# Isomorphism

Last updated
The group of fifth roots of unity under multiplication is isomorphic to the group of rotations of the regular pentagon under composition.

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".

## Contents

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 the same up to an isomorphism.[ citation needed ]

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

The term isomorphism 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:

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.

## Examples

### Logarithm and exponential

Let ${\displaystyle \mathbb {R} ^{+}}$ be the multiplicative group of positive real numbers, and let ${\displaystyle \mathbb {R} }$ be the additive group of real numbers.

The logarithm function ${\displaystyle \log$ :\mathbb {R} ^{+}\to \mathbb {R} } satisfies ${\displaystyle \log(xy)=\log x+\log y}$ for all ${\displaystyle x,y\in \mathbb {R} ^{+},}$ so it is a group homomorphism. The exponential function ${\displaystyle \exp$ :\mathbb {R} \to \mathbb {R} ^{+}} satisfies ${\displaystyle \exp(x+y)=(\exp x)(\exp y)}$ for all ${\displaystyle x,y\in \mathbb {R} ,}$ so it too is a homomorphism.

The identities ${\displaystyle \log \exp x=x}$ and ${\displaystyle \exp \log y=y}$ show that ${\displaystyle \log }$ and ${\displaystyle \exp }$ are inverses of each other. Since ${\displaystyle \log }$ is a homomorphism that has an inverse that is also a homomorphism, ${\displaystyle \log }$ is an isomorphism of groups.

The ${\displaystyle \log }$ function is an isomorphism which translates multiplication of positive real numbers into addition of real numbers. This facility makes it possible to multiply real numbers using a ruler and a table of logarithms, or using a slide rule with a logarithmic scale.

### Integers modulo 6

Consider the group ${\displaystyle (\mathbb {Z} _{6},+),}$ the integers from 0 to 5 with addition modulo  6. Also consider the group ${\displaystyle \left(\mathbb {Z} _{2}\times \mathbb {Z} _{3},+\right),}$ the ordered pairs where the x coordinates can be 0 or 1, and the y coordinates can be 0, 1, or 2, where addition in the x-coordinate is modulo 2 and addition in the y-coordinate is modulo 3.

These structures are isomorphic under addition, under the following scheme:

{\displaystyle {\begin{alignedat}{4}(0,0)&\mapsto 0\\(1,1)&\mapsto 1\\(0,2)&\mapsto 2\\(1,0)&\mapsto 3\\(0,1)&\mapsto 4\\(1,2)&\mapsto 5\\\end{alignedat}}}

or in general ${\displaystyle (a,b)\mapsto (3a+4b)\mod 6.}$

For example, ${\displaystyle (1,1)+(1,0)=(0,1),}$ which translates in the other system as ${\displaystyle 1+3=4.}$

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 ${\displaystyle \mathbb {Z} _{m}}$ and ${\displaystyle \mathbb {Z} _{n}}$ is isomorphic to ${\displaystyle (\mathbb {Z} _{mn},+)}$ if and only if m and n are coprime, per the Chinese remainder theorem.

### Relation-preserving isomorphism

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 ${\displaystyle f:X\to Y}$ such that: [1]

${\displaystyle \operatorname {S} (f(u),f(v))\quad {\text{ if and only if }}\quad \operatorname {R} (u,v)}$

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 ${\displaystyle \scriptstyle \sqsubseteq ,}$ then an isomorphism from X to Y is a bijective function ${\displaystyle f:X\to Y}$ such that

${\displaystyle f(u)\sqsubseteq f(v)\quad {\text{ if and only if }}\quad u\leq v.}$

Such an isomorphism is called an order isomorphism or (less commonly) an isotone isomorphism.

If ${\displaystyle X=Y,}$ then this is a relation-preserving automorphism.

## Applications

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

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 ${\displaystyle f(u)}$ to ${\displaystyle f(v)}$ in H. See graph isomorphism.

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.

## Category theoretic view

In category theory, given a category C, an isomorphism is a morphism ${\displaystyle f:a\to b}$ that has an inverse morphism ${\displaystyle g:b\to a,}$ that is, ${\displaystyle fg=1_{b}}$ and ${\displaystyle gf=1_{a}.}$ For example, a bijective linear map is an isomorphism between vector spaces, and a bijective continuous function whose inverse is also continuous is an isomorphism between topological spaces, called a homeomorphism.

Two categories C and D are isomorphic if there exist functors ${\displaystyle F:C\to D}$ and ${\displaystyle G:D\to C}$ which are mutually inverse to each other, that is, ${\displaystyle FG=1_{D}}$ (the identity functor on D) and ${\displaystyle GF=1_{C}}$ (the identity functor on C).

### Isomorphism vs. bijective morphism

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).

## Relation with equality

In certain areas of mathematics, notably category theory, it is valuable to distinguish between equality on the one hand and isomorphism on the other. [2] 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

${\displaystyle A=\left\{x\in \mathbb {Z} \mid x^{2}<2\right\}\quad {\text{ and }}\quad B=\{-1,0,1\}}$

are equal; 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 ${\displaystyle \{A,B,C\}}$ and ${\displaystyle \{1,2,3\}}$ are not equal—the first has elements that are letters, while the second has elements that are numbers. These are isomorphic as sets, since finite sets are determined up to isomorphism by their cardinality (number of elements) and these both have three elements, but there are many choices of isomorphism—one isomorphism is

${\displaystyle {\text{A}}\mapsto 1,{\text{B}}\mapsto 2,{\text{C}}\mapsto 3,}$ while another is ${\displaystyle {\text{A}}\mapsto 3,{\text{B}}\mapsto 2,{\text{C}}\mapsto 1,}$

and no one isomorphism is intrinsically better than any other. [note 1] [note 2] On this view and in this sense, these two sets are not equal because one cannot consider them identical: one can choose an isomorphism between them, but that is a weaker claim than identity—and valid only in the context of the chosen isomorphism.

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 isomorphism (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.

Another example is more formal and more directly illustrates the motivation for distinguishing equality from isomorphism: the distinction between a finite-dimensional vector space V and its dual space ${\displaystyle V^{*}=\left\{\varphi :V\to \mathbf {K} \right\}}$ of linear maps from V to its field of scalars ${\displaystyle \mathbf {K} .}$ These spaces have the same dimension, and thus are isomorphic as abstract vector spaces (since algebraically, vector spaces are classified by dimension, just as sets are classified by cardinality), but there is no "natural" choice of isomorphism ${\displaystyle \scriptstyle V\mathrel {\overset {\sim }{\to }} V^{*}.}$ If one chooses a basis for V, then this yields an isomorphism: For all ${\displaystyle u,v\in V,}$

${\displaystyle v\mathrel {\overset {\sim }{\mapsto }} \phi _{v}\in V^{*}\quad {\text{ such that }}\quad \phi _{v}(u)=v^{\mathrm {T} }u.}$

This corresponds to transforming a column vector (element of V) to a row vector (element of V*) by transpose, but a different choice of basis gives a different isomorphism: the isomorphism "depends on the choice of basis". More subtly, there is a map from a vector space V to its double dual ${\displaystyle V^{**}=\left\{x:V^{*}\to \mathbf {K} \right\}}$ that does not depend on the choice of basis: For all ${\displaystyle v\in V{\text{ and }}\varphi \in V^{*},}$

${\displaystyle v\mathrel {\overset {\sim }{\mapsto }} x_{v}\in V^{**}\quad {\text{ such that }}\quad x_{v}(\phi )=\phi (v).}$

This leads to a third notion, that of a natural isomorphism: while ${\displaystyle V}$ and ${\displaystyle V^{**}}$ are different sets, there is a "natural" choice of isomorphism between them. This intuitive notion of "an isomorphism that does not depend on an arbitrary choice" is formalized in the notion of a natural transformation; briefly, that one may consistently identify, or more generally map from, a finite-dimensional vector space to its double dual, ${\displaystyle \scriptstyle V\mathrel {\overset {\sim }{\to }} V^{**},}$ for any vector space in a consistent way. Formalizing this intuition is a motivation for the development of category theory.

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 "the 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 ${\displaystyle \,\approx \,}$ for an unnatural isomorphism and for a natural isomorphism, as in ${\displaystyle V\approx V^{*}}$ and ${\displaystyle V\cong V^{**}.}$ This convention is not universally followed, and authors who wish to distinguish between unnatural isomorphisms and natural isomorphisms will generally explicitly state the distinction.

Generally, saying that two objects are equal is reserved for when there is a notion of a larger (ambient) space that these objects live in. Most often, one speaks of equality of two subsets of a given set (as in the integer set example above), but not of two objects abstractly presented. For example, the 2-dimensional unit sphere in 3-dimensional space

${\displaystyle S^{2}:=\left\{(x,y,z)\in \mathbb {R} ^{3}\mid x^{2}+y^{2}+z^{2}=1\right\}}$

and the Riemann sphere ${\displaystyle {\widehat {\mathbb {C} }}}$

which can be presented as the one-point compactification of the complex plane ${\displaystyle \mathbb {C} \cup \{\infty \}}$or as the complex projective line (a quotient space)

${\displaystyle \mathbf {P} _{\mathbb {C} }^{1}:=\left(\mathbb {C} ^{2}\setminus \{(0,0)\}\right)/\left(\mathbb {C} ^{*}\right)}$

are three different descriptions for a mathematical object, all of which are isomorphic, but not equal because they are not all subsets of a single space: the first is a subset of ${\displaystyle \mathbb {R} ^{3},}$ the second is ${\displaystyle \mathbb {C} \cong \mathbb {R} ^{2}}$ [note 3] plus an additional point, and the third is a subquotient of ${\displaystyle \mathbb {C} ^{2}.}$

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 ${\displaystyle \hom(X,Y),}$ hence equality is the proper relationship), particularly in commutative diagrams.

## Notes

1. ${\displaystyle A,B,C}$ 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
${\displaystyle {\text{A}}\mapsto 1,{\text{B}}\mapsto 2,{\text{C}}\mapsto 3.}$
More formally, as sets these are isomorphic, but not naturally isomorphic (there are multiple choices of isomorphism), while as ordered sets they are naturally isomorphic (there is a unique isomorphism, given above), since finite total orders are uniquely determined up to unique isomorphism by cardinality. 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.
2. In fact, there are precisely ${\displaystyle 3!=6}$ 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 ${\displaystyle \operatorname {Iso} (A,B),}$ is a torsor for the automorphism group of A,${\displaystyle \operatorname {Aut} (A)}$ and also a torsor for the automorphism group of 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.
3. Being precise, the identification of the complex numbers with the real plane,
${\displaystyle \mathbb {C} \cong \mathbb {R} \cdot 1\oplus \mathbb {R} \cdot i=\mathbb {R} ^{2}}$
depends on a choice of ${\displaystyle i;}$ one can just as easily choose ${\displaystyle (-i),}$ which yields a different identification—formally, complex conjugation is an automorphism—but in practice one often assumes that one has made such an identification.

## Related Research Articles

In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism group. It is, loosely speaking, the symmetry group of the object.

In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two groups, then the groups are called isomorphic. From the standpoint of group theory, isomorphic groups have the same properties and need not be distinguished.

In mathematics, especially in category theory and homotopy theory, a groupoid generalises the notion of group in several equivalent ways. A groupoid can be seen as a:

In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the polynomials that give rise to them via Galois groups is called Galois theory, so named in honor of Évariste Galois who first discovered them.

In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism group of the structure. It is said that the group acts on the space or structure. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures drawn in it. In particular, it acts on the set of all triangles. Similarly, the group of symmetries of a polyhedron acts on the vertices, the edges, and the faces of the polyhedron.

In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type. The word homomorphism comes from the Ancient Greek language: ὁμός meaning "same" and μορφή meaning "form" or "shape". However, the word was apparently introduced to mathematics due to a (mis)translation of German ähnlich meaning "similar" to ὁμός meaning "same". The term "homomorphism" appeared as early as 1892, when it was attributed to the German mathematician Felix Klein (1849–1925).

In mathematics, a Lie group is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract, generic concept of multiplication and the taking of inverses (division). Combining these two ideas, one obtains a continuous group where points can be multiplied together, and their inverse can be taken. If, in addition, the multiplication and taking of inverses are defined to be smooth (differentiable), one obtains a Lie group.

In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if R and S are rings, then a ring homomorphism is a function f : RS such that f is:

In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed, this intuition can be formalized to define so-called functor categories. Natural transformations are, after categories and functors, one of the most fundamental notions of category theory and consequently appear in the majority of its applications.

In mathematics, specifically abstract algebra, the isomorphism theorems are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and various other algebraic structures. In universal algebra, the isomorphism theorems can be generalized to the context of algebras and congruences.

In category theory, an epimorphism is a morphism f : XY that is right-cancellative in the sense that, for all objects Z and all morphisms g1, g2: YZ,

In mathematics, an algebra homomorphism is a homomorphism between two associative algebras. More precisely, if A and B are algebras over a field K, it is a function such that for all k in K and x, y in A,

In mathematics, in particular algebraic geometry, a moduli space is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spaces frequently arise as solutions to classification problems: If one can show that a collection of interesting objects can be given the structure of a geometric space, then one can parametrize such objects by introducing coordinates on the resulting space. In this context, the term "modulus" is used synonymously with "parameter"; moduli spaces were first understood as spaces of parameters rather than as spaces of objects. A variant of moduli spaces are formal moduli.

In mathematics, an algebraic torus, where a one dimensional torus is typically denoted by , , or , is a type of commutative affine algebraic group commonly found in projective algebraic geometry and toric geometry. Higher dimensional algebraic tori can be modelled as a product of algebraic groups . These groups were named by analogy with the theory of tori in Lie group theory. For example, over the complex numbers the algebraic torus is isomorphic to the group scheme , which is the scheme theoretic analogue of the Lie group . In fact, any -action on a complex vector space can be pulled back to a -action from the inclusion as real manifolds.

In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often by means of an involution operation: if the dual of A is B, then the dual of B is A. Such involutions sometimes have fixed points, so that the dual of A is A itself. For example, Desargues' theorem is self-dual in this sense under the standard duality in projective geometry.

In mathematics, the Grothendieck group construction constructs an abelian group from a commutative monoid M in the most universal way, in the sense that any abelian group containing a homomorphic image of M will also contain a homomorphic image of the Grothendieck group of M. The Grothendieck group construction takes its name from a specific case in category theory, introduced by Alexander Grothendieck in his proof of the Grothendieck–Riemann–Roch theorem, which resulted in the development of K-theory. This specific case is the monoid of isomorphism classes of objects of an abelian category, with the direct sum as its operation.

In mathematics, in particular the subfield of algebraic geometry, a rational map or rational mapping is a kind of partial function between algebraic varieties. This article uses the convention that varieties are irreducible.

In algebraic geometry, a function from a quasi-affine variety to its underlying field , is a regular function if for an arbitrary point in there exists an open neighborhood U around that point such that f can be expressed by fraction like in which are polynomials on the ambient affine space of such that h is nowhere zero on U. A regular map from an arbitrary variety to affine space is a map given by n-tuple of regular functions. A morphism between two varieties is a continuous map like such that for every open set and every regular function like , the function is regular. The composition of morphisms are morphisms, so they constitute a category. In this category, a morphism which has an inverse is called isomorphism. We say that two varieties are isomorphic if there exist isomorphism between them or equivalently if their coordinate rings are isomorphic as algebras over their underlying fields.

In mathematics, Lie group–Lie algebra correspondence allows one to correspond a Lie group to a Lie algebra or vice versa, and study the conditions for such a relationship. Lie groups that are isomorphic to each other have Lie algebras that are isomorphic to each other, but the converse is not necessarily true. One obvious counterexample is and which are non-isomorphic to each other as Lie groups but their Lie algebras are isomorphic to each other. However, by restricting our attention to the simply connected Lie groups, the Lie group-Lie algebra correspondence will be one-to-one.

In mathematics, the automorphism group of an object X is the group consisting of automorphisms of X. For example, if X is a finite-dimensional vector space, then the automorphism group of X is the group of invertible linear transformations from X to itself. If instead X is a group, then its automorphism group is the group consisting of all group automorphisms of X.

## References

1. Vinberg, Ėrnest Borisovich (2003). A Course in Algebra. American Mathematical Society. p. 3. ISBN   9780821834138.