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. S

    Isomorphism to subspaces of different dimensions

    Homework Statement Given the linear transformations f : R 3 → R 2 , f(x, y, z) = (2x − y, 2y + z), g : R 2 → R 3 , g(u, v) = (u, u + v, u − v), find the matrix associated to f◦g and g◦f with respect to the standard basis. Find rank(f ◦g) and rank(g ◦ f), is one of the two compositions an...
  2. TheMathNoob

    Proof about isomorphism (Graph Theory)

    Homework Statement 1. Prove or disprove up to isomorphism, there is only one 2-regular graph on 5 vertices. Homework EquationsThe Attempt at a Solution I am making this thread again hence I think I will get more help in this section old thread...
  3. TheMathNoob

    Proof about isomorphism (Graph Theory)

    Homework Statement 1. up to isomorphism, there is only one 2-regular graph on 5 vertices. Homework EquationsThe Attempt at a Solution I am still working on the problem, but I don't understand what up to isomorphism means. Does it mean without considering isomorphism?. I just need help with...
  4. marcus

    Graph isomorphism problem-advance in complexity research

    There seems to have been a major step forward in complexity research. somebody wrote a pleasant understandable piece about it in Quanta magazine. https://www.quantamagazine.org/20151214-graph-isomorphism-algorithm/ I gave the title an "intermediate" tag because the graph isomorphism problem is...
  5. jedishrfu

    Landmark Algorithm for Graph Isomorphism

    Quanta Magazine published this article on a potentially new algorithm for graph isomorphism by Prof Laszlo Babai of the University of Chicago: https://www.quantamagazine.org/20151214-graph-isomorphism-algorithm/ There's a reference to the Arxiv preprint here: http://arxiv.org/abs/1512.03547v1
  6. S

    Linear Transformation and Isomorphism

    Homework Statement Given the transformation fh : R 3 → R 3 defined by fh(x, y, z) = (x−hz, x+y −hz, −hx+z), where h ∈ R is a parameter. a) Find, for all possible values of h, Ker(fh), Im(fh), their bases and dimensions. b) Is fh an isomorphism for some value of h? Homework Equations Ax=o The...
  7. Sarah00

    Graph Isomorphism Homework: Are 2 Graphs Isomorphic?

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  8. S

    Matrix of a Linear Transformation Example

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  9. J

    Proving Isomorphism of Linear Operator with ||A|| < 1

    Hi, I have some trouble with the following problem: Let E be a Banach space. Let A ∈ L(E), the space of linear operators from E. Show that the linear operator φ: L(E) → L(E) with φ(T) = T + AT is an isomorphism if ||A|| < 1. So the idea here is to use the Neumann series but I can't really...
  10. DeldotB

    Using the Second Isomorphism (Diamond Isomorphism) Theorem

    Homework Statement Good day all, Im completely stumped on how to show this: |AN|=(|A||N|/A intersect N|) Here: A and N are subgroups in G and N is a normal subgroup. I denote the order on N by |N| Homework Equations [/B] Second Isomorphism TheoremThe Attempt at a Solution Well, I know...
  11. E

    Linear Transformation and isomorphisms

    Homework Statement Suppose a linear transformation T: [P][/2]→[R][/3] is defined by T(1+x)= (1,3,1), T(1-x)= (-1,1,1) and T(1-[x][/2])=(-1,2,0) a) use the given values of T and linearity properties to find T(1), T(x) and T([x][/2]) b) Find the matrix representation of T (relative to standard...
  12. Avatrin

    Understanding the isomorphism theorems

    I am currently trying to understand the isomorphism theorems. The issue I am having is that I am struggling to find a way to think about them. In Stillwell's Elements of Algebra, I found a way to understand the first theorem (\frac{G}{ker \phi} \simeq I am \phi for any homomorphism...
  13. M

    Can a Projection Be an Isomorphism If It Maps to a Proper Subset?

    Pre-knowledge If V and W are finite-dimensional vector spaces, and dim(V) does not equal dim(W) then there is no bijective linear transformation from V to W. An isomorphism between V and W is a bijective linear transformation from V to W. That is, it is both an onto transformation and a one...
  14. MidgetDwarf

    Why Is Isomorphism Confusing in Linear Algebra?

    I am having problems in my linear algebra class. The class is taught rather poorly. There is only but 3 students left. The instructor is of no help. I tried reading my txtbook and following a few videos (even went to office hours). However I am not understanding Isomorphism. I know that a...
  15. K

    Derive gen sol of non-homogeneous DEs through linear algebra

    Hello, I noticed that the solution of a homogeneous linear second order DE can be interpreted as the kernel of a linear transformation. It can also be easily shown that the general solution, Ygeneral, of a nonhomogenous DE is given by: Ygeneral = Yhomogeneous + Yparticular My question: Is it...
  16. HaLAA

    Show (H,+) is isomorphic to (C,+)

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  17. S

    Graph (regular) Isomorphism in n^(O(log2(n))) .

    I am trying to construct an algorithm which is combinatorial in nature. I have shared a link- https://www.academia.edu/11354697/Graph_regular_Isomorphism_in_n_O_log2_n_ which depicts the idea simply using an example. I claim (if it is correct) n^(O(log2(n))) time complexity. happy to have...
  18. caffeinemachine

    MHB Natural Isomorphism b/w Dual Spaces Tensor Prod & Multilinear Form Space

    I am trying to prove the following. Let $V_1, \ldots, V_k$ be finite dimensional vector spaces over a field $F$. There is a natural isomorphism between $V_1^*\otimes\cdots\otimes V_k^*$ and $\mathcal L^k(V_1, \ldots, V_k;\ F)$. Define a map $A:V_1^*\times\cdots\times V_k^*\to \mathcal L^k(V_1...
  19. metapuff

    Not understanding the isomorphism R x R = C

    Now ℝxℝ≅ℂ, seen by the map that sends (a,b) to a + bi. ℂ is a field, so the product of any two non-zero elements is non-zero. However, this doesn't seem to hold in ℝxℝ, since (1,0) * (0,1) = (0,0) even though (1,0) and (0,1) are non-zero. What am I missing? Also, the zero ideal is maximal in ℂ...
  20. PsychonautQQ

    Attempting to better understand the group isomorphism theorm

    The homomorphism p:G-->H induces an isomorphism between G/Ker(p) and H (if p is onto). I am trying to understand why this must be true. I understand why these groups have the same magnitude and so a bijection is possible, but there is something that I am not able to understand. What seems to be...
  21. W

    Groups of Order 16 with 4-Torsion, Up to Isomorphism

    Hi, I am trying to find all groups G of order 16 so that for every y in G, we have y+y+y+y=0. My thought is using the structure theorem for finitely-generated PIDs. So I can find 3: ## \mathbb Z_4 \times \mathbb Z_4##, ## \mathbb Z_4 \times \mathbb Z_2 \times \mathbb Z_2 ## , and: ##...
  22. T

    Isomorphism under differentiation

    Is 《sinx》under differentiation a valid cyclic group.
  23. mnb96

    Isomorphism between Clifford algebras CL(4,2) and CL(2,4)

    Hi, I was reading a paragraph of a book (you can find it here) where the author seems to suggest that the Clifford algebras \mathcal{C}\ell_{2,4}(\mathbb{R}) and \mathcal{C}\ell_{4,2}(\mathbb{R}) are isomorphic. In particular, at the third line after Equation (10.190), when he talks about the...
  24. J

    Check for Isomorphism in Hypergraphs

    Hi, I was trying to check whether two hypergraphs are isomorphic to each other using MATLAB. I did the brute force method by permuting the vertices and check all the permutations one by one. This method is pretty slow. An idea suggested by my friend was to represent the hypergraphs as...
  25. Math Amateur

    MHB Second Isomorphism Theorem for Groups

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  26. Math Amateur

    MHB First Isomorphism Theorem for Modules - Cohn Theorem 1.17

    I am reading "Introduction to Ring Theory" by P. M. Cohn (Springer Undergraduate Mathematics Series) In Chapter 1: Basics we find Theorem 1.17 (First Isomorphism Theorem for Modules) regarding module homomorphisms and quotient modules. I need help with some aspects of the proof. Theorem 1.17...
  27. Math Amateur

    MHB Isomorphism Between Hom_F (V,W) and M_nxn(F) - theory of vector spaces

    I am spending time revising vector spaces. I am using Dummit and Foote: Abstract Algebra (Chapter 11) and also the book Linear Algebra by Stephen Freidberg, Arnold Insel and Lawrence Spence. I am working on Theorem 10 which is a fundamental theorem regarding an isomorphism between the space of...
  28. Math Amateur

    MHB Proof of Fourth or Lattice Isomorphism Theorem for Modules

    Dummit and Foote give the Fourth or Lattice Isomorphism Theorem for Modules on page 349. I need some help with the proof of Fourth or Lattice Isomorphism Theorem for Modules ... hope someone will critique my attempted proof ... (I had considerable help from the proof of the theorem for groups...
  29. Math Amateur

    MHB Fourth or Lattice Isomorphism Theorem for Modules - clarification

    Dummit and Foote give the Fourth or Lattice Isomorphism Theorem for Modules on page 349. The Theorem reads as follows:https://www.physicsforums.com/attachments/2981In the Theorem stated above we read: " ... ... There is a bijection between the submodules of M which contain N and the submodules...
  30. Math Amateur

    MHB Isomorphism Between External and Internal Direct Sums - Knapp Proposition 2.30

    I am reading Chapter 2: Vector Spaces over \mathbb{Q}, \mathbb{R} \text{ and } \mathbb{C} of Anthony W. Knapp's book, Basic Algebra. I need some help with some issues regarding Theorem 2.30 (regarding an isomorphism between external and internal direct sums) on pages 59-60. Theorem 2.27 and...
  31. Math Amateur

    MHB First Isomorphism Theorem for Vector Spaces - Knapp, Theorem 2.27

    I am reading Chapter 2: Vector Spaces over \mathbb{Q}, \mathbb{R} \text{ and } \mathbb{C} of Anthony W. Knapp's book, Basic Algebra. I need some help with some issues regarding Theorem 2.27 (First Isomorphism Theorem) on pages 57-58. Theorem 2.27 and its proof read as follows...
  32. F

    Isomorphism Transformation: Onto and One-to-One Concept Explanation

    Homework Statement Is the following transformation an isomorphism: a_0+bx+cx^{2}+dx^{3} \rightarrow \begin{bmatrix} a & b\\ c & d \end{bmatrix} Homework Equations A transformation is an isomorphism if: 1. The transformation is one-to-one 2. The transformation is onto The Attempt at a...
  33. J

    Cyclic Group - Isomorphism of Non Identity Mapping

    Homework Statement Prove that if G is a cyclic group with more than two elements, then there always exists an isomorphism: ψ: G--> G that is not the identity mapping. Homework Equations The Attempt at a Solution So if G is a cyclic group of prime order with n>2, then by Euler's...
  34. E

    Showing that a linear transformation from P3 to R4 is an isomorphism?

    I have a linear transformation, T, from P3 (polynomials of degree ≤ 3) to R4 (4-dimensional real number space). I have a second linear transformation, U, from R4 back to P3. In the first step of this four-step problem, I have shown that the composition TU from R4 to R4 is the identity linear...
  35. F

    Is the map from l^infinite to L(l^2,l^2) a bijection?

    Homework Statement Let L(l^2,l^2) be the space of bounded linear operators K:l^2->l^2. Now I define a map from l^infinite to L(l^2,l^2) as a->Ta(ei) to be Ta(ei)=aiei where ei is the orthonormal basic of l^2 and a=(a1,a2,...) is in l^infinte I want to prove this map is bijection can...
  36. Whovian

    A detail in a proof about isomorphism classes of groups of order 21

    Homework Statement While reading through my textbook on abstract algebra while studying for a test, I ran across the following statement: There are two isomorphism classes of groups of order 21: the class of ##C_{21}##, and the class of a group ##G## generated by two elements ##x## and...
  37. S

    Finding Self Isomorphisms in Graphs

    Homework Statement I'm given a graph and am told to find non-trivial self isomorphisms. Non-trivial meaning that at least 1 node is "not mapped onto itself." I've tried looking for self isomorphism but I can't find anything. I can tell when two graphs are isomorphic through inspection but...
  38. Math Amateur

    MHB Simple Problem - Establishing an Isomorphism

    In Example 7 in Dummit and Foote, Section 10.4. pages 369-370 (see attachment) D&F are seeking to establish an isomorphism: S \otimes_R R \cong S They establish the existence of two S-module homomorphisms: \Phi \ : \ S \otimes_R R \to S defined by \Phi (s \otimes r ) = sr and...
  39. J

    Abstract Algebra - Isomorphism

    1. Show that S42 contains multiple subgroups that are isomorphic to S41. Choose one such subgroup H and find σ1,...,σ42 such that How can you solve this?? I am confused if anyone can help me to solve this!
  40. N

    Linear Algebra Composition Isomorphism Question

    Homework Statement Let S : U →V and T : V →W be linear maps. Given that dim(U) = 2, dim(V ) = 1, and dim(W) = 2, could T composed of S be an isomorphism? Homework Equations If Dim(v) > dim(W), then T is 1-1 If Dimv < dim(w), then T is not onto. The Attempt at a Solution So...
  41. Math Amateur

    MHB Ideals of a Residue Class Ring- Ring Isomorphism

    I am reading R. Y. Sharp: Steps in Commutative Algebra. In Chapter 2: Ideals on page 32 we find Exercise 2.40 which reads as follows: ----------------------------------------------------------------------------------------------- Let I, J be ideals of the commutative ring R such that I...
  42. NATURE.M

    Function that is an isomorphism

    Homework Statement So my text states the proposition: If V and W are finite dimensional vector spaces, then there is an isomorphism T:V→W ⇔ dim(V)=dim(W). So, in an example the text give the transformation T:P_{3}(R)→P_{3}(R) defined by T(p(x)) = x dp(x)/dx. Now I understand T is not...
  43. B

    Construct an explicit isomorphism

    $\Bbb{R}P^1$ bundle isomorphic to the Mobius bundle I'm trying to construct an explicit isomorphism from ##E = \{([x], v) : [x] ∈ \Bbb{R}P^1, v ∈ [x]\}## to ##T = [0, 1] × R/ ∼## where ##(0, t) ∼ (1, −t)##. I verified that ##\Bbb{R}P^1## is homeomorphic to ##\Bbb{S}^1## which is homeomorphic to...
  44. E

    Question about normal subgroups/Lattice Isomorphism Theorem

    I was just brushing up on some Algebra for the past couple of days. I realize that the lattice isomorphism theorem deals with the collection of subgroups of a group containing a normal subgroup of G. Now, in general, if N is a normal subgroup of G, all of the subgroups of larger order than N do...
  45. G

    Consequence of the First isomorphism theorem

    From Wikipedia: Consider the map f: G \rightarrow Aut(G) from G to the automorphism group of G defined by f(g)=\phi_{g}, where \phi_{g} is the automorphism of G defined by \phi_{G}(h)=ghg^{-1} The function f is a group homomorphism, and its kernel is precisely the center of G, and its...
  46. F

    MHB Show we have an isomorphism

    let $x=\begin{bmatrix}i&0\\0&0 \end{bmatrix}$ and $y=\begin{bmatrix}0&1\\0&0 \end{bmatrix}$. Define $A={{\begin{bmatrix}a&b\\0&c\\ \end{bmatrix}}where c\in\mathbb{R}}$ Show that A is isomorphic to $\dfrac{R<X,Y>}{((X^2+1)X),(X^2+1)Y,YX)}$ My work: Define $f:R<X,Y>\implies A$ by $f(X)=x$...
  47. L

    Is this homomorphism, actually isomorphism of groups?

    Example: ##\mathcal{L}[f(t)*g(t)]=F(s)G(s) ## Is this homomorphism, actually isomorphism of groups? ##\mathcal{L}## is Laplace transform.
  48. Sudharaka

    MHB Canonical Isomorphism and Tensor Products

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  49. J

    Are Z7[x] and Z Isomorphic?

    Homework Statement Let R = Z7[x]. Show that R is not isomorphic to Z.Homework Equations The Attempt at a Solution One of the necessary conditions for an isomorphism f is that f be one to one. So consider 8x in Z. f(8x) = x, f(1x) = x. So f cannot be an isomorphism. I'm clearly missing...
  50. P

    Show Isomorphism btwn H x G_1 & H x G_2

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