isomorphism (one that can be done consistently), one may write

≈

*V* ≈ *V*^{*}

*V**\cong**V*^{**}*.*

*\widehat{\Complex}*

*\Complex**\cup**\{*inf*ty**\}*

*\R*^{3,}

*\Complex**\cong**\R*^{2}

*\Complex*^{2.}

In the context of category theory, objects are usually at most isomorphic—indeed, a motivation for the development of category theory was showing that different constructions in homology theory yielded equivalent (isomorphic) groups. Given maps between two objects *X* and *Y*, however, one asks if they are equal or not (they are both elements of the set

*\hom(X,**Y),*

- Bisimulation
- Equivalence relation
- Heap (mathematics)
- Isometry
- Isomorphism class
- Isomorphism theorem
- Universal property
- Coherent isomorphism

- Book: Vinberg, Ėrnest Borisovich. A Course in Algebra. American Mathematical Society. 2003. 9780821834138. 3.
- have a conventional order, namely alphabetical order, and similarly 1, 2, 3 have the order from the integers, and thus one particular isomorphism is "natural", namely
*A,**B,**C*More formally, as these are isomorphic, but not naturally isomorphic (there are multiple choices of isomorphism), while as they are naturally isomorphic (there is a unique isomorphism, given above), since finite total orders are uniquely determined up to unique isomorphism by cardinality.A

*\mapsto*1*,*B*\mapsto*2*,*C*\mapsto*3*.*This intuition can be formalized by saying that any two finite totally ordered sets of the same cardinality have a natural isomorphism, the one that sends the least element of the first to the least element of the second, the least element of what remains in the first to the least element of what remains in the second, and so forth, but in general, pairs of sets of a given finite cardinality are not naturally isomorphic because there is more than one choice of map—except if the cardinality is 0 or 1, where there is a unique choice.

- In fact, there are precisely different isomorphisms between two sets with three elements. This is equal to the number of automorphisms of a given three-element set (which in turn is equal to the order of the symmetric group on three letters), and more generally one has that the set of isomorphisms between two objects, denoted
3

*!*=6is a torsor for the automorphism group of*\operatorname{Iso}(A,B),**A,*and also a torsor for the automorphism group of*\operatorname{Aut}(A)**B.*In fact, automorphisms of an object are a key reason to be concerned with the distinction between isomorphism and equality, as demonstrated in the effect of change of basis on the identification of a vector space with its dual or with its double dual, as elaborated in the sequel. - Being precise, the identification of the complex numbers with the real plane,$$\backslash C\; \backslash cong\; \backslash R\; \backslash cdot\; 1\; \backslash oplus\; \backslash R\; \backslash cdot\; i\; =\; \backslash R^2$$depends on a choice of one can just as easily choose
*i;*which yields a different identification—formally, complex conjugation is an automorphism—but in practice one often assumes that one has made such an identification.*(*-*i),*