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. stripes

    Prove Isomorphism: R x S & S x R

    Homework Statement Show that for any rings R and S, R x S and S x R are isomorphic, and A x B is the cartesian product, or ordered pairs. So an element of R x S can be written as (r1, s1). Homework Equations The Attempt at a Solution So I have to show that there is a bijection...
  2. L

    Do Hilbert Space Isomorphism Map Dense Sets to Dense Sets?

    Suppose that H, K are Hilbert spaces, and A : H -> K is a bounded linear operator and an isomorphism. If X is a dense set in H, then is A(X) a dense set in K? Any references to texts would also be helpful.
  3. A

    MHB Using isomorphism and permutations in proofs

    I have trouble using isomorphism and permutation in proofs for combinatorics. I don't know when I can assume "without loss of generality". What are some guidelines to using symmetry in arguments. One problem I'm working on that uses symmetry is to "prove that any (7, 7, 4, 4, 2)-designs must be...
  4. caffeinemachine

    MHB Degree of extension invariant upto isomorphism?

    Let $K$ be a field and $F_1$ and $F_2$ be subfields of $K$. Assume that $F_1$ and $F_2$ are isomorphic as fields. Further assume that $[K:F_1]$ is finite and is equal to $n$. Is it necessary that $[K:F_2]$ is finite and is equal to $n$?? ___ I have not found this question in a book so I don't...
  5. N

    How Do You Prove a Function Is Isomorphic to the Cartesian Product of X?

    Homework Statement See Attachment: https://www.physicsforums.com/attachment.php?attachmentid=59074&d=1369708771 Homework Equations As shown in the attachment, I am slightly confused as to where to begin this problem. I know that I need to prove that a function, f, is 1-1 and onto...
  6. L

    Prove Isomorphism When Columns of C are Linearly Independent

    Homework Statement Let L:R->R be a linear operator with matrix C. Prove if the columns of C are linearly independent, then L is an isomorphism. Homework Equations The Attempt at a Solution Assume the columns of C are linearly independent. Then, the homogenous equation Cx=0 is...
  7. Bruce Wayne1

    MHB Help Proving Isomorphism of a group

    Hi! I'm trying to prove a cyclic group is isomorphic to ring under addition. What the strategy I would take? How would I get it started? Here's what I know so far: I need to meet 3 conditions-- 1 to 1, onto, and the operation is preserved. I also know that isomorphic means that the group is...
  8. L

    How Is the Quotient Group G/H Isomorphic to G'?

    How can one prove that for homomorphism G \xrightarrow{\rho} G' and H as kernel of homomorphism, quotient group G/H is isomorphic to G'? Thanks.
  9. R

    Isomorphism between groups and their Lie Algebra

    I must apologize if this question sounds dump but if an isomorphism is established between two groups, is it true that their lie algebra is an isomorphism too? My idea is that since the tangent space is sent to tangent space also in the matrix space, both groups' lie algebra will be isomorphism...
  10. A

    Exploring the Isomorphism Between Z_k and Aut(Z_n)

    I was a bit confused the last paragraph before "Corollary 4.6.4". It says that we have the isomorphism \alpha : Z_k \rightarrow Aut(Z_n) but then says that \alpha(a^j)(b^i)=b^{m^ji}. In a regular function f: X \rightarrow Y, we take one element from X and end up with an element in Y, right...
  11. P

    Using First Isomorphism Theorem with quotient rings

    Homework Statement Try to apply the First Isomorphism Theorem by starting with a homomorphism from a polynomial ring R[x] to some other ring S. Let I = \mathbb{Z}_2[x]x^2 and J = \mathbb{Z}_2[x](x^2+1). Prove that \mathbb{Z}_2[x]/I is isomorphic to \mathbb{Z}/J by using the homomorphism...
  12. T

    Is there a way to transform a polynomial into a vector?

    Ok for the longest while I've been at war with polynomials and isomorphisms in linear algebra, for the death of me I always have a brain freeze when dealing with them. With that said here is my question: Is this pair of vector spaces isomorphic? If so, find an isomorphism T: V-->W. V= R4 ...
  13. I

    MHB Identify isomorphism type for each proper subgroup of (Z/32Z)*

    The question is to identify isomorphism type for each proper subgroup of $(\mathbb{Z}/32\mathbb{Z})^{\times }$. (what's the "isomorphism type" means? Does the question mean we need to list all the ismorphism between of each subgroup and the respectively another group that is isomorphic to the...
  14. Y

    Isomorphism symmetry group of 6j symbol

    Hi everyone, I read in 'Angular momentum in Quantum Mechanics' by A.R Edmonds that the symmetry group of the 6j symbol is isomorphic to the symmetry group of a regular tetahedron. Is there an easy way of seeing this? I've tried working out what the symmetry relations of the 6j symbol do...
  15. M

    Group isomorphism (C,+) to (R,+)

    Homework Statement Prove (\mathbb{R},+) and (\mathbb{C},+) are isomorphic as groups.Homework Equations An isomorphism is a bijection from one group to another that preserves the group operation, that is \phi(ab)=\phi(a)\phi(b)The Attempt at a Solution I'm trying to find a bijection, but I can...
  16. B

    Mapping an isomorphism b/w 2 grps

    I googled this but couldn't find a clear answer. Is every invertible mapping an isomorphism b/w 2 grps or does it have to be linear?
  17. STEMucator

    Proving Ring Isomorphism using the First Isomorphism Theorem

    Homework Statement The question : http://gyazo.com/5372336302b5ef289b305172bcd16a2a Homework Equations First Isomorphism theorem. The Attempt at a Solution Define \phi : \mathbb{Q}[x]/<x^2-2> → Q[ \sqrt{2} ] \space | \space \phi (f(x)) = f( \sqrt{2}) So showing phi is a homomorphism is...
  18. M

    Can you explain me why this is also isomorphism?

    Homomorphism is defined by ##f(x*y)=f(x)\cdot f(y)##. One interesting example of this is logarithm function ##log(xy)=\log x+\log y##. Can you explain me why this is also isomorphism?
  19. B

    Abstract Algebra Proof Using the First Isomorphism Theory

    Homework Statement See attatchment. I couldn't upload the picture. 2. The attempt at a solution I have the following: Define mapping f: ℝ2 -> ℝ as follows: f(x,y) = 3x - 4y Claim: f is a homomorphism Pick any (x,y) in ℝ2. Then f(x,y) = f(x)*f(y) = 3x - 4y = (x+x+x)-(y+y+y+y) =...
  20. Fantini

    MHB Proving Field Isomorphism: $\sigma (x) = x$ for $x \in K$

    The following question appeared in my last Rings and Fields exam. Let $\alpha \in \mathbb{R}$ be a root of $x^3 -2$. Let $K = \mathbb{Q}(\alpha) \subseteq \mathbb{R}$ and $\sigma: K \to K$ a field isomorphism. Prove that $\sigma (x) = x$ for all $x \in K$. My attempt is as follows: since this...
  21. caffeinemachine

    MHB Direct product of abelian groups. Isomorphism.

    Let $A,B,C$ be finite abelian groups. Assume that $A\times B\cong A\times C$. Show that $B\cong C$. I observed that $(A\times B)/(A\times\{e\})\cong B$ and $(A\times C)/(A\times\{e\})\cong C$. So I need to show that $(A\times B)/(A\times\{e\})\cong (A\times C)/(A\times\{e\})$. Let...
  22. mnb96

    Question on isomorphism between addition and multiplication

    Hello, I want to find a family of functions \phi:\mathbb{R} \rightarrow \mathbb{C} that have the property: \phi(x+y)=\phi(x)\phi(y) where x,y\in \mathbb{R}. I know that any exponential function of the kind \phi(x)=a^x with a\in\mathbb{C} will satisfy this property. Is this the only choice...
  23. C

    Isomorphism types of abelian groups

    wrtie down the possible isomorphism types of abelian groups of orders 74 and 800 then for 74=2*37 then Z(74) is isomorphism to Z2 * Z37 (by chinese remainder theorem) then for 74 , 2 we have Z74 and Z2*Z37 (i am not sure it is right or wrong then for 800 i know i should apply the fundamental...
  24. P

    MHB How can I prove that the homomorphism defined by f(gH)=gJ is well defined?

    Let G be a group and H, J be normal in G with J containing H. I can prove all of the theorem except showing that the homomorphism f: G/H-> G/J defined by f(gH)=gJ is well defined! This means I need to show that gH=bH for b,g in G implies that gJ=bJ.
  25. matqkks

    MHB Isomorphism Between Vector Spaces: A Real Life Analogy

    What is the best way of describing isomorphism between two vector spaces? Is there a real life analogy of isomorphism?
  26. matqkks

    What is the best way of describing isomorphism between two vector

    What is the best way of describing isomorphism between two vector spaces? Is there a real life analogy of isomorphism?
  27. N

    If Ker T = 0 then T is not isomorphism

    Homework Statement Show that if ker T != 0 then T is not an isomorphism. Homework Equations The Attempt at a Solution If Ker T != 0 that means that there are multiple solutions for which T=0 meaning it is not injective and hence not isomorphic? Is that correct? I don't think it...
  28. K

    What is a Liouville Isomorphism in the Context of Symplectization?

    I shall use Seidel's definition of a Liouville domain; in particular, a Liouville domain is a compact manifold M with boundary together with a one-form \theta \in \Omega^1(M) such that \omega = d\theta is a symplectic form and the vector field Z defined by \iota_Z \omega = \theta is...
  29. T

    Product of Quotient Groups Isomorphism

    Homework Statement I have attached the problem below. Homework Equations The Attempt at a Solution I have tried to use the natural epimorphism from G x G x ... x G to (G x G x ... x G)/(K1 x K2 x ... x Kn), but I do not believe that this is an injective function. Then I tried...
  30. D

    Isomorphism of L_A: Orthogonal Matrix, ℝ^n -> ℝ^n

    Homework Statement if L_A: ℝ^n -> ℝ^n : X-> A.X is a linear transformation, and A is an orthogonal matrix, show that L_A is an isomorphism. also given is that (ℝ,ℝ^n,+,[.,.]) , the standard Euclidian space which has inproduct [X,Y]= X^T.Y Homework Equations ortogonal matrix, so A^T=A^{-1}...
  31. G

    Isomorphism and Binary operation

    Homework Statement (i) If (X,*) is a binary operation, show that the identity function Id_X : X \rightarrow X is an isomorphism. (ii) Let (X_1, *_1) and (X_2, *_2) be two binary structures and let f : X_1 \rightarrow X_2 be an isomorphism of the binary structures. Show that f^-1 : X_2...
  32. F

    Isomorphism of P4 and R5 in a given inner product space

    The isomorphism of ℝ5 and P4 is obvious for the "standard" inner product space. The following question arise from an example in my course literature for a course in linear algebra. The example itself is not very difficult, but there is a statement without any proof, that if the inner product...
  33. Chris L T521

    MHB How Can Matrix Powers and Group Isomorphisms Illuminate Group Theory?

    Thanks to those who participated in last week's POTW! Here's this week's problem (I'm going to give group theory another shot). ----- Problem: (i) Prove, by induction on $k\geq 1$, that \[\begin{bmatrix}\cos\theta & -\sin\theta\\ \sin\theta & \cos\theta\end{bmatrix}^k =...
  34. J

    Proving the Non-Isomorphism of D(p(x))

    Homework Statement I have D(p(x)) = (the second derivative of p with respect to x ) - (2 derivative of p with respect to x) + p Proof that D(p(x)) is not an isomorphism Homework Equations The Attempt at a Solution Just by watching the problem it seems I can assign one unique...
  35. C

    Can a group have monomorphisms in both directions and still not be isomorphic?

    Is it true that if there is monomorphism from group A to group B and monomorphism from group B to group A than A and B are isomorphic? i need some explanation. thx
  36. C

    Isomorphism between divisible groups

    proove that if G and H are divisible groups and there is monomorphisms from G to H and from H to G than G and H are isomorphic
  37. I

    Proving that Z2 X Z2 X Z2Z2 is a isomorphic (ring isomorphism) to P(N)

    Proving that Z2 X Z2 X Z2... Z2 is a isomorphic (ring isomorphism) to P(N) Homework Statement I wish to prove that the ring of Cartesian product Z2 X Z2 X Z2...X Z2 (here we have n products) under addition and multiplication (Z2 is {0,1}) is isomorphic to P(N) where P(N) is the ring of power...
  38. T

    Abstract Algebra: isomorphism proof

    Homework Statement Let G be an abelian group of order n. Define phi: G --> G by phi(a) = a^m, where a is in G. Prove that if gcd(m,n) = 1 then phi is an isomorphism Homework Equations phi(a) = a^m, where a is in G gcd(m,n) = 1 The Attempt at a Solution I know since G is an...
  39. S

    Implicit isomorphism involved in extension/sub fields/structures?

    Implicit isomorphism involved in extension/sub fields/structures? This has been bugging me for a while. I'm pretty sure I'm correct but I'd just like to verify to put my mind at ease. I'd like to know if there is an implicit isomorphism involved when we say, for example, F is a substructure of...
  40. D

    MHB Analytic Isomorphism: Annulus vs. Punctured Unit Disc

    Why can't an annulus be analytically isomorphic to the punctured unit disc? $A_{r,R}$ is an annulus Theorem: $A_{r,R}$ is analytically isomorphic to $A_{s,S}$ iff $R/r = S/s$. If our annulus $A_{1,2}$, then $R/r = 2$ and the punctured disc would be $\lim\limits_{s\to 0}1/s = \infty$. So...
  41. P

    First Isomorphism Theorem Question

    I am having a hard time using or applying the theorem . Anyways Prove that there is no homomorphism from Z_{8}\oplusZ_{2} onto Z_{4}\oplusZ_{4} Im guessing its the First Isomorphsim Theorem because its in the chapter. But I am not sure how to use it.
  42. P

    Isomorphism of relatively prime groups

    Homework Statement Allow m,n to be two relatively prime integers. You must prove that Z(sub mn) ≈ Z(sub m) x Z(sub n) Homework Equations if two groups form an isomorphism they must be onto, 1-1, and preserve the operation. The Attempt at a...
  43. P

    Isomorphism and Generators in Z sub P

    Homework Statement Let P be a prime integer, prove that Aut(Z sub P) ≈ Z sub p-1 Homework Equations none The Attempt at a Solution groups must preserve the operation, be 1-1, and be onto and they can be called an isomorphism. Z sub p-1 has one less element in it so and all the...
  44. quasar987

    Algebra Isomorphism: Complex Numbers C over Reals R

    Consider the complex numbers C as an algebra over the reals R. The author of the book I have in front of me (Dirac operators in Riemannian Geometry, p.13) writes \mathbb{C}\otimes_{\mathbb{R}}\mathbb{C}=\mathbb{C}\oplus\mathbb{C} (as real algebras). Does anyone know what this canonical algebra...
  45. M

    Commute Isomorphism & Friends problems

    Homework Statement F:P2->R5 F(xn) = en+1 Consider the linear function D:P4 -> P4 p(x) -> p'(x) Find the matrix of the linear function T:R5 -> R5 such thatHomework Equations ( T ° F ) p(x) = ( F ° D ) ( p(x) )The Attempt at a Solution T ° F ° F-1 = F ° D ° F-1 T = F ° D ° F-1 then what should...
  46. B

    How to show an isomorphism between groups?

    Is the only way to show an isomorphism between groups is to just define a map which has the isomorphism properties? So for example for a group G with order 15 to show that G \cong C_3 \times C_5 would I just have to define all the possible transformations to define the isomorphism...
  47. T

    Is the Proof for Same Dimension and Isomorphism Correct?

    Homework Statement A theorem in my book states: If V, W are finite dimensional vector spaces that are isomorphic, then V, W have the same dimension. I wrote a proof but it is different from the proof given in my book, and I'd like to know if it's right. The Attempt at a Solution Let \left\{A_1...
  48. B

    Isomorphism of the Dihedral group

    We're doing isomorphisms and I was just wondering, is the dihedral group D_{12} isomorphic to the group of even permutations A_4?
  49. B

    Isomorphism between groups of real numbers

    Apparently there is an isomorphism between the additive group (ℝ,+) of real numbers and the multiplicative group (ℝ_{>0},×) of positive real numbers. But I thought that the reals were uncountably infinite and so don't understand how you could define a bijection between them?! Thanks for...
  50. J

    Prove Order Isomorphism: α=β If (α,∈) & (β,∈)

    I am trying to prove the following results: If α and β are ordinals, then the orderings (α,∈) and (β,∈) are isomorphic if and only if α = β. So far, I have only proved that the class Ord is transitive and well-ordered by ∈. I can prove this result with the following lemma: If f:α→β is an...
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