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Bookmark or Share- 3.2: Definitions of Homomorphisms and IsomorphismsAn isomorphism is a homomorphism that is also a bijection. Intuitively, you can think of a homomorphism \(\phi\) as a “structure-preserving” map: if you multiply and then apply \(\phi\text{,}\) you get the same result as when you first apply \(\phi\) and then multiply.
- Isomorphism -- from Wolfram MathWorldIsomorphism is a very general concept that appears in several areas of mathematics. The word derives from the Greek iso, meaning "equal," and morphosis, meaning "to form" or "to shape." Formally, an isomorphism is bijective morphism.
- Group Isomorphisms and Automorphisms - GeeksforGeeksIn the study of algebraic structures, group isomorphisms and automorphisms play a fundamental role. By defining internal symmetries inside a group (automorphisms) and when two groups have the same structure (isomorphisms), these ideas aid in our understanding of the structure and symmetry of groups.
- Isomorphism | Group Theory, Algebraic Structures, Equivalence ...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.
- 5.6: Isomorphisms - Mathematics LibreTextsDefinition \(\PageIndex{1}\): Isomorphism. A linear map \(T\) is called an isomorphism if the following two conditions are satisfied. \(T\) is one to one. That is, if \(T(\vec{x})=T(\vec{y}),\) then \(\vec{x}=\vec{y}.\) \(T\) is onto. That is, if \(\vec{w}\in W,\) there exists \(\vec{v}\in V\) such that \(T(\vec{v})=\vec{w}\).
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