Group Definition and 1000 Threads

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...

Can there be repeated elements within a group?

http://i49.tinypic.com/2wqu986.png I don't understand this example. It's from an SAT math study guide. I understand that to find the fraction of the group that is both girls and seniors, 2/3 is multiplied times 2/5. Why is A + B equal to 4/9? Same with A + C.

Homework Statement Given the element: G(x,θ,θ*)= ei(xμpμ+θ Q+θ*Q*) Show that: G(0,ξ,ξ*) G(x,ζ,ζ*)= ei(x'μpμ+(ξ+ζ) Q+(ξ*+ζ*)Q*) and what does x'μ stand for? Homework Equations Some comutation formulas. [ξQ,ξ*Q*]=0 [pμ, ξQ]= ξpμξ* The Attempt at a Solution I tried to use Baker...

While studying the brillouin zone I came across the dispersion relation and the group velocity. The group velocity is given by v=dω/dκ, I understand this in the sense of beats where it is Δω/Δκ and I understand that the group velocity is the propagation speed of the envelope function. However...

On page 333 in Section 52: The Fundamental Group (Topology by Munkres) Munkres writes: (see attachement giving Munkres pages 333-334) "Suppose that h: X \rightarrow Y is a continuous map that carries the point x_0 of X to the point y_0 of Y. We denote this fact by writing: h: ( X...

im having my undergrad math thesis right now, and the topic that i want to choose is the existence of torsion-free simple linear groups. (which i found here: http:///www.sci.ccny.cuny.edu/~shpil/gworld/problems/probmat.html) i just want to know if this topic is recommended for undergrad?

Hello, I'm reading Weinberg's vol.1 on Quantum Theory of Fields and stuck on the following problem. In the massless case Wigner's little group is the group of Lorentz transformations that keep the vector (0,0,1,1) invariant. (I'm going with Wigner's notations, where the vector is denoted...

i need a help in solving ,using 1 group approxmation , estimate the critical size of cube consisting of 75% zirconium-91 and 25% plutonium--239 by volume , when the cube is surrounded by a vacumm. zr-91 microscopic cross section (capture)=0.00335 microscopic cross section (scattering )=5.89...

Homework Statement Given some group G with generators g_{1},g_{2},...,g_{n} as well as a description of the action of g on the elements of some set S={s_{1},s_{2},...,s_{k}}, how in general does one go about finding a complete defining relations (and showing they are complete)? Homework...

Does anybody know? It takes courage to let go of strict Dirac constraint quantization because maybe your chute will not open. But look at these recent papers from Thomas Thiemann and other members of the Erlangen group! Something is happening there: http://arxiv.org/abs/1206.3807 Scalar...

Hi, I have a question from "A first course in abstract algebra" by J. Rotman, Hi, this is a question from " A first course in abstract algebra" by J. Rotman define d(G) = dim(G/pG) chapter 5, lemma 5.8 (P392), Let G be a finite p primary abelian group. If S<=G, then d(G/S) <= d(G)...

It seems rather straight forward that if you have an abelian group G with \# G = p_1 p_2 \cdots p_n (these being different primes), that it is cyclic. The reason being that you have elements g_1, g_2, \cdots g_n with the respective prime order (Cauchy's theorem) and their product will have to...

How would you prove that < x_1,x_2 ... | [ x_i , x_j ] =1, i,j \in N , x_1 ^ p = 1, x_{i+1} ^p = x_i , i \in N > is presentation of Z_{p^ \infinity}

Homework Statement Let G be a group of order n, and let m be an integer such that gcd(m,n) = 1. Prove that g^m = 1 => g = 1 and show that each g \in G has an mth root, that is g = a^m, for some a \in G The Attempt at a Solution Now by Lagrange's theorem, g^n = 1. Since gcd(m,n) =...

Homework Statement I am working through MacLane/Birkhoff's Algebra, and in the section on Symmetric and Alternating groups, the last few exercises deal with generators and Defining relations for Sn (the symmetric group of degree n). These read: 11. Prove that Sn is generated by the cycles (1...

I've been doing some exercises in introductory Galois theory (self-study hence PF is the only avaliable validator :) ) and a side-result of some of them is surprising to me, hence I would like you to set me straight on this one if I'm wrong. Homework Statement Let K(x) be the field of rational...

Suppose E and D are both finite extensions of F, with K being the Galois closure of \langle D,E \rangle (is this the correct way to say it?) Is it correct that E and D are conjugate fields over F iff G,H are conjugate subgroups, where G,H\leqslant \text{Aut}(K/F) are the subgroups which fix...

In a lot of places, I can read that the roots of unity form a cyclic group, however I can find no proofs. Is the reasoning as follows: Let's work in a field of characteristic zero (I think that's necessary). Let's look at the nth roots of unity, i.e. the solutions of x^n - 1. There are n...

Homework Statement In order to determine the infinitesimal generators of the conformal group we consider an infinitesimal coordinate transformation: x^{\mu} \to x^\mu+\epsilon^\mu We obtain \partial_\mu\epsilon_\nu+\partial_\nu\epsilon_\mu=\frac{2}{d}(\partial\cdot\epsilon)\eta_{\mu\nu}...