Group Definition and 1000 Threads

i used to get pauli matrices by the following steps it uses the symmetry of a complex plane sphere i guess so..? however i can't get the 8 gell mann matrices please help ! method*: (x y) * (a b / c d ) = (x' y') use |x|^2 + |y|^2 = |x'|^2 + |y'|^2 and |x| = x * x(complex conjugate) this way...

Homework Statement Describe the structure of the Galois group for G(Q(c):Q) where c is a 14th primitive root of unity Homework EquationsThe Attempt at a Solution the minimal polynomial for c over Q is f(x)=x^6-x^5+x^4-x^3+x^2-x+1. Is the galois group isomorphic to Z*_14? Or maybe that's only...

Indiana officials are trying to block almost 45,000 black citizens from voting Seems the voter's rights group wants the Justice department to look into this and the local police want to leave it with the local county prosecutors they work with. Either way, Police spokesman Bursten said the...

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...

Let $G$ be a group such that for all $a$, $b$, $c$, $d$, and $y\in G$ if $ayb=cyd$ then $ab=cd$. Show that $G$ is an Abelian group. HINTS ONLY as this is an assignment problem.

I'm in my 3rd year of a physics degree, with plans to study further in graduate school. I am currently enrolled in a group theory class, as I have heard it can be useful in many fields (particularly solid state physics, which I am interested in learning more about). However, so far all we have...

I am very confused about that in some literature the Maurer Cartan forms for a matrix group is written as ##{\omega _g} = {g^{ - 1}}dg## what is ##dg## here? can anyone give an example explicitly? My best guess is ## dg = \left( {\begin{array}{*{20}{c}} {d{x^{11}}}& \ldots &{d{x^{1m}}}\\...

Is it possible to do a Friedel-Crafts reaction on a benzene and get a substituent on the benzene ring from that, and then afterwards add an NH2 group ortho? I know you can't add NH2 first and then do Friedel-Crafts because anilines cannot undergo Friedel-Crafts. Please advice. Thanks in advance.

Could you please explain the derivation of group velocity = dw/dk I read ut here https://en.m.wikipedia.org/wiki/Group_velocity Is it approximation, if so under what circumstances

Hello, I was building some test crystal structures with VESTA software and I noticed when generating the POSCAR output files to be read by VASP that for a simple monoatomic basis such as in elemental silicon crystals some distinct space groups produce the same exact POSCAR files (i.e. same...

Hello, what are some good books to learn group theory for physicists at an undergraduate level? Is Zee's Group Theory in a Nutshell good? Thanks in advance

Homework Statement Note: I did not get this problem from a textbook. Let denote the (nonabelian) free group on the generators , and let be arbitrary. My question is, does there exist a such that , besides (the identity); is such an equation in the free group possible? Obviously this...

Hi, I want to find the number of parameters needed to define an orthogonal transformation in Rn. As I suppose, this equals the dimension of the orthogonal group O(n,R) - but, correct me if I'm wrong. I haven't been able to figure out how to do this yet. If it helps, I know that an orthogonal...

micromass submitted a new PF Insights post Groups and Geometry Continue reading the Original PF Insights Post.

Homework Statement The elements lead, silicon, germanium, tin and carbon all lie in the same group of the periodic table. Which of the following is true? A) Like carbon, all the elements are non-metalsB) Carbon is closer in properties to silicon than tin is to leadC) Tin is closer in...

I'm looking for an online American college for a bachelor's degree, with video lectures, and no group work. What are my options? I have difficulty with reading. This kind of a college would work best for me.

Homework Statement Here is my problem : I want to know where does group delay equation come form? Homework Equations As I checked reference 21 it only indicates The Attempt at a Solution I'm following how can I start and solve this problem

While investigating various aspects of generalised least action principles over the last several years I have come across an algebraic mathematical group that I am finding hard to classify but whose root vectors should relate to the standard model (no it is not E8 ! nor any exceptional group I...

Hi, I saw that group theory is a significant asset for some physics, and math topics. I had some fundamental knowledge, but I am really keen on learning group theory deeply , so Is there a nice source( video links, books... whatever comes to your mind ) to leap further in this topic remarkably?

I'm currently studying the Landau-Wilson model for critical phenomena (Statistical Mechanics, Kerson Huang) where the renormalization group is a central object. In the end, the calculations lead to a set of coupled differential equations that describe the (metaphorical) evolution of the...

I learned a lie group is a group which satisfied all the conditions of a diferentiable manifold. that is the real rigour definition or just a simplified one? thanks

Homework Statement .[/B] I am attempting to determine the group refractive index of a laser cavity at it's resonance frequency. Homework Equations .[/B] \begin{align*} \frac{2\omega n L}{c} &= 2m\pi \end{align*} \begin{align*} n_g &= n + \omega \frac{dn}{d\omega} \end{align*} 3. The attempt...

The normal subgroup of a normal subgroup need not be normal in the original group (normalcy is not transitive). Could somebody provide me with an example of where this is the case? Thanks :D

I recently got confused about Lie group products. Say, I have a group U(1)\times U(1)'. Is this group reducible into two U(1)'s, i.e. possible to resepent with a matrix \rho(U(1)\times U(1)')=\rho_{1}(U(1))\oplus\rho_{1}(U(1)')=e^{i\theta_{1}}\oplus e^{i\theta_{2}}=\begin{pmatrix}e^{i\theta_{1}}...

Advice before joining PhD research group, if I have a personal problem with one of the group members? One more student from my cohort wants to join the same research group as me. We will be doing our research rotation together. He has an overbearing attitude and is insolent towards anyone...

Homework Statement I want to put these in order of increasing wavenumber for the absorptions of the carbonyl group in an nmr spectrum Homework EquationsThe Attempt at a Solution I know that the third one is the highest, the NH ketoamine is the lowest and last one is the second lowest but I'm...

I applied to an undergraduate program at a nearby national lab that is tailored for students in the region. I was advised by one of the program staff to contact a research group that I'm interested in working with to let them know that I'm in the applicant pool, and that I am interested in...

Homework Statement Let x and y be conjugate elements of a Group G. Prove that x^n = e if and only if y^n = e, hence x and y have the same order. Homework Equations Conjugate elements : http://mathworld.wolfram.com/ConjugateElement.html The Attempt at a Solution Since y is a conjugate of x...

Homework Statement Let ##x \in G## and ##a,b \in \mathbb{Z}^+## Prove that ##x^{a+b} = x^a x^b##. Homework EquationsThe Attempt at a Solution If I am not mistaken, we would have to do multiple induction on ##a## and ##b## for the statement/proposition ##P(a,b) : x^{a+b} = x^a x^b##. First we...

Homework Statement Determine whether the set of rational numbers with denominator equal to 1 or 2 is a group under addition. Homework EquationsThe Attempt at a Solution Please have a look at the closure proof of part 5. I don't quite understand how ##q/2## implies that the denominator of...

[Mentor's Note: Thread moved from homework forums] Where can I start to research this question? I did not take any course on Group theory before and I know almost nothing about the relationship with this pure maths and physics. I've decided to start with Arfken's book but I'm not sure. 1...

What is the most motivating way to introduce group theory to first year undergraduate students? I am looking for some real life motivation or something which has a real impact.

What is the most motivating way to introduce group theory to first year undergraduate students? I am looking for some real life motivation or something which has a real impact.

Dear All I have a project about exceptional group as a candidate group for grand unified theories. Can anyone suggest me any paper or reference to use. Thank you

The generators ##\{ L^1, L^2 , L^3 , K^1 , K^2 , K^3 \}## of the Lorentz group satisfy the Lie algebra: \begin{array}{l} [L^i , L^j] = \epsilon^{ij}_{\;\; k} L^k \\ [L^i , K^j] = \epsilon^{ij}_{\;\; k} K^k \\ [K^i , K^j] = \epsilon^{ij}_{\;\; k} L^k \end{array} It has the Casimirs C_1 =...

Hey! :o We have that $D_n=\langle a,s\mid s^n=1=a^2, asa^{-1}=s^{-1}\rangle$. I want to show the following: $s^2\in D_n'$ $D_n'\cong \mathbb{Z}_n$ if $n$ is odd $D_n'\cong \mathbb{Z}_{\frac{n}{2}}$ if $n$ is even $D_n$ is nilpotent if and only if $n=2^k$ for some $k=1,2,\dots $ I...

Homework Statement If G is a group with n elements and g ∈ G, show that g^n = e, where e is the identity element. Homework EquationsThe Attempt at a Solution I feel like there is missing information, but that cannot be. This seems too simple: The order of G is the smallest possible integer n...

Homework Statement If n ≥ 3, show that Z(D_n) = C(x) ∩ C(y). Homework Equations G is a group, g∈G C(g) = {h∈G: hg = gh } The Centralizer of g Z(G) = {h∈G: hg = gh for all g∈G} The center of G ∩ means the set of all points that fall in C(x) and C(y). Every element of D_n can be uniquely...

Hey! :o I want to show that: If $G$ contains a subgroup with index at most $4$ and $G$ has not a prime order, then $G$ is not a simple group. In my notes I found the following proposition: $$H\leq G, \ [G:H]=m \text{ and } |G|\not\mid m! \text{ then } G \text{ is not simple. }$$ We have...

Hi. I have the following question: Let G be a finite group. Let K be a subgroup of G and let N be a normal subgroup of G. Let P be a Sylow p-subgroup of K. Is PN/N is a Sylow p-subgroup of KN/N? Here is what I think. Since PN/N \cong P/(P \cap N), then PN/N is a p-subgroup of KN/N. Now...

How is loop quantum gravity related to the principle of relativity, in particular to the diffeomorphism invariance of general relativity?

Hey! :o Let $M$ be a field and $G$ the multiplicative group of matrices of the form $\begin{pmatrix} 1 & x & y \\ 0 & 1 & z \\ 0 & 0 & 1 \end{pmatrix}$ with $x,y,z\in M$. I want to show that $G$ is nilpotent. Could you maybe give me some hints what we could do in this case? (Wondering)...

As part of physical chemistry I am reading up group theory for molecular symmetries. I realize the way chemistry textbooks treat this must be very different from what mathematicians do. So I want to know how I take a point group, find the matrix operations and get the character table.For an C2...

Hey! :o I want to show that if $G$ is of order $2p$ with $p$ a prime, then $G\cong \mathbb{Z}_{2p}$ or $G\cong D_p$. I have done the following: We have that $|G|=2p$, so there are $2$-Sylow and $p$-Sylow in $G$. $$P\in \text{Syl}_p(G) , \ |P|=p \\ Q\in \text{Syl}_2(G) , \ |Q|=2$$ Let $x\in...

Hey! :o I am looking at the following exercise: Each group of order $p^2q$, with $p,q$ different primes, is solvable. To show this can we just say that it holds according to Burnside's Theorem ? (Wondering)

Homework Statement ℤ_n → D_n sending z modn → g^z where g is rotation by an nth of a turn. Homework Equations Group homomorphism imply θ(g_1*g_2)=θ(g_1)*θ(g_2) The Attempt at a Solution Before anything, I'd like to know if Group homomorphism imply θ(g_1+g_2)=θ(g_1)xθ(g_2) I've seen...

Say an element $g$ in a group has order $28$. How do I find the order of say $g^8$?

Homework Statement Let f,h ∈S_4 be described by: f(1) = 1 f(2) = 4 f(3) = 3 f(4) = 2 h(1) = 4 h(2) = 3 h(3) = 2 h(4) = 1 Express (f°h) in terms of its behavior on {1,2,3,4} and then find the order of (f°h). Homework EquationsThe Attempt at a Solution So, first I express (f°h) in terms of...

I'm looking for examples of some interesting compounds that contain alcohol functional group (please, no joke suggestions about various liquor). Bonus points for creativity/thinking-outside-the-box. Thanks.

Hello everyone, I have been reading around that when performing the analytic continuation to Euclidean space (t\to-i\tau) one also has to continue the gauge field (A_t\to iA_4) in order to keep the gauge group compact. I already knew that the gauge field had to be continued as well but I didn't...