Group Definition and 1000 Threads

I know the physical meaning of phase as well as the group velocity. I want to know the mathematical relationship between the phase velocity and group velocity. Thanks a lot.

Monoid of "specifications" for a group The question of whether there is any standard math associated with specifications of ordered pairs on a group went nowhere (https://www.physicsforums.com/showthread.php?t=640395), so I will spell out what I have in mind. It appears possible to define a...

It is a well known fact that the Lorentz group of transfornations are linear. Now reading the wiki entry on the LG it spends a good deal explaining its identity component subgroup, the restricted LG group, and it turns out it is isomorphic to the linear fractional transformation group, which are...

Homework Statement Let H be a subgroup of G Prove xH=yH ⇔ x-1.y\inH Homework Equations The Attempt at a Solution If x.H = y.H then x,y\inH since H is a subgroup x-1,y-1\inH and the closure of H means x-1.y\inH Proving the reverse is my problem despite the fact that I'm sure...

Hello, Please help in solving the four set of problems, i will be very happy explaining comment as really want to understand. The problem will spread to the extent of understanding preduduschey. 1 Problems: The set M, M = {e^(j*2*pi*k/n) , k= 0,1,2...n-1} denotes the set of the nth...

I am trying to understand phase and group velocities in a few different contexts, but require some assistance. From pictures I have seen how these speeds can be different, and I have come to understand that the phase velocity can be greater than the speed of light because it does not actually...

Homework Statement I'm trying to find the galois group of x^5 - 1 over Q, and then for each subgroup of the galois group identify which subfield is fixed. Homework Equations The Attempt at a Solution If w = exp(2*I*PI/5), then the roots not in Q are w, w^2, w^3, w^4. Its fairly...

I have heard, but never seen a representation, that we know the characteristics of photons because of their reaction with other charges particles. So what exact physical motions are induced on the particles when a photon travels through them. Does the magnitude of the electric and magnetic...

Is there a technical term in group theory for (what I would call) partially specified elements of a group? I mean "partially specified" in the following sense: An elements of a group acts as permuation on the set of elements of the group. So a group element can be considered to be a function...

1. The problem statement, all variables and given/known data Let (G,*) be a group, and denote the inverse of an element x by x'. Show that f: G to G defined by f(x) = x' is a bijection, by explicitly writing down an inverse. Given x, y in G, what is f(x *y)? Homework Equations...

Homework Statement There are two groups, group 1 and 2. Group 1 has a 0.5 chance of losing $36, while group 2 has a 0.1 chance of losing $36 dollars. The groups are of equal size. Now an insurance company is willing to cover the losses for a payment. However, the insurance company has an...

Homework Statement The only thing I need to do now is show that isometric functions commute. I've shown the 3 properties that prove the the set G of isometric functions is a group. Homework Equations If f:Z-->Z is bijective and preserves distances, then f is isometric. The Attempt...

I'm working on a problem where I have to find the little co-group and star of two wave vectors for a diamond structure (space group 227). I know I have to act on the vector by the symmetry operations in the group (perhaps only the ones in the isogonal point group, Oh?) and see if it remains the...

A little bit of background I have passed all my qualifiers and my even passed my preliminary exam (research plan). My research area was particle searches at the CMS detector. The issues that led to me being dismissed was due to my computing ability. I am able to program however had very little...

Homework Statement I'm given the refractive index of a piece of glass: $$n(\omega)=A+B\omega$$ And I have to find the speed at which a pulse of radiation will travel through the glass at an angular frequency $$\omega = 1.2 \times 10^{15} s^{-1}$$ I also have A = 1.4, B=3.00 x 10^-17. Homework...

32 point group system "mmm" in Orthorhombic crystal Hello, I am trying to understand the mmm symmetry in a orthorhombic crystal. Looking from the diagram I know there are 5 diads which will give me 10 unique planes in the same form. But how do I know the exact planes that are in the form given...

Homework Statement A wave packet in a dispersive medium is given as : y(t) =cos(x-5t)cos(.2x-.4t)cos(.1x-.2t) Find group velocity and phase velocity for the wave packet. Hence plot w-k variation for the calculated values. 2. The attempt at a solution We know that for the wave, Vg = dw/dk and...

what is an infinite group that has exactly two elements with order 4? i let G be an infinite group for all R_5 ( multiplication modulo 5) within this interval [1,7) so i got |2|=|3|=4. i'm not sure this is the right answer but i couldn't think of anything else at a moment. help please.

Can anyone recommend some litterature on representations of the Lorentz group. I'm reading about the dirac equation and there the spinor representation is used, but I would very much like to get a deeper understanding on what is going on.

Homework Statement CH3CN [ in presence of aq. H2SO4 and 2H2O] → CH3COOH + NH3 The Attempt at a Solution There is a triple bond present between Carbon and nitrogen, hence the first will be sp hybridized, while there is double bond between carbon and oxygen in the second, hence it should...

I have to choose a total of 12 modules for my 3rd year. I've everything decided except four of them. I want to eventually do research either General Relativity, quantum mechanics, string theory, something like that. I'm torn between Group Representations, with one of Practical numerical...

I have a simple question but I'm putting down the whole derivation as it is relevant. There is a point that I don't understand, or seems wrong to me. This is a derivation of Group Velocity followed by simplifying(approximating it) for long wavelength waves in shallow water. This appears in a...

In my notes on waves (specifically water waves) there is a derivation of Group Velocity. They consider two waveforms with the same amplitude, that differ slightly in wavelength and frequency, which are then superimposed to give wave groups. kis wavenumber, \delta k is how much the wavenumbers...

You know that the current theories in particle physics are expressed in the language of group theory and the symmetries of the theory describe its properties I don't know how is that but my question is,can we do that to classical physics too? I mean,can we use maxwell's equations and derive a...

Homework Statement (a) Suppose a belongs to a group and lal=5. Prove that C(a)=C(a3). (b) Find an element a from some group such that lal=6 and C(a)≠C(a3). Homework Equations The Attempt at a Solution For (a) I know I need to show that every element in the set C(a) is...

Homework Statement If G is a group, a is in G, and a*b=b for some b in G (* is a certain operation), prove that a is the identity element of G Homework Equations The Attempt at a Solution I feel like you should assume a is not the identity element and eventually show that a= the...

What's an example of a group that has finitely many generators, but cannot be presented using only finitely many relations? Are there any nice groups? They do exist, right?

I understand that the special orthogonal group consists of matrices x such that x\cdot x=I and detx=1 where I is the identity matrix and det x means the determinant of x. I get why the matrices following the rule x\cdot x=I are matrices involved with rotations because they preserve the dot...

Is the relation v_{\varphi }v_{g}=v^{2}=\frac{1}{\mu \varepsilon } always true in a plasma ? Where v_{\varphi }, v_{g} are respectively the phase and group velocity of the electromagnetic wave that is propagating in the plasma.

Homework Statement Prove that the set of all rational numbers of the form 3n6m, m,n\inZ, is a group under multiplication. Homework Equations The Attempt at a Solution For this problem I attempted to show that the given set has 1. an Identity element, 2. each element has an...

What is the relationship between topology, functional analysis, and group theory? All three seem to overlap, and I can't quite see how to distinguish them / what they're each for.

I'm trying to understand what a one-parameter group of transformations really is. At one lecture I was told that they are trivial lie groups. In Arnold's "Ordinary Differential Equations" they are defined as an action by the group of real numbers; a collection of transformations parametrised by...

Homework Statement See attached photo Homework Equations The Attempt at a Solution I figured I would use the parallel axis theorem. I'm stuck between two different methods of doing the question, both of which are choices in the answers. My gut instinct says to take the...

Hi all! I'm trying to study the Poincare group and I have one problem. I'm reading a book: Gross D. Lectures on Quantum Field Theory (there is section about it). So I do not understand how the second part of (3.26 and 3.27) folows from the first part i.e I do not understand how was obtained...

What does it mean to give a group structure? I'm working on a problem and part of it asks for the structure of the group. The law of composition and generators seem to be given already (and an expression that says that a^2 = 1 for any elt a of the group). Is there anything to do other than...

My question is about the shaded area in the attachment? How did the author get that all the elements of order p or 4 of L are contained in K? I mentioned the abstract but I do not think there is a need for that. Help?

Hello everyone, I am working on the group delay of the front end filter of a GPS system. I am given the measurements of the S parameters of the filter in a touchstone file (s2p) in the following format. ! S-Parameter for B3521 in Touchstone format with Magnitude (lin) and Phase ! Normalised...

Let p be a prime. Let H_{i}, i=1,...,n be normal subgroups of a finite group G. I want to prove the following: If G/H_{i}, i=1,...,n are abelian groups of exponent dividing p-1, then G/N is abelian group of exponent dividing p-1 where N=\bigcap H_{i} ,i=1,...,n. Proof: Since G/H_{i}...

Hi i need a little help i was given group (Z3 x Z3,+) and i should find order of every elements so the elements are {(0,0),(0,1),(0,2),(1,0),(1,1),(1,2),(2,0),(2,1),( 2,2)} and the order of every element is (0,0) has order 1 (0,1)*3=(0(mod 3),3(mod 3)) = (0,0) order 3 (0,2)*3=(0(mod...

A dihedral group of an n-gon denoted by Dn, whose corresponding group is called the Dihedral group of order 2n? What I gather from that is a square has 8 symmetries, an octagon has 16, a hexagon 12, etc?

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 Prove that for a finite group A, the order of any element in A divides the order of A. Homework Equations The order of an element a of a group A is the smallest positive interger n such that an = 1. The Attempt at a Solution Well, I know that the order of a...

Heres the question: http://img442.imageshack.us/img442/484/sarr.png Firstly does anyone know what Ki is? Ionisation constant? I'm guessing its proportional to the drugs potency. I could take a good guess by observing the differences between each isostere but I'm guessing there's some kind of...

In Particle Data Group booklet, many refined determinations of the masses and decay widths processes are collected and listed in a fabulous way. But it is still (at least for me) to find the sources of these Data, I mean from which experiments, FOCUS, or BELLE..., are taken? is there other...

Let $\mathbb{G}$ be a set with a map $(\xi, ~ \eta) \mapsto f(\xi, ~\eta)$ from $\mathbb{G}\times\mathbb{G}$ into $\mathbb{G}$. For every pair $(\xi, ~ \eta)$ in $\mathbb{G}$ let $f(\xi, ~\eta) = f(\eta, ~ \xi)$. Suppose there are elements $\omega$ and $\xi'$ in $\mathbb{G}$ such that for every...

Hello! I have a question regarding measurement uncertainties. This is not a homework problem. Let's say that I want to measure some quantity, and I want to measure it multiple times using multiple identical but separate instruments. That is, -First measurement taken using equipment 'A'...

I am totally confused about the Lorentz Group at the moment. According to wikipedia, the Lorentz group can be defined as the General Orthogonal Lie Group##O(1,3)##. However, the definition of the GO Lie Group that I know only works when there is a single number inside the bracket, not 2, e.g...

Homework Statement If G is a finite abelian group, and x is an element of maximal order, then <x> is a direct summand of G. Homework Equations The Attempt at a Solution I claim that the hypothesis implies that A = G\<x> \bigcup {e} is a subgroup of G. If so, then since G = < <x> \bigcup A>...

Hi, can anyone explain me why (mathematically) the braid group is infinte? I guess it's infinite because you can do every braid you want and even if you braid two particles interchanging them twice in a clock (or counterclock) wise manner, (so you bring them back at the original positions), the...

Homework Statement Let G be a group and e its identity. This group has the property that a^3 = e, for every a in G. What I need to do is verify if this condition is sufficient for G to be abelian. 2. The attempt at a solution I found a non-trivial group for which this is true, namely the...