Group Definition and 1000 Threads

Hi all. I have a question about the properties of the generators of the SO(N) group. What kind of commutation relation they satisfy? Is it true that the generators λ are such that: $$\lambda^T=-\lambda$$ ?? Thank you very much

SU(2) a double cover for Lorentz group? I'm presently reading the new book, "Symmetry and the Standard Model", by Matthew Robinson. On page 120, he writes, "the Lorentz group (SO(1,3), pg 117) is actually made up of two copies of SU(2). We want to reiterate that this is only true in 1+3...

Largest Structure in Universe Discovered http://news.yahoo.com/largest-structure-universe-discovered-093416167.html "An international team led by academics from the University of Central Lancashire (UCLan) has found the largest known structure in the universe. The team, led by Dr Roger...

You are given a group as a quotient of the free group on two letters, a and b. the kernel of the surjective homomorphism $F_2 \to G$ is generated by: $\{a^7,b^6,a^4ba^{-1}b^{-1}\}$ a) prove $G$ is solvable by identifying the derived series: $G' = [G,G] > G^{\prime \prime} = [G',G'] > \dots $...

Say we have the symmetric group S_5. The permutations of \{2,5\} are the identity e and the transposition (25). But what are all the permutations of \{3\}? Is it e and the 1-cycle (3)?

Hi, I am reading J.P. May's book on "A Concise Course in Algebraic Topology" and have approached the calculation where \pi_{1}(S^{1})\congZ He defines a loop f_{n} by e^{2\pi ins} I want to show that [f_{n}][f_{m}]=[f_{m+n}] I understand this as trying to find a homotopy between...

I'm having some difficulty understanding the representation theory of the Lorentz group. While it's a fundamentally mathematical question, mathematicians and physicists use very different language for representation theory. I think a particle physicist will be more likely than a mathematician to...

I'm looking for a text that covers group theory and its applications for QM and QFT, targeted towards an audience that knows their QM but is ignorant of everything quantum fieldy. Any recommendations?

A group of 45 Computer Science students at a particular University had to take their first Discrete Mathematics course in the Autumn, Winter or Spring quarter of their Freshman year. How many possible ways were there for this group to meet this requirement? i came across this question on...

So I'm revamping the question I had posted here, after a bit of work. I'm concerned with the homomorphism induced by the inclusion of the Figure 8 into the Torus, and why it is surjective. There seem to be a lot of semi-explanations, but I just wanted to see if the one I thought of makes...

Hello. I have been looking at some questions from old exams that I am preparing for, and I have some trouble with the kind of problems that I will now give an example of. Homework Statement Let G = (a,b,c | a^4 = 1, b^2 = a^2, bab^{-1} = a^{-1}, c^3 = 1, cac^{-1} = b, cbc^{-1} = ab)...

Given a group G acting on a set X we get an equivalence relation R on X by xRy iff x is in the orbit of y. My question is, does some form of "reciprocal" always work in the following sense: given a set X with an equivalence relation R defined on it, does it always exist some group G with some...

I need help with this theorum, please. How is this (the attachment) true? It's for my cryptology class. The rest of the day's notes are here: http://crypto.linuxism.com/thursday_december_13_2012

Can someone please explain to me, in as simple words as possible, what a quotient group is? I hate my books explanation, and I would love it if someone can tell me what it is in english?

Let G be a group, |G|=n and m an integer such that gcd(m,n)=1. (i) show that $x^m=y^m$ implies $x=y$ (ii)Hence show that for all g in G there is a unique x such that $x^m=g$ (i) there exist a, b such that am+bn=1 so that $m^{-1}=a (mod n)$. Hence $x^m=y^m ->x=y$ ok? (ii) (i) shows...

everyone knows, there exists the relation between the group velocity and energy dispersion. a question is how to expression the relation between the velocity and momentum k? it seems that the Dirac electron in graphene is massless.

[FONT=CMMI12]Let G [FONT=CMR12]be a finite group and [FONT=CMMI12]N [FONT=CMR12]a normal subgroup of [FONT=CMMI12]G[FONT=CMR12]. Assume further that [FONT=CMMI12]N [FONT=CMR12]is a p [FONT=CMR12]-group for some prime [FONT=CMMI12]p[FONT=CMR12]. 1) By considering G/N, show that there is a...

I'm pursuing a degree in nuclear physics. However, I have a huge interest in particle physics (i know they are closely related). I am wondering how much a math course in group theory will help me understand particle physics. I want to minor in math, so I'm going to take some extra math...

Homework Statement Let (G,*) be a finite group of even order. Prove that there exists some g in G such that g≠e and g*g=e. [where e is the identity for (G,*)] Homework Equations Group properties The Attempt at a Solution Let S = G - {e}. Then S is of odd order, and let T={g,g^-1...

Homework Statement From a bag containing 4 white and 6 black balls, 2 balls are drawn at random. If the balls are drawn one after the other, without replacement, find the probability that The first ball is white and the second ball is black Homework Equations The Attempt at a...

Dear Colleagues, Is there anyone here who would be interest in sharing articles, data, theories, and discussions on risks of EMF, RF, radiation emission from devices and man made structures and health consequences and epidemiology. Please contact or send a follow up post here. I am...

[FONT=CMR12]Let [FONT=CMMI12]G [FONT=CMR12]be a group with normal subgroups [FONT=CMMI12]H[FONT=CMR8][FONT=CMR8]1 [FONT=CMR12]and [FONT=CMMI12]H[FONT=CMR8][FONT=CMR8]2 [FONT=CMR12]with H[FONT=CMR8][FONT=CMR8]2 not a subset of H[FONT=CMR8][FONT=CMR8]1. [FONT=CMR12][FONT=CMR12]Let...

Let $G$ be a finite group of order $4n+2$ for some integer $n$. Let $g_1, g_2 \in G$ be such that $o(g_1)\equiv o(g_2) \equiv 1 \, (\mbox{mod} 2)$. Show that $o(g_1g_2)$ is also odd. I found a solution to this recently but I think that solution uses a very indirect approach. Not saying that that...

A portion of a homework problem was given me to solve for practice. I have solved some but not all of the homework problem and I hope you all can help. Here is the problem: 1. For each x \in R it is conventional to write cis(x) = cos(x) + i sin(x). Prove that cis(x+y) = cis(x) cis(y)...

Homework Statement A group G of order 12 contains a conjugacy class C(x) of order 4. Prove that the center of G is trivial.Homework Equations |G| = |Z(x)| * |C(x)| (Z(x) is the centralizer of an element x\inG, the center of a group will be denoted as Z(G)) The Attempt at a Solution Let G...

Homework Statement I have read the following text in a textbook(look the attaxhement) ,and i have a simple question .WHY every 2x2 hermitian matrix would have to satisfy this Equation.It is not obvious to me why.Does anyone know the answer? The textbook stops there without giving any...

Assume that G of order 48 has 3 sylow 2-subgroups. Let G act on the set of such subgroups by conjugation. How do I know that this action is onto? I know that all 3 subgroups are conjugate but I'm not sure this is enough.

Homework Statement Prove that any subgroup of a finitely generated abelian group is finitely generated. Homework Equations The Attempt at a Solution I've attempted a proof by induction on the number of generators. The case n=1 corresponds to a cyclic group, and any subgroup of a...

first , if p is prime , show that an element has order p in Sn iff it's cycle decomosition is a product of commuting p-cycles my solution is very diffrent about the one in the book and I don't know if my strategy is right my proof ______ let T is an element of Sn and the cycle...

Homework Statement I am trying to prove the following: Let G be a finite group and let \{p,q\} be the set of primes dividing the order of G. Show that PQ=QP for any P Sylow p-subgroup of G and Q Sylow q-subgroup of G. Deduce that G=PQ. Homework Equations The set PQ=\{xy: x \in P \text{ and }...

Hi guys, I have quastion about groups: G is abelian group with an identity element "e". If xx=e then x=e. Is it true or false? I was thinking and my feeling is that it's true but I just can't prove it. I started with: (*) ae=ea=a (*) aa^-1 = a^-1 a = e those from the...

So I was working through some problems in Herstein's Algebra on my own time, and I came across something I wasn't so sure about. The question was, Find a non-abelian group of order 21 (Hint: let a3=e and b7=e and find some i such that a-1ba=bi≠b which is consistent with the assumptions that...

Consider θ:Z -> Z is a mapping where θ(n) = n^3 and it's homomorphism under multiplication. In this case, it's not a homomorphism under addition. So my question is this. In general, if we show that a group is homomorphic under multiplication, does this imply that it is not under addition...

Suppose G is a group and g,h are elements of g. Does (g.h)n=gn.hn if we don't know what the groups operation is.

Homework Statement Let (G,.) be an non-abelian group. Choose distinct x and y such that xy≠yx. Show that if x2≠1 then x2\notin{e,x,y,xy,yx} The Attempt at a Solution If x2=x would imply x.x.x-1=x.x-1 and x=e which cannot be. If x2= xy or x2=yx would imply x=y which also cannot be...

the polynomial x^4+8x+12=0 has the Galois group A4. I have all its roots, but can't figure out its splitting field. The roots are \alpha_1=\sqrt{2}(\sqrt{\cos{(\pi/9)}}+i\sqrt{\cos{(2\pi/9)}}+i\sqrt{\cos{(4\pi/9)}}) \alpha_2=\sqrt{2}(\sqrt{\cos{(\pi/9)}} -...

I have difficulty understanding the exact concept of group velocity. Consider a wave packet as a linear combination of a number of eigenstates of a 1-D particle in box. The dispersion curve(\omegaversus k) is composed of discrete points located on a parabola. Well, for each point one can...

Homework Statement Let G = \langle x,y \ | \ x^2, y^3, (xy)^3 \rangle, and f: G \rightarrow A_4 the unique homomorphism such that f(x) = a, f(y) = b, where a = (12)(34) and b = (123). Prove that f is an isomorphism. You may assume that it is surjective. Homework Equations N/A The...

I was wondering if there are any theorems that specify an exact number of subgroups that a group G has, maybe given certain conditions.The closest thing I know is a theorem that says if G is finite and cyclic of order n it has exactly one subgroup of order d for each divisor d or n. I am not...

hi , let (G,*) be a semigroup with the property that for any two elements a,b belongs to G , the equations: a*x=b , y*a=b have solutions x,y in G , verify that (G,*) forms a group. --- my attempt first one , * is associative as (G,*) is a semigroup assume a=b ,then ...

Hi guys, This is a general question that I'm thinking about now. Imagine that I've been given a set which is a group and we have defined a topology on it. how can I show that the group operation is continuous? Actually to begin with, how can I know if the group operation is really continuous...

I am currently reading a paper discussing the convexity of the image of moment maps for loop groups. In particular, if G is a compact Lie group and S^1 is the circle, the paper defines the loops group to be the set of function f: S^1 \to G of "Sobolev class H^1 ." Now in the traditional...

Let G be a finite group. Under what circumstances ... Homework Statement ... is that map δ:G→G defined by δ(x)=x2 an automorphism of F. Homework Equations And automorphism δ:G→G is a bijective homomorphism. The Attempt at a Solution The only circumstance I've so far found is...

Does anyone have any intuitive idea of how to visualize a group. The closest thing I know of in terms of a group visualization tool is a Cayley graph. I was wondering if anybody knows of a better method to visualise a group? And slightly different question what is the use of Cayley graph?

Say we have a cyclic group G, and a generator a in G. This means [a] = G. We know the order of an element a, is the order of the group it generates, [a], and also this is the smallest integer s such that as=e, where e is the identity element. In this case, [a]=G, so s is just the order of G...

What about in a ring where we have two binary operations defined. I get super confused when I see someone just switch from something like n*a to a + a + ... + a (n times, where * is the binary operation on the "semigroup" part of the ring, and + is the operation on the "group" part of the ring)...

1) Since Wigner it is well known that for massless particles of spin s the physical states are labelled by helicity h = ±s; other states are absent. So e.g. for photons the physical states are labelled by |kμ, h> with kμkμ = 0 and h = ±1 and we have two d.o.f. 2) For gauge theories with...

How we speak about the coil of primary in phase of a versus secondary coil this is accepted (as there is on same core) But question about we say this transformer (ex) is vector group Dy11 that means phase difference between primary and secondary voltage is 330 angle ! It's conflicted...

Can group have a "volume"? For example, SL(n, R) is a subgroup in a linear matrix space with det A = 1. So can this equation represent a certain "region" in the n-dimensional linear space and therefore that it has a "volume"?

Homework Statement Had this question on a test today and now I'm having second thoughts. We were asked to assign the point group of symmetry for the compound XeO2F4. Homework Equations NoneThe Attempt at a Solution I had initially thought that it fell under D4h because it contains 1 C4 axis...