Isomorphism Definition and 321 Threads

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...

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.

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...

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...

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...

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...

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...

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.

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...

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...

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...

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 ...

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...

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...

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...

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?

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...

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?

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) =...

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...

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...

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...

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...

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.

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

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

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...

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...

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}...

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...

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...

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 =...

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...

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

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

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...

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...

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...

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...

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.

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...

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...

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...

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...

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...

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...

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

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...

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...