This article is about mathematics. For other uses, see Isomorphism (disambiguation).
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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".
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 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:
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 symplectomorphism is an isomorphism of symplectic 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.
isomorphic, with a unique isomorphism. The isomorphism theorems provide canonical isomorphisms that are not unique. The term isomorphism is mainly used for algebraic...
in accordance with the general notion of isomorphism being a structure-preserving bijection. If an isomorphism exists between two graphs, then the graphs...
the mathematical field of topology a uniform isomorphism or uniform homeomorphism is a special isomorphism between uniform spaces that respects uniform...
mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship...
specifically, in differential geometry—the musical isomorphism (or canonical isomorphism) is an isomorphism between the tangent bundle T M {\displaystyle \mathrm...
bijective correspondence. Thus, the definition of an isomorphism is quite natural. An isomorphism of groups may equivalently be defined as an invertible...
of order theory, an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism for partially ordered sets...
science: Can the graph isomorphism problem be solved in polynomial time? (more unsolved problems in computer science) The graph isomorphism problem is the computational...
that subgraph isomorphism remains NP-complete even in the planar case. Subgraph isomorphism is a generalization of the graph isomorphism problem, which...
In mathematics, an isomorphism class is a collection of mathematical objects isomorphic to each other. Isomorphism classes are often defined as the exact...
Isomorphism problem may refer to: graph isomorphism problem group isomorphism problem This disambiguation page lists mathematics articles associated with...
isomorphism entails elementary equivalence, however the converse is not generally true, but it holds for ω-saturated models. A potential isomorphism between...
{\displaystyle \eta _{X}} is an isomorphism in D {\displaystyle D} , then η {\displaystyle \eta } is said to be a natural isomorphism (or sometimes natural equivalence...
two are isometrically anti-isomorphic. The (anti-) isomorphism is a particular natural isomorphism. Let H {\displaystyle H} be a Hilbert space over a...
homomorphism is an isomorphism if and only if it is bijective as a function on the underlying sets. If there exists a ring isomorphism between two rings...
In mathematics, an exceptional isomorphism, also called an accidental isomorphism, is an isomorphism between members ai and bj of two families, usually...
B} be a real vector bundle of rank n. Then there is an isomorphism, now called a Thom isomorphism Φ : H k ( B ; Z 2 ) → H ~ k + n ( T ( E ) ; Z 2 ) , {\displaystyle...
topological vector space isomorphism (abbreviated TVS isomorphism), also called a topological vector isomorphism or an isomorphism in the category of TVSs...
geometric version of the Satake isomorphism, proved by Ivan Mirković and Kari Vilonen (2007). Classical Satake isomorphism. Let G {\displaystyle G} be a...
An isomorph is an organism that does not change in shape during growth. The implication is that its volume is proportional to its cubed length, and its...
notion of computability on a set. By the Myhill isomorphism theorem, the relation of computable isomorphism coincides with the relation of mutual one-one...
Diffeomorphism – Isomorphism of differentiable manifolds Uniform isomorphism – Uniformly continuous homeomorphism is an isomorphism between uniform spaces...
In graph theory and theoretical computer science, a maximum common subgraph may mean either: Maximum common induced subgraph, a graph that is an induced...
statement that "there are two different groups of order 4 up to isomorphism", or "modulo isomorphism, there are two groups of order 4". This means that, if one...
In mathematics, a Borel isomorphism is a measurable bijective function between two standard Borel spaces. By Souslin's theorem in standard Borel spaces...
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