Group Definition and 1000 Threads

Homework Statement Show that the group velocity vg=dω/dk can be written as vg=v-λ*dv/dλ where v = phase velocity Homework Equations n=n(k)=c/v k=2∏/λ ω=2∏f=kv fλ=c The Attempt at a Solution dω/dk = d(kv)/dk= v+k(dv/dk)= v+ck(d(n^-1)/dk) =v-(ck/n^2)(dn/dk)...

Homework Statement Homework Equations The Attempt at a Solution Is my initial assumption wrong?

Homework Statement Prove that in SL(2) group the matrix ## \begin{pmatrix} -1 & \lambda \\ 0 & -1 \end{pmatrix} ## can not be presented as a single exponentail but instead as product of two exponentials of ##sl(2)## algebra. ##\lambda \in \mathbb{R} ## Homework Equations I don't understand...

Hi all, Here i ask the fisrt serie of questions i couldn't solve; A basic knowledge of group theory is supposed for solving them! ------------------------------------------------------------ 1- Can you find 3 subgroups H, k and L of a group G such that H U k U L = G ;and no one of the 3...

Homework Statement Homework Equations The Attempt at a Solution For the first question, since [f(a)][g(a)] is in C, can I just say that since C is a ring, it is an abelian group, then the four axioms are proven? Then just show closure? Probably not I'm guessing. Associativity...

Hi! I'm trying to prove a cyclic group is isomorphic to ring under addition. What the strategy I would take? How would I get it started? Here's what I know so far: I need to meet 3 conditions-- 1 to 1, onto, and the operation is preserved. I also know that isomorphic means that the group is...

Homework Statement Let G be an ableian group of order mn, where m and n are relativiely prime. If G has has an element of order m and an element of order n, G is cyclic. The Attempt at a Solution ok so we know there will be some element a that is in G such that a^m=e where e is the...

How can one prove that for homomorphism G \xrightarrow{\rho} G' and H as kernel of homomorphism, quotient group G/H is isomorphic to G'? Thanks.

Hey, just trying to get my head around the logic of this. I can see that if composition factors are cyclic then clearly the group is soluble, since there exists a subnormal series with abelian factors, but I am struggling to see how the converse holds. If a group is soluble, then it has a...

Hello everyone, I read in Edmond's 'Angular momentum in Quantum Mechanics' that the symmetry group of the 9j symbol is isomorphic to the group S_3 \times S_3 \times S_2. Why is this? Can anyone shed some light on this?

I need to read about finite abelian groups. I searched 'finite abelian group' on amazon and the closest search result was 'finite group theory'. Googling didn't help either. Does there exist a book dedicated to finite abelian groups? If yes, and if you know of a good one then please reply...

Is there a general algorithm for taking the presentation of a group and get the permutation generators for the subgroup of A(S) to which the group is isomorphic? For example, given x^5=y^4=e, xy=f(c^2) how do I find (12345) and (1243), the permutations corresponding to x and y? BTW, the example...

Hello! I'm currently reading Ryder - Quantum Field Theory and am a bit confused about his discussion on the correpsondence between Lorentz transformations and SL(2,C) transformations on 2-spinor. He writes that the Lie algebra of Lorentz transformations can be satisfied by setting \vec{K}...

Homework Statement Prove that the proper orthochronous Lorentz group is a linear group. That is SOo(3, 1) = {a \in SO(3, 1) | (ae4, e4) < 0 } where (x,y) = x^T\etay for \eta = [1 0 0 0; 0 1 0 0; 0 0 1 0; 0 0 0 -1] (sorry couldn't work out how to properly display a matrix). Homework...

Homework Statement Let's say I have two compounds in an identical solvent. The compounds are also identical except for the following: One has a Br bonded to a tertiary carbon, and the other has an I bonded to a secondary carbon. Which would react first in an Sn1 nucleophilic reaction...

The phase velocity of ocean waves is (gλ/2∏)1/2,where g is the acceleration of gravity.Find the group velocity of ocean waves. Relevant equations: λ=h/γmv phase velocity= c2/v(velocity of particle) group velocity=v (velocity of particle). thnxx in advance

Homework Statement Are all subgroups of a cyclic group cyclic themselves? Homework Equations G being cyclic means there exists an element g in G such that <g>=G, meaning we can obtain the whole group G by raising g to powers. The Attempt at a Solution Let's look at an arbitrary...

I am reading about group theory in particle physics and I'm slightly confused about the word "representation". Namely, it is sometimes said that the three lightest quarks form a representation of SU(3), or that the three colors do. But at the same time, it is said that a group can be...

Thanks! I started a new thread for new questions. Let me start with one I don't even know makes sense: are there counterparts to the braid group for higher dimensional objects like membranes?

I am very confused by the treatment of Peskin on representations of Lorentz group and spinors. I am confronted with this stuff for the first time by the way. For now I just want to start by asking: If, as usual Lorentz transformations rotate and boost frames of reference in Minkowski...

Suppose we have a group with presentation G = <A|R> i.e G is the quotient of the free group F(A) on A by the normal closure <<A>> of some subset A of F(A). Is it true that that fundamental group of the Cayley graph of G (with respect to the generating set A) will be isomorphic to the subgroup...

Hello, In my abstract algebra class, my teacher really stresses that when you show that a set is a group by satisfying the axioms of a group (law of combination, associativity, identity element, inverse elements) these axioms MUST be proved in order. This makes some amount of sense to me...

Homework Statement Let G be a finite group in which every element has a square root. That is, for each x in G, there exists a y in G such that y^2=x. Prove every element in G has a unique square root. Homework Equations G being a group means it is a set with operation * satisfying...

Homework Statement True or False? Every infinite group has an element of infinite order. Homework Equations A group is a set G along with an operation * such that if a,b,c \in G then (a*b)*c=a*(b*c) there exists an e in G such that a*e=a for every a in G there exists an a' such...

I am reading my textbook of QFT (Maggiore, Modern Introduction in QFT), and there is this statement: "If T^a_R is a representation of the algebra and V a unitary matrix of the same dimension as T^a_R , then V T^a_R V^\dagger is still a solution o the Lie algebra and therefore provides...

Homework Statement There is a group of 7 people. How many groups of 3 people can be made from the 7 when 2 of the people refuse to be in the same group? Homework Equations nCr The Attempt at a Solution Here is what I know: 7C3 gives the total number of groups that can be...

Is there an easy way to see if a unitary group is cyclic? The unitary group U(n) is defined as follows U(n)=\{i\in\mathbb{N}:gcd(i,n)=1\}. Cyclic means that there exits a element of the group that generates the entire group.

I just studied group theory. Its all nice with all the definitions and rules that are supposed to be followed for a set with a given operation to be called a group. But I fail to see the importance of defining such an algebraic structure. What are its uses?

Does the permutation group $S_8$ contain elements of order $14$?My answer: If $\sigma =\alpha \beta$ where $\alpha$ and $\beta$ are disjoint cycles, then $|\sigma|=lcm(|\alpha|, |\beta|)$ . Therefore the only possible disjoint cycle decompositions for a permutation $\sigma \in S_8$ with...

Why Denote Group Operation with Multiplication?? When groups are introduced in most abstract algebra texts, the operation is denoted by multiplication or juxtaposition and addition notation is reserved for abelian groups. This seems to cause a lot of unnecessary confusion. Professors often...

Hi everyone, I read in 'Angular momentum in Quantum Mechanics' by A.R Edmonds that the symmetry group of the 6j symbol is isomorphic to the symmetry group of a regular tetahedron. Is there an easy way of seeing this? I've tried working out what the symmetry relations of the 6j symbol do...

Let's consider a single frequency signal of frequency say 'f'. If the wave is propagating through a medium (EM wave with a velocity of 'c') then what will be the phase and group velocity? I believe that we can't find out the phase velocity and that the group velocity should be equal to the...

Homework Statement Prove (\mathbb{R},+) and (\mathbb{C},+) are isomorphic as groups.Homework Equations An isomorphism is a bijection from one group to another that preserves the group operation, that is \phi(ab)=\phi(a)\phi(b)The Attempt at a Solution I'm trying to find a bijection, but I can...

Homework Statement Let R be a subring of ℂ such that the group of invertible elements U(R) is finite, show that this group is a cyclic group. (Group operation being multiply). Homework Equations The Attempt at a Solution I have the answer, and I got very close to getting there myself before...

Homework Statement 1)Fix ##n \in \mathbb{N}##. Consider multiplication mod ##n##. Let G be the subset of {1,2,...,n-1} = ##\mathbb{Z}_n## \ ##\left\{0\right\}## consisting of all those elements that have a multiplicative inverse (under multiplication mod n). Show that G is a group under...

Homework Statement Prove the collection of all finite order elements in an abelian group, G, is a subgroup of G. The Attempt at a Solution Let H={x\inG : x is finite} with a,b \inH. Then a^{n}=e and b^{m}=e for some n,m. And b^{-1}\inH. (Can I just say this?) Hence...

Homework Statement So we have this operation x*y=x+2y+4 and then our 2nd one is x*y=x+2y-xy I need to check if it is commutative,associative, and if it has a identity and an inverse. The Attempt at a Solution y*x=y+2x+4 so it is not commutative x*(y*z)=x+2(y+2z+4)+4=x+2y+4z+12...

Homework Statement Im looking at this example and trying to figure out how they showed it was associative. They start out with x*y=x+y+1 then they add in z to show it is associative. x*(y*z)=x*(y+z+1)=x+(y+z+1)+1=x+y+z+2 I don't know how they go from this x*(y+z+1)=x+(y+z+1)+1 and...

What exactly is a graded group, Is it just the direct decomp. of the group, or space? is it a way of breaking a group/space into its generators? how do these entities work? Help, please!

ello everybody, how can I calculate the group velocity of a wave package in an infinite square well? I know only how it can be calculated with a free particle, the derivation of the dispersion relation at the expectation value of the moment. But in the well, there are only discrete...

Let $G$ be a group. Let $a ∈ G$. An inner automorphism of $G$ is a function of the form $\gamma_a : G → G$ given by $\gamma_a(g) = aga^{-1}$. Let $Inn(G)$ be the set of all inner automorphisms of G. (a) Prove that $Inn(G)$ forms a group. (starting by identifying an appropriate binary...

Homework Statement Let ##\theta : G \mapsto H## be a group homomorphism. A) Show that ##\theta## is injective ##\iff## ##\text{Ker}\theta = \left\{e\right\}## B) If ##\theta## is injective, show that ##G \cong I am \theta ≤ H##. The Attempt at a Solution A)The right implication is...

Do adults in school constitute an "underrepresented group?" I know we have a few of us here - "returning adults," "non traditional students," or whatever appellation is currently P.C. I'm wondering if, given that I'm not a member of any sort of minority group, I should be considered...

An infinitely long "mass-spring transmission line", consisting of masses (m) connected by springs (spring constant s) obeys the following dispersion relation: ω = \sqrt{4s/m} sin(kd/2). The group velocity is dω/dk = d/2 \sqrt{4s/m} cos(kd/2). What does zero group velocity "mean" for...

Homework Statement Let G be a finite group, a)Prove that if ##g\,\in\,G,## then ##\langle g \rangle## is a subgroup of ##G##. b)Prove that if ##|G| > 1## is not prime, then ##G## has a subgroup other than itself and the identity. The Attempt at a Solution a) This one I would just like...

hi , this result is from text , Abstract Algebra by Dummit and foote . page 120 the result says , if G is a finite group of order n , p is the smallest prime dividing the order of G , then , any subgroup H of G whose index is p is normal and the text gave the proof of this result ...

Hello, it is known that "Every regular G-action is isomorphic to the action of G on G given by left multiplication". Is this true also when G is a Lie group? There is an ambiguous sentence in Wikipedia that is confusing me. It says: "The above statements about isomorphisms for regular, free...

My IR spectra correlation chart for organic chemistry says that the stretch for a nitro N-O bond occurs at "1550 and 1400cm-1" and that it will look like "teeth". Why does N-O have two peaks? The rest of the functional groups on my chart list a range in which a single peak should appear, but why...

Hello ladies and gentelmen. I am currently studying mathematics in middle-east, I want to study Twistor Theory in master. the problem is most of twistor specialists in UK(including Roger Penrose) and my scholarship allow me to choose only university in USA . I did quick search here are...

Recently, over in the relativity forum, Micromass contributed a post: https://www.physicsforums.com/showpost.php?p=4168973&postcount=89 giving a proof that the most general coordinate transformation preserving the property of zero acceleration (i.e., maps straight lines to straight lines) is...