Group Definition and 1000 Threads

Hello, I've been reading a book on QCD on I have a question: what is the purpose of the renormalization group? Is it to remove the large logs so that we can use pertubation theory (at least for large -q^2)? And what is the physical significance of the renormalization scale \mu^2?

For any prime p how do I show that there is a nonabelian group of order p^3? Since we are dealing with a p-group (call it G), its center is nontrivial (i.e., of order p,p^2, or p^3). Obviously, the center cannot have order p^3 (otherwise it's abelian). Also, if its center has order p^2, then...

Dear Folks: Is there a general method to find all subgroups in a given abstract group?? Many Thanks! This question came into my classmates' mind when he wants to find a 2 sheet covering of the Klein Bottle. This question is equivalent to find a subgroup of index 2 in Z free product...

iam currently studying undergraduate abstract algebra and i have reached to the permutation group topic i understand every thing till now but iam having trouble understanding the proof of "IF the identity permutation I of {1,2...n} is represented by m transpositions then m is even" I...

I am working on an assignment here; A linear chain with a two-atom primitive basis, both atoms of the same mass but different nearest neighbor separation and thus different force constants. I have made a rigorous calculation in order to find the dispersion relation ω(k), with extensive...

Let G be a finite group, H and K subgroups of G such that G=HK. Show that there exists a p-Sylow subgroup P of G such that P=(P∩H)(P∩K). I found this proof and it is clear http://math.stackexchange.com/questions/42495/sylow-subgroups but I do not understand step 3 which is "It is clear in this...

Hey I've been working on this question, How that the following is a homomorphism \theta :{{D}_{2n}}\to {{D}_{2n}}\,\,\,givenby\,\,\,\theta ({{a}^{j}}{{b}^{k}})={{b}^{k}}\,\,\, \theta ({{a}^{j}}{{b}^{k}})\theta ({{a}^{m}}{{b}^{n}})={{b}^{k}}{{b}^{n}} \theta...

To show that a non-abelian group G, has elements x,y,z such that xy = yz where y≠z, Is it enough to simply state for non-abelian groups xy≠yx so if you have xy=yz then it is not possible for x=z due to xy≠yx? Or is more detail required?

Hey, I'm just trying to grasp ordering of groups and subgroups a little better, I get the basics of finding the order of elements knowing the group but I have a few small questions, If you have a group of say, order 100, what would the possible orders of an element say g^12 in the...

There are various methods to slow and even stop visible light. Most of these methods appear to be slowing the group velocity of light, not the phase velocity. Can you slow the velocity of propagation of electricity? I have seen estimates that electricity, in a wire, propagates at 2/3 the...

Probability question: "From a group of 8 women, 6 men, ..." Homework Statement Homework Equations n choose k = n!/((n-k)! * k!) multiplication rule The Attempt at a Solution Clearly the number of total committees that can be formed from 3 of 8 women and 3 of 6 men is (8...

Let K be a finite group and H be a finite simple group. (A simple group is a group with no normal subgroups other than {1} and itself, sort of like a prime number.) Then the group extension problem asks us to find all the extensions of K by H: that is, to find every finite group G such that...

I was doing one of the proofs for my abstract algebra class, and we had to prove that the cardinality of the image of G, [θ(G)] is a divisor lGl. I'm trying to intuitively understand why G and it's image don't necessarily have the same cardinality. I'm thinking it's because there isn't...

While studying groups, Is there a common pattern/arrangement of the group elements represented in a table? Is there a pattern for the associativity axiom? Thanks.

Hi Everyone, I am kind of looking some online text to understand Lie Algebra, Group Theory and so forth. I usually need application (everyday/science context how it is used) and intuition more than mere mathematical definition to understand topics. So I need some text that gives very deep...

Hi, I am having trouble with this question so it would be really nice if anyone could provide some help. Let $$\phi: G \to G'$$ be a group homomorphism, and let $$H' \le G'$$ be a subgroup of G'. a) Show that $$H=\phi^{-1}(H')$$ is a subgroup of G. b) Now suppose H’ is a normal subgroup of...

Hello, I was reading these notes on supersymmetry, and in the appendix (which as he says is just to establish his conventions), talks about a lot of group theory and stuff that I don't know. Can someone please recommend a book/lecture notes where I can learn this?

Homework Statement I need to find an isomorphic group for the following group F A B C D E F G H < these are the rotations/reflections, f is the operation followed by A G D E B C H A F B D G F A H C B E C E F G H A B C D D B A H G F E D C E C H A F G D E B F H C B E D G F A G A B C D E...

I've been thinking about complex residues and how they relate to the topology of a function's Riemann's surface. My conclusion is this: it definitely tells us something, but it relates more directly to the Riemann surface of its antiderivative. Specifically: A closed contour in the plane is...

This is not really a homework questions, rather a concept based one. I am studying from Fraleigh's ''Intro to abstract algebra'' and in chapter 15 it states, that for a group G and normal non-trivial subgroup of N of G, the factor group G/N will be smaller than G. I am not sure how he counts the...

Hi, I am told to give the subgroup H=<α,β> with α,β\inS3 α = (1 2) β = (2 3) So I know that H={αkβj|j,k\in(the integers)} However, would αβα or βαβ (in this case, they're equal) be in H? The set H={ε,(1 2), (2 3), (1 2 3), (1 3 2)} (or {ε,α,β,αβ,βα}) would not be closed because (1 2...

Homework Statement Clasify the group Z4xZ2/{0}xZ2 using the fundamental theorem of finitely generated abelian groups. Homework Equations FTOFGAG: In short it states that every finitely generated abelian group G is isomorphic to a direct product of cyclic groups of the form...

Hey, I've been trying to solve this question, Show that <a in D8 : a2=1> is a not a group. I might not be processing it properly, but my interpretation of the question is that <a in D8 : a2=1> = <(a,b): a2=e, b2=2, ab=a-1b> Which is just D4, a group, the set of dihedral where...

Hey, I have a small question about groups, If you have a comunitative 'group' H = <a in H : a2=1>, Is that enough information to show that it is a group, without knowing the binary operation? say b is also in H then a*b=b*a (a*b)*(b*a) = (a*a)*(b*b) = 1 (since its...

I have just read my first course on Quantum Field Theory (QFT) and have followed the book by Srednicki. I have peeked a bit in the books by Peskin & Schroeder and Ryder also but mostly Srednicki as this was the main course book. Now, I have to do a project in a topic not covered in the course...

I am currently in my second undergraduate quantum course and just finished studying the addition of angular momenta. I am also in my third abstract algebra course and am now covering product groups and group actions. In my QM book (griffiths) there was a reference made to group theory. it said "...

Are there two separate renormalization group equations? One for how the physical coupling constants change with time, and one for how the bare parameters/coupling constants change with cutoff? Is there a relationship between the two? It just seems that textbooks use the term renormalization...

Hey, I just have a small question regarding the conjugation of permutation groups. Two permutations are conjugates iff they have the same cycle structure. However the conjugation permutation, which i'll call s can be any cycle structure. (s-1 a s = b) where a, b and conjugate...

Homework Statement In set theory, i have a two part question, the first is showing that the system S={set of all real numbers( \Re )}, #} where a#b=a+b-ab we have to show that it's not a group. and then find what c is so that the system = { \Re \cap\overline{c}, # } is a group.Homework...

Hello! My book here states that for a medium where the index of refraction n increases with increasing frequency (or wavenumber), "the group velocity is less than the phase velocity". This is stated for a wave which is the sum of two waves with equal amplitude and differing frequency...

I was just wondering what the word is that describes the group of Metals and Non-Metals and Metalloids. like Hydrogen, Calcium, Carbon are all Elements Metals, Metalloids and Non-Metals are all _______ If there happens to be no such word please comment so.

find a group isomorphic to ℤ[i] 1. Knowing the below proof, The group of units of ℤ[i] is isomorphic to a familiar group. Which one? 2. We have already shown: "Let R be a ring with unity, and let U denote the set of units in R. Show that U is a group under the multiplication in R."...

G finite abelian group WTS: There exist sequence of subgroups {e} = Hr c ... c H1 c G such that Hi/Hi+1 is cyclic of prime order for all i. My original thought was to create Hi+1 by reducing the power of one of the generators of Hi by a prime p. Then the order of Hi/Hi+1 would be p, but...

Homework Statement Show that the general linear group GL(3,R) with matrix multiplication is not an abelian group. Homework Equations The Attempt at a Solution A group to be abelian we have to show that it satisfies the Commutativity. a*b=b*a How are we going to show...

Sorry if I formatted this thread incorrectly as its my first post ^^ Homework Statement For every integer n greater than 2, prove that the group U(n^2 - 1) is not cyclic. Homework Equations The Attempt at a Solution I've done a problem proving that U(2^n) is not cyclic when...

Homework Statement Determine the orders of all the elements for the symmetric group on 3 symbols S3. Homework Equations _______________________________________ The Attempt at a Solution 3 symbols : e,a,b I don't know how to do the S3 table using just these 3 letters I can do...

For arbitrary natural numbers a and b, I don't think the direct sum of Z_a and Z_b (considered as additive groups) is isomorphic to Z_ab. But I think if p and q are distinct primes, the direct sum of Z_p^m and Z_q^n is always isomorphic to Z_(p^m * q^n). Am I right? I've been freely using...

Hello, Let's suppose that I have a Lie group G parametrized by one real scalar t and acting on ℝ2. Is it generally correct to say that the orbits of the points of ℝ2 under the group action are one-dimensional submanifolds of ℝ2, because G is parametrized by one single scalar? If so, how can I...

I need to show a group with $g^{2} = 1$ for all g is Abelian. This is all the information given, I do know what Abelian is, and I know that this group is but I don't know how to 'show' it. Can someone help? Gracias, GreenGoblin

Hi all, I'm looking for basic strategies for identifying the subgroups of a group. I believe I have to use conjugacy classes and cycle types, but I'm not sure how to apply those concepts. Let me pose a specific problem: Let $G$ be a subgroup of the symmetric group $S_5$, with $|G| = 4$. By...

Homework Statement You walk into your class the first day of classes, and you notice that there are 30 men and 20 women in the class already. Let's suppose you decide to choose two people from the class to be your study partners. If you choose your study partners at random, and given...

Let H be a subgroup of G and define the core of H as such core H={g\inG| g\inaHa^-1 for all a\inG}= \bigcap{aHa^-1|a\inG} Prove that the core of H is normal in G and core H\subsetH. I am having a hard time proving this because isn't the definition of core H basically saying the the core...

I would like to study the path components (isotopy classes) of the diffeomorphism group of some compact Riemann surface. To make sense of path connectedness, I require a notion of continuity; hence, I require a notion of an open set of diffeomorphisms. What sort of topology should I put on the...

Would it be correct to call mixture of three chemical solutions, namely salt water, salt water with sodium hydroxide, and salt water with HCl, a group? As I understand this, (which is not entirely realistic) mixture of solutions is associative and closed, salt water would be the identity which...

How does one compute the modular group of the torus? I see how Dehn twists generate the modular group, and I see how Dehn twists are really automorphisms of isotopy classes. Based on this, it seems that the modular group should be Aut(pi1(T^2))=Aut(Z^2)=GL(2,Z). But I've read that the modular...

Hello, if I have a set of functions of the kind \{ f_t | f_t:\mathbb{R}^2 \rightarrow \mathbb{R}^2 \; ,t\in \mathbb{R} \}, where t is a real scalar parameter. The operation I consider is the composition of functions. What should I do in order to show that it forms a Lie Group?

The general linear group of a vector space GL(V) is the group who's set is the set of all linear maps from V to V that are invertible (automorphisms). My question is, why is this a group? Surely the zero operator that sends all vectors in V to the zero vector is not invertible? But isn't it...

What is "Little Group"? In my Quantum Field Theory class, I too often meet with the term "Little Group". Unfortunately, I cannot find a good description of Little Group until now. I just know it is a subgroup of Lorentz Group. Can anyone have any brief description of this concept? Or any...

Homework Statement Is the set of a single element {e} with the multiplication law ee = e a group?Homework Equations none.The Attempt at a Solution Yes, it is a group. But that is not my question. My question is how do you ask the question? If I were face to face with you and wanted to ask you...

Renormalization Group concept is rarely given in laymen book on QM and QFT and even Quantum Gravity book like Lisa Randall Warped Passages. They mostly described about infinity minus infinity and left it from there. So if you were to write about QFT for Dummies. How would you share it such...