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. jfy4

    Isomorphism & Isometry: Hilbert Spaces

    Hi, I am wondering if all isomorphisms between hilbert spaces are also isometries, that is, norm preserving. In another sense, since all same dimensional hilbert spaces are isomorphic, are they all related by isometries also? Thank you,
  2. L

    Abstract Algebra: Ring Isomorphism Construction

    Homework Statement Homework Equations The Attempt at a Solution
  3. B

    What Does Up to Isomorphism Really Mean?

    what does it really mean? for instance if asked to list all abelian grps of order 12 up to iso, then do we include Z12 or not?
  4. B

    Can K⊕K ≅ N⊕N Imply K ≅ N in Finite Abelian Groups?

    All groups are finite abelian if K⊕K ≅ N⊕N, prove that K≅N I'm thinking of constructing bijection, but I don't know if my argument makes sense! since K⊕K ≅ N⊕N, there exists a bij between the two assume ψ: K⊕K ----> N⊕N (k,k') |---> (n,n') where n = f(k) for some...
  5. B

    Isomorphism: subspace to subspace?

    Homework Statement We're looking at a mapping from P2 (polynomials of degree two or less) to M2(R) (the set of 2x2 real matrices). The nuance here is that the transformation into the matricies is such that its basis consists of only three independent matrices, making its dimension 3. This...
  6. H

    Finding a group isomorphism under modulo multiplication

    Homework Statement I am given the group (G,\bullet) consisting of all elements that are invertible in the ring Z/20Z. I am to find the direct product of cyclic groups, which this group G is supposedly isomorphic to. I am also to describe the isomorphism. Homework Equations The...
  7. J

    Linear Algebra Question on Isomorphism

    Homework Statement Let U be a finite dimensional vector space and suppose that U and W are nonzero subspaces of V prove that (U+W)/W is isomorphic to U/(U \cap W). Homework Equations Here the use of / denotes a quotient space. The Attempt at a Solution Not even sure where to begin.
  8. E

    Isomorphism from group to a product group

    Homework Statement Determine whether or not G is isomorphic to the product group HXK. G=ℂx H={unit circle} K={Positive real numbers} Homework Equations Let H and K be subroups of G, and let f:HXK→G be the multiplication map, defined by f(h,k)=hk. Its image is the set HK={hk...
  9. L

    Group of p-power order isomorphism

    Homework Statement Let G be a group of order p2, where p is a positive prime. Show that G is isomorphic to either Z/p2 or Z/p × Z/p. The Attempt at a Solution Am I completely wrong here or is this just the definition of a p-Sylow subgroup? what I mean is that if g is of order p2...
  10. A

    Does Matrix M Prove T is an Isomorphism Between Vector Spaces?

    Information: The vector-space \mathcal{F}([0,\pi],\mathbb{R}) consists of all real functions on [0,\pi]. We let W be its subspace with the basis \mathcal{B} = {1,cost,cos(2t),cos(3t),...,cos(7t)}. T: W \rightarrow \mathbb{R} ^8 is the transformation where: T(h) = (h(t_1), h(t_2),...,h(t_8))...
  11. B

    Is There an Isomorphism Between ℤ and ℤ[x]?

    There exists none. What's the easiest way to prove this? Can we state that all elements of ℤ are in ℤ[x] but not the other way around?
  12. T

    Is this necessary for showing g≅h? (isomorphism)

    • \mathfrak{g} is the Lie algebra with basis vectors E,F,G such that the following relations for Lie brackets are satisfied: [E,F]=G,\;\;[E,G]=0,\;\;[F,G]=0. • \mathfrak{h} is the Lie algebra consisting of 3x3 matrices of the form \begin{bmatrix} 0 & a & c \\ 0 & 0 & b \\ 0 & 0 & 0...
  13. T

    Proving Isomorphism between Lie Algebras

    Homework Statement Let \mathfrak{g} , \mathfrak{h} be Lie algebras over \mathbb{C}. (i) When is a mapping \varphi : \mathfrak{g} \to \mathfrak{h} a homomorphism? (ii) When are the Lie algebras \mathfrak{g} and \mathfrak{h} isomorphic? (iii) Let \mathfrak{g} be the Lie algebra with...
  14. S

    Anyone up to double check an isomorphism proof?

    Homework Statement *Attached is the problem statement, along with a definition which is to be used (I feel like some of the definitions in this text are somewhat untraditional, so I am including this one for clarity). Edit* sorry, it's problem #31. Homework Equations The...
  15. A

    Nessary and sufficient condition for homomorphism to be isomorphism.

    The necessary and sufficient condition for homomorphisim f of a group G into a group G' with kernel K to be isomorphism of G into G' is that k={e} ... THOUGH I AM ABLE TO PROVE THAT f IS ONE-ONE AND f IS HOMOMORPHISM (in converse part) BUT CAN'T GET ANY IDEA TO PROVE THAT f IS ONTO. PLEASE...
  16. M

    Why can't Dn be isomorphic to the direct product of its subgroups?

    The dihedral group Dn of order 2n has a subgroup of rotations of order n and a subgroup of order 2. Explain why Dn cannot be isomorphic to the external direct product of two such groups. Please suggest how to go about it. If H denotes the subgroup of rotations and G denotes the subgroup of...
  17. M

    Difference between isomorphism and equality in graph theory?

    As the title suggest, I do not understand what the difference between isomorphism and equality is in terms of graph theory. I have tried searching the internet but the few definitions I could find for each did not really shed light on the difference. I know that an isomorphism is when there is a...
  18. Shackleford

    Proving Find isomorphism: What Else Needed?

    Is what I did all I need to do? Is there anything else I need to prove? http://i111.photobucket.com/albums/n149/camarolt4z28/IMG_20110728_175901.jpg?t=1311905852 http://i111.photobucket.com/albums/n149/camarolt4z28/IMG_20110728_175917.jpg?t=1311905865
  19. T

    Is it possible to find an isomorphism between two groups with a new operation?

    Homework Statement Let (G,\cdot) be a group. Defining the new operation * such that a*b = b \cdot a it is pretty easy to show that (G,*) is a group. Show that this new group is isomorphic to the old one. Homework Equations The Attempt at a Solution I have been experimenting...
  20. P

    SOLVE: Isomorphism Problem for Z252 X Z294 and Z42 X Z1764

    Homework Statement is Z252 X Z294 isomorphic to Z42 X Z1764? Explain. Homework Equations The Attempt at a Solution I checked that the highest order of the element in both group are 1764, but don't really know how to justify if there is an isomorphism...Can anyone give me some hints?
  21. Z

    Partitioning into Isomorphism Classes: Exam Q&A

    Just had an exam there, one of the questions was Partition the list of groups below into isomorphism classes 1.\mathbb{Z}_8 2.\mathbb{Z}_8^* (elements of Z_8 relatively prime to 8) 3.\mathbb{Z}_4 \times \mathbb{Z}_2 4.\mathbb{Z}_{14} \times \mathbb{Z}_5 5.\mathbb{Z}_{10} \times...
  22. M

    Isomorphism only if it is a linear transformation?

    Hi, I've come across this result which says that if there are two isomorphic vector spaces with a transformation between them, then that transformation must be linear. Can anyone help me prove this? For instance, if I have a transformation T: Z -> Z where Z is the set of integers, T(z) =...
  23. quasar987

    Order Formula Under Second Isomorphism Theorem: Does it Hold?

    In the conditions where the second isomorphism theorem applies, one has H/HnK = HK/K so in particular, taking orders (in the finite case), one has the order formula |HK| = |H|*|K|/|H n K|. Does anyone know if this formula holds in general, or under lesser hypotheses? Thx.
  24. D

    An isomorphism maps a zero vector to a zero vector?

    (Apologies for ascii art math, I don't know latex. Also apologies if this is in the wrong forum.) Homework Statement Why, in this lemma, must there be a vector v in V? That is, why must V be nonempty? An isomorphism maps a zero vector to a zero vector. Where f:V->W is an isomorphism, fix any...
  25. Y

    Natural isomorphism from V to V

    natural isomorphism from V to V** It is known that there is a natural isomorphism \epsilon \rightleftharpoons \omega^\epsilon from V to V**, where \omega: V \times V* \rightarrow R is a bilinear mapping. So given a certain \epsilon \in V, its image under the isomorphism is actually a set of...
  26. D

    Rings Isomorphism: Proving R & R_2 Subrings of Z & M_2(Z)

    3. Let R = a+b \sqrt{2} , a,b is integer and let R_{2} consist of all 2 x 2 matrices of the form [\begin{array}{cc} a & 2b \\ b & a \\ \end{array} }] Show that R is a subring of Z(integer) and R_{2} is a subring of M_{2} (Z). Also. Prove that the mapping from R to R_{2} is a isomorphism.
  27. L

    Mobius transformation (isomorphism)

    Im having difficulty understanding this satement - can someone please explain it to me... let M be the class of mobius transformations M is isomorphic to GL2/Diag isomorphic to SL2/Id, where GL2 is the group of non-degenerate matrices of size 2 x 2 with complex entries, SL2 = A in GL2 ...
  28. M

    Finding a Function from (G x K) to (G/H) x (K/L) for First Isomorphism Theorem

    Homework Statement Suppose H is a normal subgroup G and L is a subgroup of K. Then (G x K)/(H x L) is isomorphic to (G/H) x (K/L) Homework Equations The Attempt at a Solution I know that I have to use the First Isomorphism Theorem, but in order to do that I need some function phi...
  29. R

    First isomorphism theorem for rings

    Consider: \varphi:R\rightarrow S is a homomorphism. Also,\hat{\varphi}:\frac{R}{ker\varphi}\rightarrow \varphi(R). How can I show \hat{\varphi} is bijective? Most textbooks say it is obvious. I see surjectivity obvious but not injectivity. Could anyone provide a proof for injectivity?
  30. M

    Is f a Surjective and Injective Isomorphism from HxN to HN in G?

    Let G be a group, H a normal subgroup, N a normal subgroup, and H intersect N = {e}. Let H x N be the direct product of H and N. Prove that f: HxN->G given by f((h,n))=hn is an isomorphism from HxN to the subgroup HN of G. Hint: For all h in H and n in N, hn=nh.
  31. I

    Is the transformation t' from V/ker(t) to W injective?

    Homework Statement Let t:V -> W be a linear transformation. Then the transformation t':V/ker(t) -> W defined by: t'(v + ker(t)) = tv is injective and V/ker(t) \approx im(t) Homework Equations A previous theorem: Let S be a subspace of V and let t satisfy S <= ket(t). Then there is...
  32. I

    Prove Basis to Basis Isomorphism

    Homework Statement Let t \in L(V,W). Prove that t is an isomorphism iff it carries a basis for V to a basis for W.Homework Equations L(V,W) is the set of all linear transformations from V to WThe Attempt at a Solution So I figured I would assume I have a transformation from a basis for V to a...
  33. K

    Checking Ring Isomorphism: Z_9 and Z_3 + Z_3

    I was asked to decide if Z_9 and the direct sum of Z_3 and Z_3 are isomorphic. Do I check to see if they are 1-1 and onto?
  34. M

    Isomorphism classes of groups of order 21

  35. D

    Infinitely generated and isomorphism

    Homework Statement Consider the groups Q+ and Q* (rational under addition and ration under multiplication). Prove that neither of these groups is finitely geneated by using the fact that there are infinitely many primes. And prove that Q+ is not isomorphic to Q*. 2. The attempt at a...
  36. K

    Proving Isomorphism of R^x/<-1> and Positive Real Numbers

    Homework Statement Show that R^x/<-1> is isomorphic to the group of positive real numbers under multiplication. Homework Equations The Attempt at a Solution I know I need to show we have a homomorphism, and is one - to one and onto in order to be isomorphic. I know all...
  37. E

    Isomorphism and Cyclic Groups: Proving Generator Mapping

    Homework Statement I need to prove that any isomorphism between two cyclic groups maps every generator to a generator. 2. The attempt at a solution Here what I have so far: Let G be a cyclic group with x as a generator and let G' be isomorphic to G. There is some isomorphism phi: G...
  38. C

    Is R Isomorphic to S? Finding an Explicit Isomorphism

    Homework Statement Determine whether R is isomorphic to S for each pair of rings given. If the two are isomorphic, find an explicit isomorphism (you do not need to show the formal proof). If not, explain why. Homework Equations R= 2x2 matrix, a 0, 0 b, for some integers a,b S= Z x Z...
  39. C

    Using the First Isomorphism Theorem

    I'm trying to understand the first isomorphism theorem for groups. Part of the examples given in the book is showing that Q[x]/(x^3-3) is isomorphic to {a+b*sqrt(3)} As I understand it, by finding a homomorphism from Q[x] to {a+b*sqrt(3)} in which the kernel is x^3-3, the two are...
  40. G

    Here f:Z36 to Z4 x Z9 is group isomorphism given by f(n+36Z)= (n+4Z,

    here f:Z36 to Z4 x Z9 is group isomorphism given by f(n+36Z)= (n+4Z, n+9Z) then what is the inverse of f ?
  41. D

    Proving Isomorphism of Z4 / (2Z4) and Z2

    Homework Statement Why does it make sense (when considering Z4)to form the factor group Z4 / (2Z4) where kZn = {0, k mod n, 2k mod n, ..., nk mod n}? I believe that this above factor group is isomorphic to Z2, but how can I prove this?
  42. D

    Proving Isomorphism of Z4 / (2Z4) to Z2

    Why does it make sense ( when considering Z4)to form the factor group Z4 / (2Z4) where kZn = {0, k mod n, 2k mod n, ..., nk mod n}? I believe that this above factor group is isomorphic to Z2, but how can I prove this?
  43. N

    Is there an isomorphism between O(2n) and SO(2n)xZ2?

    Homework Statement Is there an isomorphism between O(2n)\simeq SO(2n)\times \mathbb{Z}_2 O(2n+1)\simeq SO(2n+1)\times \mathbb{Z}_2 Homework Equations First isomorphism theorem The Attempt at a Solution I think, if I can show a homomorphism between SO(2n)\times\mathbb{Z}_2...
  44. K

    Isomorphism of C(x)-axa^-1 Function in Group G

    Homework Statement Let G be any group and let a be a fixed element of G. Define a function c_{a}:G-->G by c_{a}(x)=axa^{-1} for all x in G. Show that c is an isomorphism The Attempt at a Solution Need to show 1-1, onto and c(ab)=c(a)c(b) I guess my biggest problem is starting because I...
  45. M

    Isomorphism and Binary Structures

    Let J be a set of all linear functions. Consider the set R^2 in the Euclidean plane. Define a binary operation * on R^2 in such a way that the two binary structures <J, +> and <R^2, *> will be isomorphic. Any thoughts? If something is not clear please ask. Thank you.
  46. I

    Isomorphism between Order Ideals and Distributive Lattices

    The poset on the set of order ideals of a poset p, denoted L(p), is a distributive lattice, and it is pretty clear why this is since the supremum of two order ideals and the infimum of 2 order ideals are just union and intersection respectively, and we know that union and intersection are...
  47. K

    How to prove that a group of order prime number is cyclic without using isomorphism?

    How to prove that a group of order prime number is cyclic without using isomorphism/coset? Can i prove it using basic knowledge about group/subgroup/cyclic(basic)? I just learned basic and have not yet learned morphism/coset/index. Can you guys kindly give me some hints or just answer...
  48. D

    Determining whether the map is an isomorphism

    Homework Statement Let F be the set of all functions f mapping R into R that have derivatives of all orders. Determine whether p is an isomorphism of the first binary structure with the second. 1. <F, +> with <R, +> where p(f) = f'(0) 2. <F, +> with <F, +> where p(f)(x) = \int^{x}_{0}...
  49. A

    Isomorphism of Hom_K(V,K) and Hom_K(V⊗V,K)

    Why these two tensor products are isomorphic? Hom_{K}(V,K) \otimes Hom_{K}(V,K) and Hom_{K}(V \otimes V,K) where K is a field and V is a vector space over K.
  50. K

    Why Is There an Isomorphism on Sets If They Lack Algebraic Structure?

    Hi i just start learning algebra. Here are some definitions and examples given in Wikipedia: 1.An isomorphism is a bijective map f such that both f and its inverse f^{-1} are homomorphisms, i.e., structure-preserving mappings. 2.A homomorphism is a structure-preserving map between two algebraic...
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