# 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. ### 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. ### 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. ### Fixed point free automorphism of order 2

I did not use the hint for this problem. Here is my attempt at a proof: Proof: Note first that ##σ(σ(x)) = x## for all ##x \in G##. Then ##σ^{-1}(σ(σ(x))) = σ(x) = σ^{-1}(x) = σ(x^{-1})##. Now consider ##σ(gh)## for ##g, h \in G##. We have that ##σ(gh) = σ((gh)^{-1}) = σ(h^{-1}g^{-1})##...
4. ### 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. ### 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. ### MHB Is G/G isomorphic to the trivial group? A proof for G/G\cong \{e\}

Reorder the statements below to give a proof for G/G\cong \{e\}, where \{e\} is the trivial group. The 3 sentences are: For the subgroup G of G, G is the unique left coset of G in G. Therefore we have G/G=\{G\} and, since G\lhd G, the quotient group has order |G/G|=1. Let \phi:G/G\to \{e\} be...
7. ### I Understanding the concepts of isometric basis and musical isomorphism

Im very new to the terminologies of isometric basis and musical isomorphism, will appreciate a lot if someone could explain this for me in a simple way for a guy with limited experience in this field. The concrete problem I want to figure out is this: Given: Let ##v_1 = (1,0,0) , v_2 = (1,1,0)...
8. ### F is an isomorphism from G onto itself,...., show f(x) = x^-1

i) Proof: Let ##a, b \in G## ##(\Rightarrow)## If ##G## is abelian, then ## \begin{align*} f(a)f(b) &= a^{-1}b^{-1} \\ &= b^{-1}a^{-1} \\ &= (ab)^{-1} \\ &= f(ab) \\ \end{align*} ## So ##f## is a homomorphism. ##(\Leftarrow)## If ##f## is a homomorphism, then ## \begin{align*}...
9. ### I Classify the isomorphism of a graph

N and k are positive integers satisfying $$1<=k < n$$ An undirected graph $$G_{n,k}= (V_{n,k} ,E_{n,k})$$ is defined as follows. $$V_{n,k}={1,2,3,...n}$$ $$E_{n,k}={\{\{u,v\}|u-v \equiv k \, (mod \, n) \, or \, u-v \equiv -k \, mod \, n}$$ However, $$x \equiv y \, (mod \, n)$$ indicates...
10. ### MHB Isomorphism of logic, arithmetic, and set theory

Has anybody ever heard of this? I learned about it in a discrete math class in grad school, and I've never heard of it anywhere else !? For example, logical disjunction (OR) and set-theoretic UNION are isomorphic in this sense: 0 OR 0 = 0. {0} UNION {0} = {0}. Similarly, logical AND & set...
11. ### Isomorphisms preserve linear independence

Homework Statement Let ##T:V \rightarrow W## be an ismorphism. Let ##\{v_1, ..., v_k\}## be a subset of V. Prove that ##\{v_1, ..., v_k\}## is a linearly independent set if and only if ##\{T(v_1), ... , T(v_2)\}## is a linearly independent set. Homework EquationsThe Attempt at a Solution...
12. ### Proof of isomorphism of vector spaces

The theorem is as follows: All finite dimensional vector spaces of the same dimension are isomorphic Attempt: If T is a linear map defined as : T : V →W : dim(V) = dim(W) = x < ∞ & V,W are vector spaces It would be sufficient to prove T is a bijective linear map: let W := {wi}ni like wise let...

39. M

### I A regular matrix <=> mA isomorphism

Hello all Let ##m_A: \mathbb{K^n} \rightarrow \mathbb{K^n}: X \mapsto AX## and ##A \in M_{m,n}(\mathbb{K})## (I already proved that this function is linear) I want to prove that: A regular matrix ##\iff m_A## is an isomorphism. So, here is my approach. Can someone verify whether this is...
40. ### Isomorphism to certain Galois group and cyclic groups

Homework Statement Let c be a pth root of unit where p is prime. Then the Galois group G(Q(c):Q) is isomorphic to Z_p*. Show that if there is some m that divides p-1, then there is an extension K of Q such that G(K:Q) is isomorphic to Z_q* Homework EquationsThe Attempt at a Solution I suspect...
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### I Is Every Isomorphism in Vector Spaces Reflexive?

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. ### Isomorphism between so(3) and su(2)

Homework Statement How do I use the commutation relations of su(2) and so(3) to construct a Lie-algebra isomorphism between these two algebras? Homework Equations The commutation relations are [ta, tb] = i epsilonabc tc, the ts being the basis elements of the algebras. They basically have the...
43. ### 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. ### When is this linear transformation an isomorphism?

Homework Statement Let L: ℝ2→ℝ2 such that L(x1, x2)T=(1, 2 ; 3, α)(x1, x2)T=Ax Determine at what values of α is L an isomorphism. Obviously L is given in matrix form. The Attempt at a Solution First of all a quick check, dim (ℝ2)=dim(ℝ2)=2 Ok. An isomorphism means linear transformation which...
45. ### 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. ### Proving a function is an isomorphism

Homework Statement Let G be a finite abelian group with no elements of order 2 Show that the function φ: G-> G defined as φ(g) = g^2 for all g ∈G, is an isomorphism.Homework Equations Abelian group means xy = yx for all x,y∈G Isomorphic if there exists a bijection ϒ: G_1 -> G_2 such that for...
47. ### Abstract Algebra: Bijection, Isomorphism, Symmetric Sets

Homework Statement Suppose X is a set with n elements. Prove that Bij(X) ≅ S_n. Homework Equations S_n = Symmetric set ≅ = isomorphism Definition: Let G and G2 be groups. G and G2 are called Isomorphic if there exists a bijection ϑ:G->G2 such that for all x,y∈G, ϑ(xy) = ϑ(x)ϑ(y) where the...
48. ### Isomorphism is an equivalence relation on groups

Homework Statement Prove that isomorphism is an equivalence relation on groups. Homework Equations Need to prove reflexivity, symmetry, and transitivity for equivalence relationship to be upheld. **We will use ≅ to define isomorphic to** The Attempt at a Solution Let G, H, and K be groups...
49. ### Is Isometry the Same as Isomorphism?

I have read a definition of isomorphism as bijective isometry. I was also showed a definition that isomorphism is a bijective map where both the map and its inverse are bounded (perhaps only for normed spaces??). This does not seem to be the same thing as an isometry. For example, the poisson...
50. ### Understanding Isomorphisms for Linear Transformations

Homework Statement I have a question about isomorphisms -- I'm not sure if this is the right forum to post this in though. A linear transformation is an isomorphism if the matrix associated to the transformation is invertable. This means that if the determinant of a transformation matrix = 0...