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.
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?
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}}...
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})##...
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 ##...
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...
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)...
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*}...
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...
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...
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...
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...
Homework Statement
For a commutative ring ##R## with ##1\neq 0## and a nonzerodivisor ##r \in R##, let ##S## be the set
##S=\{r^n\mid n\in \mathbb{Z}, n\geq 0\}## and denote ##S^{-1}R=R\left[\frac{1}{r}\right]##.
Prove that there is a ring isomorphism $$R\left[\frac{1}{r}\right]\cong...
Homework Statement
Let ##G## be a group. Let ##H \triangleleft G## and ##K \leq G## such that ##H\subseteq K##.
a) Show that ##K\triangleleft G## iff ##K/H \triangleleft G/H##
b) Suppose that ##K/H \triangleleft G/H##. Show that ##(G/H)/(K/H) \simeq G/K##
Homework Equations
The three...
Homework Statement
Let m ≥ 3. Show that $$D_m \cong \mathbb{Z}_m \rtimes_{\varphi} \mathbb{Z}_2 $$
where $$\varphi_{(1+2\mathbb{Z})}(1+m\mathbb{Z}) = (m-1+m\mathbb{Z})$$
Homework Equations
I have seen most basic concepts of groups except group actions. Si ideally I should not use them for this...
Homework Statement
*This is from a Group Theory class
**My secondary aim is to practice writing the math perfectly because I tend to loose a lot of points for not doing so in exams...
Let λ ∈ Q*
fλ : Q → Q defined as fλ(x) = λx
a) Show that fλ is and automorphism of the group of rationals...
i saw a proof that said “in R/{0} , the set [-1,0) has an upper bound ,but no least upper bound. no such set exists in linearly ordered R” ,but i could not understand it.
I'm trying to figure out how to prove this, but I'm unsure how to approach it.
Let G and H be groups, let ϕ: G → H be an isomorphism, and let ψ be the inverse function of ϕ. Prove that ψ is an isomorphism from H to G.
any help? thanks
Homework Statement
Determine ##\phi(R_{180})##, if ##\phi:D_n\to D_n## is an automorphism where ##n## is even so let ##n=2k##.
The solutions manual showed that since the center of ##D_n## is ##\{R_0, R_{180}\}## and ##R_{180}## is not the identity then it can only be that...
Homework Statement
Prove that ##D_\infty/\langle R^n \rangle\cong D_{2n}##, where ##D_\infty=\langle R,S \mid S^2=e, SRS=R^{-1}\rangle##.
Homework EquationsThe Attempt at a Solution
Pick ##g:\{R,S\} \to D_{2n}## such that ##g(R) = r## and ##g(S) = s##. We note that ##g(S)^2 = 1## and...
Homework Statement
##(\mathbb{C}^\times,\cdot)/\mu_m\cong (\mathbb{C}^\times,\cdot)## for any integer ##m\geq 1##, where ##\mu_m=\{z\in \mathbb{C} \mid z^m=1\}##.
Homework EquationsThe Attempt at a Solution
Here is my idea. Consider the map ##f: \mathbb{C}^{\times} \to \mathbb{C}^{\times}##...
Homework Statement
A group is called Hamiltonian if every subgroup of the group is a normal subgroup. Prove that being Hamiltonian is an isomorphism invariant.
Homework EquationsThe Attempt at a Solution
Let ##f## be an isomorphism from ##G## to ##H## and let ##N \le H##. First we prove two...
Let ##G## and ##H## be groups, and let ##\phi : G \to H## be an isomorphism. I want to show that ##\phi^{-1} : H \to G## is also an isomorphism. First, note that ##\phi^{-1}## is clearly a bijection as ##\phi## is its inverse. Second, let ##a,b \in H##. Since ##\phi## is surjective, there exist...
I am interested in the following theorem:
Every field of zero characteristics has a prime subfield isomorphic to ℚ.
I am following the usual proof, where we identify every p∈ℚ as a/b , a∈ℤ,beℕ, and define h:ℚ→P as h(a/b)=(a*1)(b*1)-1 (where a*1=1+1+1... a times) I have worked out the...
Hi physicsforums,
I am an undergrad currently taking an upper-division course in Quantum Mechanics and we have begun studying L^2 space, state vectors, bra-ket notation, and operators, etc.
I have a few questions about the relationship between L^2, the space of square-integrable complex-valued...
I am reading "The Basics of Abstract Algebra" by Paul E. Bland ... ...
I am currently focused on Chapter 3: Sets with Two Binary Operations: Rings ... ...
I need help with Bland's proof of the Third Isomorphism Theorem for rings ...
Bland's Third Isomorphism Theorem for rings and its proof...
I am reading "The Basics of Abstract Algebra" by Paul E. Bland ... ...
I am currently focused on Chapter 3: Sets with Two Binary Operations: Rings ... ...
I need help with Bland's proof of the Third Isomorphism Theorem for rings ...
Bland's Third Isomorphism Theorem for rings and its proof...
I am reading "The Basics of Abstract Algebra" by Paul E. Bland ... ...
I am currently focused on Chapter 3: Sets with Two Binary Operations: Rings ... ...
I need help with Bland's proof of the Second Isomorphism Theorem for rings ...
Bland's Second Isomorphism Theorem for rings and its proof...
I am reading "The Basics of Abstract Algebra" by Paul E. Bland ... ...
I am currently focused on Chapter 3: Sets with Two Binary Operations: Rings ... ...
I need help with Bland's proof of the Second Isomorphism Theorem for rings ...
Bland's Second Isomorphism Theorem for rings and its proof...
I have gotten the following answer to (a) and (b) which require verification on them. I have also attached the theorem for reference.
(a) Z x Z => have zero divisors
The matrix has no zero divisors (no nonzero matrix when multiplied to the matrix gives zero element)
Hence not...
Hi, I have attached the question and the solutions to part a and b of this question. Would like someone to verify if I have done anything wrong. Greatly appreciate it! Thanks.
Would also like to check if there is a simpler method to prove f is an isomorphism? Thanks
Homework Statement
Let ##A,B## be subgroups of a finite abelian group ##G##
Show that ##\langle g_1A \rangle \times \langle g_2A \rangle \cong \langle g_1,g_2 \rangle## where ##g_1,g_2 \in B## and ##A \cap B = \{e_G\}##
where ##g_1 A, g_2 A \in G/A## (which makes sense since ##G## is abelian...
My book is trying to show that the rngs ##2 \mathbb{Z}## and ##3 \mathbb{Z}## are not isomorphic. It starts by saying that if there were an isomorphism ##\mu : 2 \mathbb{Z} \to 3 \mathbb{Z}## then by group theory we would know that ##\mu (2) = \pm 3##. It then goes on to show that this leads to...
Homework Statement
T((x_0, x_1, x_2)) = (0, x_0, x_1, x_2)
Homework Equations
None
The Attempt at a Solution
I'm getting hung up on definitions. My book says that T is an is isomorphism if T is linear and invertible. But it goes on to say that for T of finite dimension, T can only be an...
Homework Statement
H and K are normal subgroups of G such that the intersection of H and K is the identity. Also, G = HK = {hk | h in H and k in K}. Find an isomorphism between G and H x K
Homework EquationsThe Attempt at a Solution
I was thinking that an isomorphism could be ##\mu : G...
Hi,
I have the following:
Let ##\Omega ## be a discrete subgroup of ##C##, the complex plane.
If:
i) ##\Omega = \{nw_1 | n \in Z\} ##, then ##\Omega ## is isomorphic to ##Z##.
ii) ##\Omega = \{nw_1 + mw_2 | m,n \in Z\} ## where ##w_1/w_2 \notin R ## , then ##\Omega## is isomorphic to ##Z## x...
Hey! :o
Let $V$ be the real vector space $\mathbb{R}[X]$ and $M \subset \mathbb{R}$ a set with $d$ elements. Let $$U_1 := \{ f \in \mathbb{R}[X] | \forall m \in M : f(m) = 0\}, \ \ U_2 := \{ f \in \mathbb{R}[X] \mid \deg(f) \leq d − 1\}$$ be two vector spaces of $V$. Let $\Phi: V\rightarrow...
I felt upon a mistake I made but cannot understand. I consider the following rotation transformation inspired from special relativity :
$$\left(\begin{array}{c} x'\\ict'\end{array}\right)=\left (\begin {array} {cc} cos(\theta) & -sin(\theta) \\ sin(\theta) & cos(\theta) \end...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...