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. P

    Show isomorphism between two groups

    Homework Statement Suppose G is a non-abelian group of order 12 in which there are exactly two elements of order 6 and exactly 7 elements of order 2. Show that G is isomorphic to the dihedral group D12. Homework Equations The Attempt at a Solution My attempt (and what is listed...
  2. H

    Isomorphism beetwenn vector space and sub space

    Hi, I have to find a vector space V with a real sub space U and a bijective linear map. Here my Ideas and my questions: If the linear map is bijective, than dim V = dim U Because U is a real sub space the only way to valid this constraint is if the dimension is infinity. I wrote...
  3. S

    Establishing uniqueness of an isomorphism

    Homework Statement Let G=[a] and G'= be cyclic groups of the same order. Show, that among the isomorphisms \theta from G to G', there is exactly one with \theta(a)=c if and only if c is a generator of G. Homework Equations The Attempt at a Solution I have managed to show the...
  4. S

    Order of groups in relation to the First Isomorphism Theorem.

    Given H,K and general finite subgroups of G, ord(HK) = [(ord(H))(ord(K))] / ord(H intersection K) I know by the first isomorphism theorem that Isomorphic groups have the same order, but the left hand side of the equation is not a group is it? I am struggling to show this.
  5. H

    Good basic isomorphism online tutorials

    How the group of symmetries of the regular pentagram is isomorphic to the dihedral group of order 10? Suggest me some good basic isomorphism online tutorial.
  6. K

    Z_2 /<u^4+u+1> isomorphism Z_2 /<u^4+u^3+u^2+u+1>

    Z_2[u]/<u^4+u+1> isomorphism Z_2[u]/<u^4+u^3+u^2+u+1> Homework Statement How to figure an isomorphism from Z_2[u]/<u^4 + u +1> to Z_2[u]/<u^4 + u^3 + u^2 + u + 1> What I can now show (after a page and a half of work) is that the two polynomials generating the ideals are irreducible...
  7. P

    What is the First Isomorphism Theorem?

    Homework Statement can someone explain the 1st isomorphism theorem to me(in simple words) i really don't get it Homework Equations The Attempt at a Solution
  8. S

    Number of Isomorphisms f from G to G' of Order 8 Cyclic Groups

    The question is this: How many isomorphisms f are there from G to G' if G and G' are cyclic groups of order 8? My thoughts: Since f is an isomorphism, we know that it prserves the identity, so f:e-->e', e identity in G, e' identity in G'. Also f preserves the order of each element...
  9. F

    How Does G/N Relate to Complex Numbers of Absolute Value 1?

    Homework Statement Let G be the group of real numbers under addition and let N be the subgroup of G consisting of all the integers. Prove that G/N is isomorphic to the group of all complex numbers of absolute value 1 under multiplication. Hint: consider the mapping f: R-->C given by...
  10. T

    Isomorphism of A, B ∩ C: Techniques

    What would be the technique to show A is isomorphic to (B intersection C)?where A, B and C are groups.
  11. D

    Proving Banach Space Property Using Topological Isomorphism

    Homework Statement http://img219.imageshack.us/img219/2512/60637341vi6.png Homework Equations I think this is relevant: http://img505.imageshack.us/img505/336/51636887dc4.png The Attempt at a Solution A topological isomorphism implies that T and T-1 are bounded and given is that all cauchy...
  12. Y

    Understanding Factor Rings F[x]/<x^3+X+1> & F[x]/<x^3+X^2+1>

    I'm lost and don't even know where to start. Let F = Z mod 2, show F[x]/<x^3+X+1> and F[x]/<x^3+X^2+1> are isomorphic. I guess fist I need help understanding what those two factor rings look like and what elements they contain. Thanks
  13. E

    Ring of Integers Isomorphism Problem

    Homework Statement Let N = AB, where A and B are positive integers that are relatively prime. Prove that ZN is isomorphic to ZA x ZB. The attempt at a solution I'm considering the map f(n) = (n mod A, n mod B). I've been able to prove that it is homomorphic and injective. Is it safe to...
  14. D

    How Can I Prove Boundedness and Continuity in Isomorphism Problems?

    Homework Statement http://img297.imageshack.us/img297/1434/25931863lt2.png Homework Equations http://img297.imageshack.us/img297/4654/35953374xl9.png I don't see how you can show that T and T-1 are bounded. Furthermore I don't understand the notation Tf is that T*f or Tf as...
  15. L

    Isomorphism from <R,+> to <R+,\times>: Proving 1-1 and Onto Function

    Homework Statement Is there an isomorphism from <R,+> to <R+,\times> where \phi(r)=0.5^{r} when r \in R? [b]2. Homework Equations For an isomorphism I know it is necessary to show there is a 1-1 and onto function. I am unsure if I can use the steps I am trying to use to show it is 1-1...
  16. D

    Natural isomorphism of Left adjoints

    Given two left adjoints F,H:\mathcal{C}\to\mathcal{D} of a functor G:\mathcal{D}\to\mathcal{C}, how do we show that F and H are naturally isomorphic? This is my idea so far (I am working with the Hom-set defenition of adjunction): We need to construct a natural isomorphism \alpha. So, for...
  17. J

    Simple question on disproving a group isomorphism

    I am trying to prove that the additive groups \mathbb{Z} and \mathbb{Q} are not isomorphic. I know it is not enough to show that there are maps such as, [tex]f:\mathbb{Q}\rightarrow \mathbb{Z}[/itex] where the input of the function, some f(x=\frac{a}{b}), will not be in the group of integers...
  18. E

    Is Q[x]/I ring-isomorphic to Q[\sqrt{2}]?

    Homework Statement Prove that Q[x]/\langle x^2 - 2 \rangle is ring-isomorphic to Q[\sqrt{2}] = \{a + b\sqrt{2} \mid a,b \in Q\}. The attempt at a solution Denote \langle x^2 - 2 \rangle by I. a_0 + a_1x + \cdots + a_nx^n + I belongs to Q[x]/I. It has n + 1 coefficients which somehow map...
  19. G

    Field on R^3 and isomorphism between C and R

    Hey Guys ; I need to discuss this problem with you. 1st of all , I'm going to post some posts about some questions with answers . ======================================================================= Q) Could we define a multiplication operation on \mathbb R^3 to have a field on it ...
  20. M

    What Are Some Examples of Isomorphic Vector Spaces with Different Dimensions?

    I came across this problem today and haven't been able to figure it out... Give an example of a vector space V which isomorphic to a proper subspace W, i.e. V != W. It seems to me that V can't have a finite basis, but can't think of any examples regardless...any thoughts?
  21. J

    O(3) sp(2) lie algebra isomorphism problem

    I'm mainly hoping that somebody else might have done the same exercise earlier. In that case it could be possible to spot where I'm going wrong. Homework Statement I'm supposed to prove that Lie algebras \mathfrak{o}(3) and \mathfrak{sp}(2) are isomorphic. Homework Equations Let's...
  22. H

    Isomorphism Question: Struggling with Engineering Maths - Need Help!

    Hi everyone. I am new to these forums. I do Computer System Engineering at Brunel university in London. I did Maths and Physics at A-level but I'm struggling with some of the maths in my Engineering Maths module. Could someone please help me with the exam question I have attached with this post...
  23. F

    Isomorphism Criteria in Linear Algebra

    Can anyone tell me clearly what the criteria for isomorphism in linear algebra is? For instance, my book gives the following reason: Transformation T is not isomoprhic because T((t-1)(t-3)) = T(t^2 - 4t +3) = zero vector. I don't get why this means T is not an isomorphism. Can anyone...
  24. S

    How to Show F is an Isomorphism for Polynomial Vectors?

    Homework Statement Let V = P2(R) be the vector space of all polynomials P : R −> R that have order less than 2. We consider the mapping F : V −> V defined for all P belonging to V , by F(P(x)) = P'(x)+P(x) where P'(x) denotes the first derivative of the polynomial P. Question is: Show...
  25. D

    Subring of Z₂₈ & Isomorphism: S={0,4,8,12,16,24}

    Show that the set S = { 0, 4, 8, 12, 16, 24} is a subring of Z subscript 28. Then prove that the map Ø: Z subscript 7 → S given by Ø(x) = 8x mod 28 is an isomorphism
  26. E

    Group Theory: Isomorphism between Z_3 x Z_4 & Z_12

    [SOLVED] group theory Homework Statement My book says that Z_3 cross Z_4 is isomorphic to Z_12, which I am confused about because Z_3 cross Z_4 has four different generators and Z_12 only has 1. EDIT: wait that is not true, Z_12 has generators 1,5,7,11 It is probably true in general that the...
  27. L

    Remember Isomorphism Theorems: Intuition Guide

    Does anybody know of a nice, intuitive way to remember the second and third isomorphism theorems?
  28. M

    Please explain isomorphism with respect to vector spaces.

    Can someone explain isomorphism to me, with respect to vector spaces. Thanks!
  29. copper-head

    Is Every Group Isomorphic to a Subgroup of GLn(R)?

    Hello. My book offers this statement with no proof, i have been searching in other books with no luck ! I'm beginning to question whether or not the statement is valid at all ! Here it goes: "Every group G of order n is isomorphic to a subgroup of GLn(R)" Could someone please help me out...
  30. A

    Isomorphism and linear independence

    I think I am missing a key info below. I have listed the problem statement, how I am approaching and why I think I am missing something. Please advise why I am wrong. Thanks Asif ============ Problem statement: Let T: U->V be an isomorphism. Let U1, U2,...,Un be linearly...
  31. T

    Isomorphism and direct product of groups

    Just wondering if there is a general way of showing that (Z, .)n isomorphic to Zm X Zp with the obvious requirement that both groups have the same order?
  32. D

    Decide whether each map is an isomorphism

    Homework Statement Decide whether each map is an isomorphism (if it is an isomorphism then prove it and if it isn’t then state a condition that it fails to satisfy). Homework Equations f : M2×2 ---- P^3 given by: a b c d --- c + (d + c)x + (b + a)x^2 + ax^3 The Attempt at...
  33. Q

    Proving Isomorphism without Explicit Functions in Abstract Algebra

    I am having a very hard time with a general concept of proving something. If I have some arbitrary function mapping one ring, let's say R, to another ring, S, and want to prove that R is isomorphic to S, then I need to show that there exists a bijective homomorphism between R and S. But how do I...
  34. K

    Proving U(8) is not Isomorphic to U(10): Insights and Techniques

    Hi. Hoping a could have a little bit of guidance with this question Show that U(8) is not isomorphic to U(10) So far, I've realized that in U(8) each element is it's own inverse while in U(10) 3 and 7 are inverses of each other. I guess that's really all I need to say that they aren't...
  35. N

    Abstract Algebra - isomorphism question

    Abstract Algebra -- isomorphism question If N, M are normal subgroups of G, prove that NM/M is isomorphic to N/N intersect M. That's how the problem reads, although I am not sure how to make the proper upside-down cup intersection symbol appear on this forum. Or how to make the curly "="...
  36. J

    Lie algebra, ideal and isomorphism

    Suppose A\subset\mathfrak{g} and I\subset\mathfrak{g} are subalgebras of some Lie algebra, and I is an ideal. Is there something wrong with an isomorphism (A+I)/I \simeq A/I, a+i+I=a+I\mapsto a+I, for a\in A and i\in I? I cannot see what could be wrong, but all texts always give a theorem...
  37. D

    Proof of Order Isomorphism Claim for Well Ordered Sets

    Hi, I am trying to prove a claim about order isomorphisms (similarity) between well ordered sets. I have an argument for it, but it seems needlessly complicated and I was wondering if anyone might have a simpler proof. Before stating the claim and my proof, I will define a few things: 1. A...
  38. X

    Group isomorphism and Polynomial ring modulo ideal

    Hi everyone. I have two questions that I hope you can help me with. First when trying to show isomorphism between groups is it enough to show that the order of each element within the group is the same in the other group? For example the groups (Z/14Z)* and (Z/9Z)*. They are both of order...
  39. B

    Group G_k: Proving Isomorphism

    Let k be a positive integer. define G_k = {x| 1<= x <= k with gcd(x,k)=1} prove that: a)G_k is a group under multiplication modulos k (i can do that). b)G_nm = G_n x G_m be defining an isomorphism.
  40. B

    Some basic algebra (using Isomorphism Theorems)

    Homework Statement Let G be a group with a normal subgroup N and subgroups K \triangleleft H \leq G. If H/K is nontrivial, prove that at least one of HN/KN and (H\cap N)/(K\cap N) must be nontrivial. Homework Equations The Three (or Four) Isomorphism Theorems. The Attempt at...
  41. happyg1

    Prove Isomorphism: Aut(Z_2⊕Z_4) = D_8

    Homework Statement Prove that Aut(Z_2\oplus Z_4) \cong D_8 Homework Equations The Attempt at a Solution To start, I wrote out all of the elements of Z_2\oplus Z_4. There are 8 of them, of course. Then I need to find the automorphisms of it. It looks to me like they would be the same as...
  42. E

    Can L Be Isomorphic to sl(2,C)?

    Homework Statement Take L = \left(\begin{array}{ccc}0 & -a & -b \\b & c & 0 \\a & 0 & -c\end{array}\right) where a,b,c are complex numbers. Homework Equations I find that a basis for the above Lie Algebra is e_1 = \left(\begin{array}{ccc}0 & -1 & 0 \\0 & 0 & 0 \\1 & 0 &...
  43. K

    Isomorphism O: L2(E) to (E,E*) for Vector Spaces over Field K

    Show that the isomorphism O:L2(E)—>(E,E*) Where E is a vectorspace over a field K E* is a dual space L2:bilinear form L: n-linear form.
  44. M

    Guided proof to the isomorphism theorems.

    Homework Statement Let G_1 and G_2 be groups with normal subgroups H_1 and H_2, respectively. Further, we let \iota_1 : H_1 \rightarrow G_1 and \iota_2 : H_2 \rightarrow G_2 be the injection homomorphisms, and \nu_1 : G_1 \rightarrow G_1/H_1 and \nu_2 : G_2/H_2 be the quotient epimorphisms...
  45. C

    How do I find an isomorphism between Sn+m and Zn x Zm?

    How do I find an isomorphism between Sn+m and Zn x Zm? provided n,m are not relatively prime? Thanks.
  46. P

    Isomorphism between finite sets

    In general if two finite sets contain exactly the same number of unique elements than the two sets are isomorphic to each other. Is this correct? An isomorphism => both 1-1 and onto. If two sets both have an equal number of unique elements than they must be onto because every element in one set...
  47. C

    Proving Isomorphism of Complex Number Conjugate: General Algebraic Systems

    I am working through this algebra book and some of the problems. The chapter this comes out of is General Algebraic Systems and the section is Isomorphisms. I am new to proofs and maths higher than calculus I so I am not sure if I am following the text or not. There aren't any solutions and this...
  48. T

    Is there a isomorphism between N and Q?

    Hi all, I wonder if there is an isomorphism between the group of \mathbb{N} and the group of \mathbb{Q} (or \mathbb{Q}+). I know there is a proof that there is a bijection between these sets, but I didn't find a way how to construct the isomorphism. What confuses me a little is that (I...
  49. JasonJo

    Does the First Isomorphism Theorem apply in this case?

    prove that there does not exist a homomorphism from G:= (integers modulo 8 direct product integers modulo 2) to H:= (intergers modulo 4 direct product integers modulo 4). Pf: i tried this route, assume that there is such a homomorphism. then by first isomorphism theorem, G/ker phi is...
  50. P

    Proving Isomorphism of (ZxZxZ)/<(2,4,8)> to (Z(index 2)xZxZ)

    Hi I'm trying to solve (find a group that is ismorphic to) (ZxZxZ)/<(2,4,8)>. (1,2,4)+<(2,4,8)> must be of order 2 in the factor group. (0,1,1)+<(2,4,8)> and (0,0,1)+<(2,4,8)> generates infinite cyclic subgroups of the factor group. So it would be reasonable to presume that (ZxZxZ)/<(2,4,8)>...
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