What is Isomorphism: Definition and 321 Discussions

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.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:

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 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.

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  1. jv07cs

    I Do Metric Tensors Always Have Inverses?

    I am reading about musical isomorphisms and for the demonstration of the index raising operation from the sharp isomorphism, we have to multiply the equation by the inverse matrix of the metric. Can we assume that this inverse always exists? If so, how could I prove it?
  2. K

    I ##\mathbb{C}\oplus\mathbb{C}\cong\mathbb{C}\otimes\mathbb{C}##

    Hello! Reading book o Clifford algebra authors claim that ##\mathbb{C}\oplus\mathbb{C}\cong\mathbb{C}\otimes_{\mathbb{R}}\mathbb{C}## as algebras. Unfortunately proof is absent and provided hint is pretty misleading As vector spaces they are obviously isomorphic since ##\dim_{\mathbb{R}}...
  3. PragmaticYak

    Fixed point free automorphism of order 2

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  4. L

    I Anyone knows why musical isomorphism is called so?

    Anyone knows why musical isomorphism is called so? Why is it musical? https://en.wikipedia.org/wiki/Musical_isomorphism
  5. C

    Existence of isomorphism ϕ:V→V s.t. ϕ(ϕ(v))=−v for all v∈V

    Problem: Let ## V ## be a vector space over ## \mathbb{F} ## and suppose its dimension is even, ## dimV=2k ##. Show there exists an isomorphism ## \phi:V→V ## s.t. ## \phi(\phi(v))=−v ## for all ## v \in V ## Generally that way to solve this is to define a basis for the vector space ## V ##...
  6. L

    MHB Is G/G isomorphic to the trivial group? A proof for G/G\cong \{e\}

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  7. K

    I Understanding the concepts of isometric basis and musical isomorphism

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  8. F

    F is an isomorphism from G onto itself,...., show f(x) = x^-1

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  9. F

    I Classify the isomorphism of a graph

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  10. E

    MHB Isomorphism of logic, arithmetic, and set theory

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  11. GlassBones

    Isomorphisms preserve linear independence

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  12. S

    Proof of isomorphism of vector spaces

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  13. Mr Davis 97

    Showing isomorphism between fractions and a quotient ring

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  14. Alex Langevub

    An exercise with the third isomorphism theorem in group theory

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  15. A

    Isomorphism of dihedral with a semi-direct product

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  16. A

    Rigorous proof of isomorphism

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  17. QuasarBoy543298

    I Why are linearly ordered R and R/{0} not isomophic?

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  18. H

    MHB Is ψ an Isomorphism from H to G?

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  19. T

    ##\phi(R_{180})##, if ##\phi:D_n\to D_n## is an automorphism

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  20. Mr Davis 97

    1st Isomorphism thm for dihedral gps

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  21. Mr Davis 97

    Is the First Isomorphism Theorem Applicable to this Complex Number Group?

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  22. Mr Davis 97

    Having all subgroups normal is isomorphism invariant

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  23. Mr Davis 97

    I Showing that inverse of an isomorphism is an isomorphism

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  24. D

    I Field of zero characteristics

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  25. A

    I State Vectors vs. Wavefunctions

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  26. Math Amateur

    MHB Third Isomorphism Theorem for Rings .... Bland Theorem 3.3.16 .... ....

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  27. Math Amateur

    I Third Isomorphism Theorem for Rings .... Bland Theorem 3.3.16

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  28. Math Amateur

    MHB Second Isomorphism Theorem for Rings .... Bland Theorem 3.3.15 .... ....

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  29. Math Amateur

    I Second Isomorphism Theorem for Rings .... Bland Theorem 3.3.1

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  30. J

    MHB Verifying Answers to "Zero Divisors & Isomorphism Theorem"

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  31. J

    MHB Verifying Solutions to Isomorphism Problem: Need Help!

    Hi, I have attached the question and the solutions to part a and b of this question. Would like someone to verify if I have done anything wrong. Greatly appreciate it! Thanks. Would also like to check if there is a simpler method to prove f is an isomorphism? Thanks
  32. M

    Show isomorphism under specific conditions

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  33. Mr Davis 97

    I Why must an isomorphism between 2Z and 3Z result in mu(2) = +/- 3?

    My book is trying to show that the rngs ##2 \mathbb{Z}## and ##3 \mathbb{Z}## are not isomorphic. It starts by saying that if there were an isomorphism ##\mu : 2 \mathbb{Z} \to 3 \mathbb{Z}## then by group theory we would know that ##\mu (2) = \pm 3##. It then goes on to show that this leads to...
  34. H

    Determine rank of T and whether it is an isomorphism

    Homework Statement T((x_0, x_1, x_2)) = (0, x_0, x_1, x_2) Homework Equations None The Attempt at a Solution I'm getting hung up on definitions. My book says that T is an is isomorphism if T is linear and invertible. But it goes on to say that for T of finite dimension, T can only be an...
  35. Mr Davis 97

    Isomorphism between HK and H x K

    Homework Statement H and K are normal subgroups of G such that the intersection of H and K is the identity. Also, G = HK = {hk | h in H and k in K}. Find an isomorphism between G and H x K Homework EquationsThe Attempt at a Solution I was thinking that an isomorphism could be ##\mu : G...
  36. binbagsss

    A Isomorphism concepts,( example periods elliptic functions )

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  37. M

    MHB Vector space isomorphism

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  38. jk22

    I Complex Isomorphism Error in Lorentz Transform

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  39. M

    I A regular matrix <=> mA isomorphism

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  40. PsychonautQQ

    Isomorphism to certain Galois group and cyclic groups

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  41. M

    I Reflexive relation question

    Hello all. I have a question about a reflexive relation. Consider ##1_V : V \rightarrow V## with ##V## a vector space. Obviously, this is an isomorphism. My book uses this example to show that V is isomorphic with V (reflexive relationship). However, suppose I have a function ##f: V\rightarrow...
  42. M

    Isomorphism between so(3) and su(2)

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  43. M

    MHB Is φ a bijective homomorphism between simple $R$-modules?

    Hey! :o Let $R$ be a commutative ring with unit and $M$ be a $R$-module. Let $\phi : M\rightarrow M'$ be a non-zero homomorphism of simple $R$-module. I want to show that $\phi$ is an isomorphism. To show that we have to show that $\phi$ is bijective, right? (Wondering) What exactly is...
  44. lep11

    When is this linear transformation an isomorphism?

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  45. Geofleur

    A Isomorphism between a linear space and its dual

    I have been trying to prove the following theorem, for a finite dimensional vector space ## X ## and its dual ## X^* ##: Let ## f:X\rightarrow X^* ## be given by ## f(x) = (x|\cdot) ##, where ## (x|\cdot) ## is linear in the first argument and conjugate linear in the second (so I am using the...
  46. RJLiberator

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  47. RJLiberator

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  48. RJLiberator

    Isomorphism is an equivalence relation on groups

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  49. E

    Is Isometry the Same as Isomorphism?

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  50. S

    Understanding Isomorphisms for Linear Transformations

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