Group Definition and 1000 Threads

Hello, Frustration in receiving timely responses from my teaching assistant has lead me to this website. Currently have a homework assignment on multiple group diffusion theory and one of the assigned questions is, Determine the thermal flux due to an isotropic point source, So fast...

Homework Statement Prove that the group velocity of a wave packet is equal to the particle’s velocity for a relativistic free particle. Homework Equations vgroup = Δω/Δk = dω/dk E = (h/2π)*ω = √(p2c2 + m2c4) The Attempt at a Solution I'll be honest..I have no idea where to...

Here is a problem from some russian book of algebra: $$\varphi(x)=y\leftrightarrow\varphi(y)=x$$ and I know $$\varphi(e)=e.$$ I can see from this that $$G$$ is a group of odd order. How I prove commutativity? Do you think I can prove first that $$\varphi(a)=a^{-1}$$?

Contemporary Abstract Algebra by Gallian This is Exercise 14 Chapter 3 Page 69 Question Let $G$ be the group of polynomials under the addition with coefficients from $Z_{10}$. Find the order of $f=7x^2+5x+4$ . Note: this is not the full question, I removed the remaining parts. Attempt...

I am reading a text about the splitting of the energy levels in crystals caused by the spin orbit interaction. In particular, the argument is treated from the point of view of the group theory. The text starts saying that a representation (TxD) for the double group can be obtained from the...

I just though of this and though "it's abstract math meeting physics, so probably not". After looking up fields in several abstract algebra books I thought that maybe fields in physics were called as such in physics because they share something with the mathematical structure of fields in group...

Well I am not sure if this thread belongs here or in mathematics/groups but since it also has to do with physics, I think SR would be the correct place. An element of the Euclidean group E(n) can be written in the form (O,\vec{b}) which acts: \vec{x} \rightharpoondown O\vec{x}+\vec{b} With O...

Can anyone give an answer (or give a web reference) to the following question: How is a group assigned to a particle? I've seen groups assigned to shapes, polynomials, permutations, rotations and transformations. But how is a group assigned to a point particle?

Homework Statement ##e = \begin{bmatrix} 1 & 0 \\[0.3em] 0 & 1 \\[0.3em] \end{bmatrix}##, ##a =\frac{1}{2} \begin{bmatrix} 1 & -\sqrt{3} \\[0.3em] -\sqrt{3} & -1 \\[0.3em] \end{bmatrix}##. ##b...

You people know that group velocity of a wave packet is calculated with the formula v_g=\frac{d \omega}{d k} .But this gives an expression which,in general,is a function of k.My problem is,I can't think of an interpretation for it.What is that wave-number appearing in the expression for group...

Homework Statement (a) For SU(N), we have: N ⊗ N = A_A + S_S where A corresponds to a field with two antisymetric fundamental SU(N) in- dices φij = −φji, and S corresponds to a field with two symmetric fundamental SU(N) indices φij = φji. By considering an SU(2) subgroup of SU(N), compute...

Homework Statement Find ##(g_ig_j)^{-1}## for any two elements of group ##G##. Homework Equations For matrices ##(AB)^{-1}=B^{-1}A^{-1}## The Attempt at a Solution I'm not sure how to show this? I could show that for matrices ##(AB)^{-1}=B^{-1}A^{-1}##. And that for numbers...

##H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+\frac{1}{2}m\omega^2x^2## Parity ##Px=-x## end ##e## neutral are group of symmetry of Hamiltonian. ## PH=H## ##eH=H## so I said it is group of symmetry because don't change Hamiltonian? And ##e## and ##P## form a group under multiplication. Is there...

Homework Statement Let G be a finite complex matrix group: G \subset M_{n\times n}. Show that, for g \in G, |\text{tr}(g)| \le n and |\text{tr}(g)| = n only for g = e^{i\theta}I. 2. The attempt at a solution Since G is finite, then every element g \in G has a finite order: g^r = I for some...

Homework Statement Obtain multiplication table for quaternion group. Homework Equations ##i^2=j^2=k^2=ijk=-1## The Attempt at a Solution I have problem with elements for example ##ji## in the table. For example when I have ##ij## I say ##ijk=-1## and ##k^2=-1## so ##ij=k##. But...

Hello, I am studying a system of a group of walkers and how they behave. There are four kind of forces: - Repulsion between walkers - A force modeled by the fact that all walkers gather into groups and walk in the same direction so their velocity are // that i will name S. -...

The lorentz group SO(3,1) is isomorphic to SU(2)*SU(2). Then we can use two numbers (m,n) to indicate the representation corresponding to the two SU(2) groups. I understand (0,0) is lorentz scalar, (1/2,0) or (0,1/2) is weyl spinor. What about (1/2, 1/2)? I don't get why it corresponds to...

Hello everyone! This is my first post here, though I've been a silent observer in PF for a long time. I'm planning on working through Spivak's "Calculus" or Wilson's "Introduction to Graph Theory" and was wondering if anyone here might be interested in joining a study group for it. There's no...

Homework Statement Prove that Klein 4 group is not isomorphic with ##Z_4##. Homework Equations Klein group has four elements ##\{e,a,b,c\}## such that ##e^2=e,a^2=e,b^2=e,c^2=e## As far as I know ##Z_4## group is ##(\{\pm 1,\pm i\},\cdot)##. Right? The Attempt at a Solution As far...

Homework Statement Let ##G## be a group of order ##n## where ##n## is an odd squarefree prime (that is, ##n=p_1p_2\cdots p_r## where ##p_i## is an odd prime that appears only once, each ##p_i## distinct). Let ##N## be normal in ##G##. If I have that ##|G/N|=p_j## for some prime in the prime...

Is it ##(\mathcal{R} without \{0\},\cdot)## Lie group?

Homework Statement Say you have a small boat moving through water, and creating a short wave-group which is a superposition of waves in the range of 0.2m-2m. If the shore 50meters away, how long will it take the fastest of the wave-components to reach shore? {assume the depth is constantly very...

Greetings, Just checking if I'm getting this ... please correct me if I'm wrong. The value of the wavefunction is 'probability amplitude' in discrete case and 'probability amplitude density' in continuous case. The former is a dimensionless complex number and the latter is the same...

This isn't a homework problem, but something from a set of notes that I'd like to better understand. My confusion starts on page 23 here: http://isites.harvard.edu/fs/docs/icb.topic1146665.files/III-9-RenormalizationGroup.pdf. I'm having trouble reproducing his calculation for the...

A group ##G## is said to act on a set ##X## when there is a map ##\phi:G×X \rightarrow X## such that the following conditions hold for any element ##x \in X##. 1. ##\phi(e,x)=x## where ##e## is the identity element of ##G##. 2. ##\phi(g,\phi(h,x))=\phi(gh,x) \ \ \forall g,h \in G##. My...

Hello, I have troubles formulating this question properly. So I will explain it through one example. If we consider the Lie group R=SO(2) of rotations on the plane, we know that we can find a manifold on which the group SO(2) acts regularly: this manifold is the unit circle in ℝ2. In fact...

In transitions in the crystals we always use conservation of wave vector of electron, not electron momentum conservation. For example in an indirect transition from top of valence band to bottom of conduction band, the group velocity of electron and hence its momentum would not change (it is...

Hello! I'm currently reading Peskin and Schroeder and am curious about a qoute on page 38, which concerns representations of the Lorentz group. ”It can be shown that the most general nonlinear transformation laws can be built from these linear transformations, so there is no advantage in...

I have no idea how to do this or where to start. Can someone please help me? Problem 4.4- Suppose n o and n e are given. In (a) you only need to find the magnitude of the group velocity. Problem #2 in HW 10 may be helpful. You can also directly use the definition of group velocity, i.e., v g =...

I am having trouble proving that my function is surjective. Here is the problem statement: Problem statement: Let T be the tetrahedral rotation group. Use a suitable action of T on some set, and the permutation representation of this action, to show that T is isomorphic to a subgroup of $S_4$...

Homework Statement Prove that any group of order ##15## is cyclic. 2. The attempt at a solution I am looking at a link here: (http://www.math.rice.edu/~hassett/teaching/356spring04/solution.pdf) and I am confused why "there must be one orbit with five elements and three orbits with three...

I want to ask how does the wave keep the same amplitude if the wave broadens ? Thank you for your time

Homework Statement Let G be a group such that |G|>6. Then there are at least 4 conjugacy classes in GHomework Equations The Attempt at a Solution Well, I tried by contradiction by using the group of order 7, which must be isomorphic to the cyclic group of order 7, which has 7 conjugacy classes...

Hello, I want to prove that the set SO(2) of orthogonal 2x2 matrices with det=1 is a Lie group. The group operation is of course assumed to be the ordinary matrix multiplication \times:SO(2)→SO(2). I made the following attempt but then got stuck at one point. We basically have to prove that...

Hello Homework Statement I have this tab http://img18.imageshack.us/img18/3317/eyph.png Anf I have to find the group and phase velocities for a wavelength λ=550nm Homework Equations v(phase)= c/n v(group)=dw/dk v(phase)*v(group)=c²/n² The Attempt at a Solution I don't know...

Homework Statement Let ##G## be a group of order ##4, G = {e, a, b, ab}, a^2 = b^2 = e, ab = ba.## Determine Aut(G). 2. The attempt at a solution How can I do these types of problems? When doing these types of problems is any automorphism of ##G## always determined by the images? And why...

I'm preparing for an upcoming exam, and as I see one of the typical questions that is frequently asked in our exams is about finding the number of elements that have a particular order in a group like Sn. I searched on google and came up with some such problems with solutions. To be honest...

Homework Statement How do I prove that the inner automorphisms is isomorphic to ##S_3##? The attempt at a solution I know ##S_3 = \{f: \{ 1,2,3 \}\to\{ 1,2,3 \}\mid f\text{ is a permutation}\}## and I know for every group there is a map whose center is its kernel so the center of of...

Hello Everyone, I am seeking some guidance in a topic related to my group dissertation subject. The exact title of the dissertation is 'Investigation on the effect of using nano-ceramics on the strength of composite plates subjected to dynamic (impact) load.'. I had initially been bestowed the...

I want to show that if G is a smooth manifold and the multiplication map m:G×G\rightarrow G defined by m(g,h)=gh is smooth, then G is a Lie group. All there is to show is that the inverse map i(g)=g^{-1} is also a smooth map. We can consider a map F:G×G\rightarrow G×G where F(g,h)=(g,gh) and...

From Artin's Algebra: "Prove that the set ##\operatorname{Aut}(G)## of automorphisms of a group ##G## forms a group, the law of composition being composition of functions." Of course, we could go through and prove that the four group axioms in the standard definition of a group hold for...

My group has given the task of modeling projection of line and my part is to construct the vertical and horizontal plane. So my idea is to have two mirrors joined like an laptop, so that we can fold them and they can also be perpendicular. I'm thinking of making hole in one mirror and a...

Homework Statement Let Q = {±1, ±i, ±j, ±k} be the quaternion group. Find all homomorphisms from Z2 to Q and from Z4 to Q. Are there any nontrivial homomorphisms from Z3 to Q? Then, find all subgroups of Q. Homework Equations The Attempt at a Solution I don't even know...

I have a question about quantum field theory. What does the phrase 'the field is in [certain, e. g. fundamental] representation of a [certain, e. g. SU(2)] group' mean? I know mathematical definitions of groups and their representations, but what does this specific phrase mean?

When I am reading about the Wilson approach to renormalization in Chapter12.1 of Peskin & Shroeder I am wondering why are you allowed only to contract the \hat{\phi} field (this is the field that carries the high-momentums degrees of freedom)as they show in equation 12.10, I thought that we...

I would like to know why $M_n$ $\not\cong$ $O_n$ x $T_n$, where $M_n$ is the group of isometries of $\mathbb R^n$, $O_n$ is the group of orthogonal matrices, and $T_n$ is the group of translations in $\mathbb R^n$. **My attempt:** Can I show that one side is abelian, while the other group is...

Homework Statement Show that for any field F , for n\ge2, the group GL_{n}(F) is not abelian. Homework Equations The Attempt at a Solution I have found a counter example for all such n. First, for n=2, consider the matrices: A = \left( \begin{array}{ccc} 1 & 1\\ 1 & 1 \end{array}...

Homework Statement Hey guys, So I have the following permutations, which are a subgroup of S3: σ_{1}=(1)(2)(3), σ_{5}=(1,2,3), σ_{6}=(1,3,2) This is isomorphic to Z3, which can be written as {1,ω,ω^{2}} Next, we have the basis for the subgroup of S3: e_{i}=e_{1},e_{2},e_{3} And we also have...

Homework Statement Hi guys, The title pretty much says it. I need to explain why: (a) an abelian group of order |G| has precisely |G| conjugacy classes, and (b) why the irreducible representations of abelian groups are one-dimensional. Also in my description below, if I make any mathematical...

I have a question that I have approached, but want to check if I'm on the right track. Let G denote the group of symmetries of a circle. There are infinitely many reflections and rotations. There are no elements besides reflections and rotations. The identity element is the rotation by zero...