Isomorphism

In mathematics, an isomorphism is an invertible map that preserves the specified structure of mathematical objects. Two objects are called isomorphic when such a map exists, commonly written $A\cong B$. They need not have identical elements or descriptions: the correspondence makes their relevant operations or relations match. What counts as relevant depends on the setting—for example, group operations, vector addition and scalar multiplication, or topological structure. Category theory expresses the common idea as a morphism possessing a two-sided inverse. (sites.math.northwestern.edu)

Structure preservation and invertibility

For algebraic objects with binary operations, a function $f:A\to B$ preserves an operation when

A map preserving the specified algebraic operations is a homomorphism. For groups, rings, and vector spaces, a bijective homomorphism is an isomorphism: its inverse automatically preserves those operations. Bijectivity requires both injectivity, meaning distinct inputs have distinct outputs, and surjectivity, meaning every target element is reached. (jmilne.org)

For structures defined using relations, preservation must generally work in both directions. For example, a relation $R$ must satisfy

Preserving a relation only in the forward direction can lose information. Likewise, a bijective structure-preserving map need not be an isomorphism in every setting: the inverse must also be an admissible structure-preserving map. This distinction is important in topology. (maths.tcd.ie)

Algebraic examples

In group theory, an isomorphism preserves multiplication, identity elements, and inverses. A basic example identifies the additive group of integers with the multiplicative group $\{2^n:n\in\mathbb Z\}$ through $n\mapsto2^n$. The correspondence respects the operations because $2^{m+n}=2^m2^n$. Isomorphic groups therefore share structural properties, including commutativity and the orders of corresponding elements. (jmilne.org)

In abstract algebra, an isomorphism of rings preserves addition and multiplication, and also the multiplicative identity when this is required by the chosen convention. An isomorphism of fields preserves their field operations. The structures being compared must thus be specified, rather than inferred merely from their underlying sets. (categorytheory.gitlab.io)

The first isomorphism theorem for groups states that a homomorphism $\varphi:G\to H$ induces

Here the quotient identifies elements whose images coincide. The theorem describes precisely how removing the information collapsed by a homomorphism produces a structure isomorphic to its image. (jmilne.org)

Vector spaces and coordinates

In linear algebra, an isomorphism between vector spaces over the same field is a bijective linear map. It preserves vector addition and scalar multiplication, and carries a basis to a basis. Finite-dimensional vector spaces over a fixed field are isomorphic exactly when they have the same dimension. Consequently, every $n$-dimensional real vector space is isomorphic to $\mathbb R^n$. (opentext.uleth.ca)

Choosing an ordered basis $B=(v_1,\ldots,v_n)$ gives the coordinate isomorphism

For example, the space of real polynomials of degree at most two is isomorphic to $\mathbb R^3$ by $a+bx+cx^2\mapsto(a,b,c)$. This concerns its vector-space structure, not polynomial multiplication. (opentext.uleth.ca)

Such coordinate identifications depend on the chosen basis. For a map between equal finite-dimensional spaces, its representing matrix is invertible exactly when the map is an isomorphism. Thus coordinate calculations can establish an abstract structural correspondence without making the vectors themselves literally equal to coordinate tuples. (opentext.uleth.ca)

Topological and metric structures

In topology, the appropriate isomorphism is a homeomorphism: a bijective continuous function whose inverse is continuous. Continuity in only one direction is insufficient. For instance, the identity from the real line with the discrete topology to the real line with its usual topology is continuous and bijective, but its inverse is not continuous. These spaces are therefore not identified by that map as topological objects. (sites.math.northwestern.edu)

For metric spaces, the relevant correspondence depends on the allowed maps. With continuous maps, isomorphisms are homeomorphisms. With distance-nonincreasing maps, isomorphisms are bijective isometries, preserving every distance exactly. A homeomorphism need not preserve distances, so topological equivalence and metric equivalence impose different requirements. (sites.math.northwestern.edu)

Graphs and invariants

A graph isomorphism between simple graphs is a bijection between their vertex sets that preserves adjacency and non-adjacency. Vertex names and the arrangement of a drawing are irrelevant; the edge pattern must correspond exactly. For finite graphs, this means their adjacency matrices can be made identical by suitable vertex orderings. (maths.tcd.ie)

Isomorphic graphs have the same numbers of vertices and edges, and corresponding vertices have the same degrees. Such properties are invariants, useful for proving non-isomorphism. However, matching vertex degrees alone does not establish isomorphism. For example, a six-vertex cycle and two disjoint triangles both have six vertices of degree two, but only the former is connected; no adjacency-preserving bijection exists between them. (people.math.sc.edu)

The categorical formulation

In a category, a morphism $f:A\to B$ is an isomorphism if there is a morphism $g:B\to A$ satisfying

The inverse is unique. Identity morphisms are isomorphisms, inverses are isomorphisms, and composites of isomorphisms are isomorphisms. Hence being isomorphic defines an equivalence relation on the objects of a category. Classification “up to isomorphism” distinguishes structural types rather than particular presentations. (categorytheory.gitlab.io)

An isomorphism from an object to itself is an automorphism. Automorphisms describe its structure-preserving symmetries and form a group under composition. An object may have many automorphisms, so knowing that two objects are isomorphic does not generally select a unique correspondence between them. (jmilne.org)