Group Definition and 1000 Threads

Hey guys! Basically, I was wondering how to prove the following statement. I've seen it in the Hamermesh textbook without proof, so I wanted to know how you go about doing it. Let's say you have a group element g_{1}, which has a corresponding inverse g_{1}^{-1}. Let's also define a linear...

Hey everyone Let's say I have two generators, a and b, with the following relations: a^{5}=b^{2}=E bab^{-1}=a^{-1}; Where E is the Identity element. What I've done so far is this - the number of elements of the group is the product of the exponents of both generators, which is 10...

I want to determine the orbits of the proper orthochronous Lorentz group SO^{+}(1,3) . If I start with a time-like four-momentum p = (m, 0, 0, 0) with positive time-component p^{0} = m > 0 , the orbit of SO^{+}(1,3) in p is given by: \mathcal{O}(p) \equiv \lbrace \Lambda p...

Hi, I am looking to find the invariants of products of fields under SU(5) and other possible gauge groups (but let's take SU(5) as an example). Take, for example, two matter fields in the 5* and 10 and two Higgses in 5 and 5* (called H_{5} and \bar{H}_{5*}). Then the term 5* 10...

Homework Statement Show that any group of order 4 or less is abelian 2. The attempt at a solution I came across this hint. Since its of order 4 we have {e,a,b,c}, where e = identity. The elements a, b, c must have order 2 or 4. There are two possibilities. 1. a, b, c all have order 2. 2...

Homework Statement The dielectric constant k of a gas is related to its index of refraction by the relation k = n^{2}. a. Show that the group velocity for waves traveling in the gas may be expressed in terms of the dielectric constant by \frac{c}{\sqrt{k}}(1 -...

Hey! From MathWorld on solvable group: But why is that a special case? The way I understand it: the normal series can always be made such that all composition factors are simple, but then the composition factors are both simple and Abelian, and hence (isomorphic to) \mathbb Z_p, i.e. the...

Let's say you have a group of 22 people, which you would like to break into 5 different groups -- 3 groups of 4 and 2 groups of 5. How many distinct ways can you form such groups? I don't want to double count groups. Let's say I number the people from A - V. The group ABCDE and ACDBE should...

My prof has been throwing around some group theory terms when talking about spin and isospin (product representations, irreducible representations, SU(3), etc.) I'm looking for a brief intro to group theory, the kind you might find in a first chapter of a physics textbook, so I can get familiar...

I was reading the derivation on Wikipedia: http://en.wikipedia.org/wiki/Group_velocity#Derivation Why is the first part before the integral sign ignored when calculating the velocity? Surely it would also cause a phase shift in some time interval and make the waves move forward (or backward)?

Homework Statement Prove that no group of order 160 is simple. Homework Equations Sylow Theorems, Cauchy's Theorem, Lagrange's Theorem.The Attempt at a Solution Because 160 = 2^5×5, by the First Sylow theorem, there is a subgroup H of order 2^5 = 32 in G. Let S be the set of all...

1. Homework Statement . Prove that the only homomorphism between Z5 and Z7 is ψ(x)=0 (the trivial homomorphism). 3. The Attempt at a Solution . I wanted to check if my solution is correct, so here it goes: Any element x in Z5 belongs to the set {0,1,2,3,4} So, I trivially start by...

1. Homework Statement . Let p be a prime number, m a natural number and G a group of order p^m. Prove that there exists an element a in G such that ord(a)=p. 3. The Attempt at a Solution . I know of the existence of Lagrange theorem, so what I thought was: I pick an arbitrary element a (I...

Hello, I'm following the proof for this theorem in my textbook, and there is one part of it that I can't understand. Hopefully you can help me. Here is the part of the theorem and proof up to where I'm stuck: Let ##N## be a normal subgroup of a group ##G##. Then every subgroup of the...

Hi! I keep hearing that in the large N limit (so I am talking in specific AdS/CFT but more general too I guess) U(N) and SU(N) are isomorphic. So if I construct, say, the ## \mathcal{N}=1 ## SYM Lagrangian in the large N limit, I can take as gauge group both of the ones mentioned above...

I couldn't find the words to summarize my question perfectly in the title so I will clarify my question here. Say we have a group G in which every element can be written in the form g_1^{e_1} g_2^{e_2}...g_n^{e_n}, 0 ≤ e_i < |g_i| . Suppose that there exists a different set g_1', g_2', ...

1. Homework Statement Let S be the set of complex numbers z such that |z|=1. Is S a cyclic group? 3. The Attempt at a Solution I think this group isn't cyclic but I don't know how to prove it. My only idea is: If G is a cyclic group, then there is an element x in G such that...

Homework Statement Prove or disprove the following assertion. Let G, H, and K be groups. If G × K \cong H × K, then G \cong H.Homework Equations G × H = \left\{ (g,h): g \in G, h \in H \right\} The Attempt at a Solution I don't even know whether the statement is true or false... I tried...

Is any given finite semigroup isomorphic to some finite semigroup S that consists of some subsets of some finite group G under the operation of set multiplication defined in the usual way? (i.e. the product of two subsets A,B of G is the set consisting of all (and only) those elements of G that...

Hi, I have a question about groups: What is the de Sitter group?? and how does it relate to poncaire's group? Thanks!

Why must the charged particle that leads to Cherenkov radiation travel faster than the phase velocity of light not the group velocity of light? One of the sides of the triangle that is used to define cosθ is v=c/n i.e. the phase velocity. I don't see why it's one rather than the other. Thanks!

Let ##G## be a set equipped with a binary associative operation ##\cdot##. In both of the following situations, we have a group: 1) ##G## is not empty, and for all ##a,b\in G##, there exists an ##x,y\in G## such that ##bx=a## and ##yb=a##. 2) There exists a special element ##e\in G##...

Hello. I have been looking into group theory for its applications to subject I am studying. I am not a mathematician by profession or training, but I find it has great use to any analytical pursuit. With that said, I have outlined below type of group that I would like to know more about. For...

Hi, I'm having trouble understanding why the follow composition table for the set \left\{ a, b, c, d \right\} with operation * doesn't define a group. \begin{array}{c|cccc} * & a & b & c & d \\ \hline a & c & d & a & b \\ b & d & c & b & a \\ c & a & b & c & d \\ d & b & a &...

I am reading QFT from Srednicki's book. In the 2nd chapter of this book and in the spin half part of this book, group theory and group representation theory is used. Can you suggest me a book from where I can learn this?

So I'm intending to teach myself some Particle Physics and Standard Model type stuff, I was wondering if someone who's already covered this could give me some advice. I did some Group Theory a few years back and looking over content pages of lecture notes I occasionally spot references to...

Hi. Today I sat my final first year Modern Physics exam. It went very well, however I got stuck in one question. It asked (i) to prove the following relation for the matter wave \omega^{2}=k^{2}c^{2}+m^{2}c^{4}/\hbar^{2} and (ii) to obtain the group velocity and phase velocity of a matter wave...

Interesting question I've happened upon: If there is an epimorphism (i.e. onto homomorphism) $\phi:G\times G \to H\times H$, is there necessarily an epimorphism $\psi:G\to H$? If not, under what conditions can we ascertain such an epimorphism given the existence of $\phi$? I would think that...

Please show me some group theory books that considering the combination of quantum mechanics and relativity theory that leads to the needing of notion of fields.I have heard that the irreducible representation of Poicare group leading to the infinite dimensions representation(meaning field...

I already know about generators, rotations, angular momentum, etc. When I see questions about SO(3), SU(3), and lie groups as it pertains to quantum mechanics, I always hold off on getting into the discussion because I think maybe I don't know what that means. It all seems really familiar...

Homework Statement The problem statement is to prove the following identity (the following is the solution provided on the worksheet): Homework Equations The definitions of L_{\mu \nu} and P_{\rho} are apparent from the first line of the solution. The Attempt at a Solution I get to the...

Hi, I'm interested in doing some self-study this summer and learning some group theory. This has come up a lot as I'm getting into graduate level physics courses, so I'd like a good solid introduction to it. Any recommendations on a book? Preferably one that's at the level of an introductory...

"Real Time Entanglement" from the Zeilinger Group And the "Gee Whiz!" article that referenced it: http://www.preposterousuniverse.com/blog/2013/05/29/visualizing-entanglement-in-real-time/ CW

I recall visiting a website that was a wiki for group theory and had many articles on specific groups, but I don't find it today doing a simple-simon search on keywords like "group theory". Anyone know the website that I'm talking about?

I Still don't understand why the group velocity has to be less than c but phase velocity not. Can you explain me this? Thank you :cry:

Homework Statement I want to find the orders of the elements in Z_8/(Z_4 \times Z_4), (Z_4 \times Z_2)/(Z_2 \times Z_2), and D_8/(Z_2 \times Z_2). Homework Equations The Attempt at a Solution The elements of Z_2 \times Z_2 are (0,0), (1,0), (0,1), (1,1), and the elements of Z_8 are of course...

I'm a condensed matter student with limited knowledge of chemistry or bond notation. In the attached paper, I'm trying to understand what is meant by \equiv\text{Ti}-\text{OH} and \equiv\text{Ti}-\text{F} All I've been able to gather is that these represent "surface groups", although I'm...

Does anyone know what this guy is on about? I understand some of the basics of group theory and I know there's a connection between Galois theory and the solving of a Rubik's cube, but I'm not sure what law he is even trying to disprove here. I'm assuming something with regards to symmetry or...

Was wondering if the only required definition for finite groups is closure (maybe associativity as well). It seems that is all that is necessary. The inverse and identity necessarily seem to follow based on the fact that if I multiply any element by itself enough times, I have to repeat back to...

Here is the question: Here is a link to the question: Quick Proof about a Square Matrix? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Homework Statement I am wondering how for determine the central atom's orbitals from the point group character tables described by group theory. For example CO3^-2 (D3h) Carbon's (central atom) p-orbitals are described by a1''+e'. The s-orbital is a1' Homework Equations The...

Homework Statement Compute the volume of the group SU(2) Homework Equations Possibly related: in a previous part of the problem I showed that any element g = cos(\theta) + i \hat{n} \cdot \vec{\sigma}sin(\theta) The Attempt at a Solution How do I compute the infinitesimal...

When proving that x^m x^n = x^{m+n} and that (x^m)^n = x^{mn} for all elements x in a group, it's easy enough to show that they hold for all m \in \mathbb{Z} and for all n \in \mathbb{N} using induction on n. The case n = 0 is also very easy. But how does one prove this for n \in...

I just want to check if there is anything wrong with my understanding... Let's say we have a group of order 42 that contains Z_6. Since the group of units of Z_6 has order (3-1)(2-1), it means that we have 2 elements of order 6 in G, right? In other words, for any cyclic subgroup of order n...

I'll be taking an Elementary Abstract algebra class in Summer B (six week session) at my University. It will likely be pretty intense. (I actually requested/petitioned the class and got it). I want to do what I can so that me and my classmates will survive do well in the class, so I'm...

I am new to group theory, and read about a "universal property of abelianization" as follows: let G be a group and let's denote the abelianization of G as Gab (note, recall the abelianization of G is the quotient G/[G,G] where [G,G] denotes the commutator subgroup). Now, suppose we have a...

Greetings! I have this friend who had synthesis of nanomaterials as his MS thesis. After talking with him, I realized that his passion is on helping the farmers in the agriculture industry. I want to help him find a research group or professor (for his Phd) in which he can apply what he...

Hello. I'm just beginning my course in algebra. I've been reading Milne, Group Theory ( http://www.jmilne.org/math/CourseNotes/GT310.pdf page 29). I've found there a very nice proof of the fact that given two elements in a finite group, we cannot really say very much about their product's...

From what I now understand of renormalization it is really a reparametrization of the theory in terms of measurable quantities instead of the 'inobservable bare quantities' that follow the Lagrangian; at least that is one interpretation of what is going on. The originally divergent physical...

Homework Statement Consider the group D_{4} = <x,y:x^2=1,y^4=1,yx=xy^3> and the homomorphism \Phi : D_{4} \rightarrow Aut(D_{4}) defined by \Phi (g) = \phi _{g}, such that \phi _{g} = g^{-1}xg. (a) Determine K = ker(\Phi) (b) Write down the cosets of K. (c) Let Inn(D_{4}) = \Phi (D_{4})...