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    Isomorphism

    In mathematics, invertible morphism

    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 is derived from Ancient Greek ἴσος 'equal' and μορφή 'form, shape'. The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties. 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. An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a canonical isomorphism if there is only one isomorphism between the two structures, or if the isomorphism is much more natural than other isomorphisms. For example, for every prime number p, all fields with p elements are canonically isomorphic, with a unique isomorphism. Wikipedia

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  2. en.wikipedia.org

    An isomorphism is a structure-preserving mapping between two mathematical structures of the same type that can be reversed by an inverse mapping. Learn about different types of isomorphisms, such as group, ring, field, and graph isomorphisms, and see examples and applications in various fields of mathematics.
  3. math.libretexts.org

    A mapping \\(T:V\\rightarrow W\\) is called a linear transformation or linear map if it preserves the algebraic operations of addition and scalar multiplication.
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  5. britannica.com

    isomorphism, in modern algebra, a one-to-one correspondence between two sets that preserves binary relationships between elements of the sets. For example, the set of natural numbers can be mapped onto the set of even natural numbers by multiplying each natural number by 2. The binary operation of adding two numbers is preserved—that is, adding two natural numbers and then multiplying the ...
    Author:William L. Hosch
  6. math.stackexchange.com

    There are two different things going on here. The simpler one is the notation $\to$, which usually means that we in some way, not necessarily an isomorphism, mapping one object to another.. An isomorphism is a particular type of map, and we often use the symbol $\cong$ to denote that two objects are isomorphic to one another. Two objects are isomorphic there is a $1$-$1$ map from one object ...
  7. Learn how to define and identify homomorphisms and isomorphisms between groups, and see examples of matrices, permutations, and exponential maps. Explore the properties and applications of these maps in group theory and representation theory.
  8. math.libretexts.org

    This page titled 3.2: Definitions of Homomorphisms and Isomorphisms is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Jessica K. Sklar via source content that was edited to the style and standards of the LibreTexts platform.
  9. math.libretexts.org

    Learn what isomorphisms are and how to recognize them in different algebraic systems. See examples of isomorphisms between sets, groups, rings and fields, and how they are used in mathematics and computer science.
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