Group Definition and 1000 Threads

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.

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}...

Homework Statement Let a and b be elements of a group,with a^2=e , b^6=e and a.b=b^4.a find its order and express its inverse in form of a^m.b^n Homework Equations The Attempt at a Solution (ab)^2=(ab)(ab)=(ab)(b^4.a)=a(b^5)a (ab)^3=a(b^5)a(ab)=a(b^5)(a^2)b=a(b^6)=ae=a it...

i am trying to find G(F_{q}(x^{\frac{1}{q - 1}}/F_{q}(x)) where q is the power of some prime. i know that F_{q}(x^{\frac{1}{q - 1}}) is an extension of F_{q}(x) so i need to find the irreducible polynomial of x^{\frac{1}{q - 1}} over F_{q}(x). i found this to be t^{q - 1} - x...

Hi, I'm looking at proposition 1.14(c) of Artin's Algebra. It says if we have K a splitting field for polynomial f from F[x], with roots a_1,...,a_n, then the Galois group G(K/F) acts faithfully on the set of roots. I look at faithful as the symmetries in the roots completely represent...

A problem asks to find an abelian group V and a field F such that there exist two different actions, call them \cdot and \odot, of F on V such that V is an F-module. A usual way to solve this is to take any vector space over a field with a non-trivial automorphism group, and define r\odot \mu...

Homework Statement Prove that a k-cycle in the group Sn has order k. Homework Equations The Attempt at a Solution I'm mostly confused on how to write this in math notation. I know it will have order k because a1 → a2 → a3 ... ak-1 → ak → a1 if we do the compositions K times. and so...

Is it correct to say that any order 4 group is only isomorphic to either C4 or C2+C2 ? where C4 is the order 4 cyclic group and C2 the order 2 cyclic group

Homework Statement A group of particles is traveling in a magnetic field of unknown magnitude and direction. You observe that a proton moving at 1.60 km/s in the + x-direction experiences a force of 2.10×10−16 N in the + y-direction, and an electron moving at 4.30 km/s in the - z-direction...

[FONT="Arial"]I'm reading a book about Group Theory (by Mario Livio: The Equation that Couldn't be Solved ). On page 46 he explains that four rules and one operation define a group: The rules are Closure, Associativity, the existence of an Identity Element and finally the existence of an...

1. Homework Statement My challenge is as follows: Let Dn be the dihedral group (symmetries of the regular n-polygon) of order 2n and let ρ be a rotation of Dn with order n. (a) Proof that the commutator subgroup [Dn,Dn] is generated by ρ2. (b) Deduce that the abelian made Dn,ab is...

I am really stuck with how to prove that the only element in Sn (with n>=3) commuting with all the other elements of this group is the identity permutation id. I have no idea what I am supposed to do with it, i know why S3 has only one element that commutes but i don't know how to prove it...

Today and tomorrow (5/11 and 5/12) an extremely large group of sunspots is directly facing the earth: http://abcnews.go.com/blogs/technology/2012/05/enormous-sunspot-could-lead-to-solar-flares/ I went outside this morning and using a ND 7.0 filter could see it by eye, so photographs could...

Homework Statement I am struggling with a proof for this. Obviously Sylow's theorems come into play. We have that |G| = 952. As sylow's first theorem only covers subgroups of order pn, we cannot directly use it to assert the existence of a subgroup of order 68. On the other hand, if we can...

Hello, I've been reading about the weak interaction. Basically, the weak interaction couples to particles that are left-handed, and we introduce the electron-electron neutrino as a (left-handed) SU(2) doublet. So, the gauge bosons (W+, W-, and Z) transform SU(2) triplet. Am I right...

Hi! I was just going through this script on Lie groups: http://www.mit.edu/~ssam/repthy.pdf At one point the following is said: (see attachment) I've spent multiple hours trying to figure out why this is a group homomorphism. Sure, once you know the theorem is correct, this follows. But...

A question in my test asked which of the acyl halide will hydrolyse faster in an aqueous solution of NaOH Well the asnwer is acyl iodide becasue iodine is a better leaving group( the solution says so) But i don't understand - fluorine will be hydrated to the greatest extent so removal of...

Homework Statement http://img515.imageshack.us/img515/5954/asdaii.jpg Homework Equations Y(a)Y(b)= Y(ab) Z10 = {1,3,7,9} Z8 = {1,3,5,7} The Attempt at a Solution Y(1)=1 Y(3)=3 Y(7)=5 Y(9)=7 Y(9.7)=Y(3)=3 Y(x)=(x-1)(x-3) + x works Y(7) = 24 + 7 =...

Hi everyone, I'm currently taking an abstract Algebra course and need a little guidance with an analysis of solving a system of linear equations. We are given two linear equations and need to solve for x and y using the method of "substitution" and again using "elimination". However, we must...

Hi! In Weinberg's book "The quantum theory of fields", chapter2, it states that the transformation of a massive particle is U(\Lambda)\Psi_{p,\sigma}= N\sum\mathcal{D}^{(j)}_{\sigma',\sigma}(W)\Psi_{\Lambda p,\sigma'} where W is an element in the little-group SO(3). But than it states that...

Can the group refractive index ng be 1>ng>0 ?

I have a pretty urgent question concerning the calculation of the class group, so any help will be very much appreciated:) I'd like to illustrate my question with an example: Calculate the ideal class group of Q(√-17), giving a representative ideal for each ideal class and a description of the...

Rational numbers form a group under addition. However, a sequence of rational numbers converges to irrational number. Presumably, group theory does not allow adding an infinite number of rational numbers. This is not indicated in the textbook definition of a group. I might be looking in vain...

Does group velocity effect long linear waves generated by a paddle generating waves in deep water? I have developed a numerical wave tank in CFD at full scale, using a bottom hinged flap paddle that oscillates to produce regular waves, the domain is roughly three wavelengths long, and a beach...

It's problem #5 on this homework set: https://docs.google.com/open?id=0B9c8sp75B5ZRMHAxYXB3MWdhYk0 I can calculate (\pi/4)(n1σ1 + n2σ2 + n3σ3) easily, but I have NO clue how a matrix M = exp[(\pi/4)(n1σ1 + n2σ2 + n3σ3)].

Is it possible to define an abelian group on the natural numbers (including 0)? It's just that for every binary operation I've tried, I can't find an inverse!

I am writing an undergraduate "thesis" on group representations (no original work, basically a glorified research paper). I was wondering if anyone could suggest interesting aspects that might be worth writing about in my paper. I have only just begun to explore the topic, and I see that it...

Hello. I have a question about ionization energy: Two hypothetical elements in the 2nd or 3rd period have the following ionization energies: Element X First: 800 kJ/mol Second: 2500 kJ/mol Third: 3900 kJ/mol Fourth: 23000 kJ/mol Element Y First: 700 kJ/mol Second: 2200 kJ/mol...

Being not an expert, my question might sound naive to students of mahematics. My question is how on Earth a Lie group helps to solve an ode. Can anyone explain me in simple terms?

I have to find if the set [1;+∞[ x [1;+∞[ with the operation (x;y)°(v;w) = (x+v-1; yw) is a group I have already proven Closure, associativity and Identity but I have some problems with invertibility :) The neutral element that I have found is (1;1) I did (x;y)°(x1;y1)= (1;1) and I have...

Can't find (or maybe recognize when I see it) anything that discusses this question: A group G is a set of members. We normally assign familiar labels on the members such as a five member group with members labeled as 0, .. , 4. Then, a group operation + is defined as GxG -> G so that a look...

I am Kimberly, newbie here. I love here!

in the appendix on Group Theory in Zee's book there is a discussion of commutations for SO(3) two questions - does [J^{ij},J^{lk}] = J^{ij}*J^{lk}-J^{lk}*J^{ij}? and there is an expression in the appendix that the commutator equals i(\delta^{ik}J^{jl} ... i don't understand the why...

The relation between two terminology "cusp" (group & algebraic curve) Dear Folks: I come across the word "cusp" in two different fields and I think they are related. Could anyone specify their relationship for me?? Many thanks! the cusp of an algebraic curve: for example: (0,0)...