Group Definition and 1000 Threads

I was rethinking about some things I learned but I came to things that seemed to be not firm enough in my mind. 1) When we want to find the unitary matrix that block-diagonalizes a certain matrix through a similarity transformation, we should find the eigenvectors of that matrix and stick them...

Homework Statement Determine all the subgroups of (A,x_85) justify. where A = {1, 2, 4, 8, 16, 32, 43, 64}.The Attempt at a Solution To determine all of the subgroups of A, we find the distinct subgroups of A. <1> = {1} <2> = {1,2,4..} and so on? <4> = ... ... is this true? are there any other...

Hello, I am learning Feynman diagrams and I still do not understad quite well the symmetry factor idea. The equation is: $$ \frac{1}{O(G)} = \frac{M}{n!(4!)^n} $$ I was trying the next example: If I am not wrong it is O(G) = 10 taking care of the asymmetry of each pair of internal lines...

In my class, we have denoted the elements of the dihedral group ##D_4## as ##\{R_0, R_{90}, R_{180}, R_{270}, F_{\nearrow}, F_{\nwarrow}, F_{\leftrightarrow}, F_{\updownarrow} \}##. Not surprising, I was rather bewildered when I searched the internet for information on this group and encountered...

The problem: Suppose G is Abelian with two representations as the internal direct product of subgroups: G=HxK1, G=HxK2. Assume K1 is a subset of K2 and show K1=K2. My attempted solution: I took the element (e_H, k_2), where e_H is the identity element of H and k_2 is an arbitrary element in K2...

Hello everyone, I am working with an arbitrary finite group ##G##, and I am trying to prove a certain property about the order of an arbitrary element ##g \in G##. Supposedly, if we are dealing with a such a group, then ##o(g)##, which is the cardinality of the set ##| \langle g \rangle |##, is...

In Ng (the group index for a range of wavelengths), there is an index of refraction n used, but if the medium is dispersive, meaning n is a function of wavelength λ, which n is used? Is it some kind of an average? Or does n not change much over this range of wavelengths? if it doesn't change...

Hello everyone, I have to demonstrate that the two groups ##(\mathbb{Q'}, \cdot )## and ##(\mathbb{R'}, \cdot )##, where ##\mathbb{Q'} = \mathbb{Q} \setminus \{0\}## and ##\mathbb{R'} = \mathbb{R} \setminus \{0\}##. While trying to solve this problem, a thought suddenly occurred to me. Here is...

In the context of the homomorphism between SL(2,C) and SO(3,1), I have that \textbf{x}=\overline{\sigma}_{\mu}x^{\mu} x^{\mu}=\frac{1}{2}tr(\sigma^{\mu}\textbf{x}) give the explicit form of the isomorphism, where \textbf{x} is a 2x2 matrix of SL(2,C) and x^{\mu} a 4-vector of SO(3,1)...

Homework Statement Exercises: https://mega.co.nz/#!YdIgjA7T!WmgIpFjCoO-elDyPtUkDNarm21sZ_xet6OTJndPGiRY Text: https://mega.co.nz/#!pVRxVKIC!RfFZiW2atRNj9ycGa4Xx_7Nu5FO4a1e6wmyQVLCcGlQ 2. Homework Equations The Attempt at a Solution This is what I made, obviously all help would be appreciated...

I am trying to do an exercise where I am showing that the set of all elements of $\Bbb{Z}_n$ that are coprime with n form a group under modular addition. So far I have shown associativity, identity, and closure, but I'm having trouble showing the existence of an inverse. I know I can't use...

Homework Statement G is a finite group. K is normal to G. If G/K has an element of order n, show that G has an element of order n. Homework Equations none. The Attempt at a Solution (Kg)^n = K for some Kg in G/K. (Kg)^n = (Kg^n) = K, hence g^n = 1 where g is an element of G. Is this...

I got a table for a simple pendulum. I have 8 lengths, from 0.20m going up by 0.01 to 0.27. For each length, I have time for 10 oscillations (10T) that I've measured, and I have repeated the measurement twice for each length. Then I got the average time for 10T. I divided this average to give me...

Hey! :o "Show that a group of even order contains an odd number of elements of order $2$." We know that the order of an element of a finite group divides the order of the group. Since, the order of the group is even, there are elements of order $2$. But how can I show that the number of...

Hey! :o Show that a cyclic group with only one generator can have at most two elements. I thought the following: When $a \neq e$ is in the group, then $a^{-1}$ is also in the group. So, when $a$ is a generator, then $a^{-1}$ is also a generator. Is this correct?? (Wondering) But I how can I...

Hello, I have a group (G,\cdot) that has a subgroup H \leq G, and I consider the action of H on G defined as follows: \varphi(h,g)=h\cdot g In other words, the action is simply given by the group operation. Now I am interested in finding a (non-trivial) invariant function w.r.t. the action of...

Does anyone knows the application and Significance of group velocity?

In Lang's book,page 39-40, he factorizes ##F_{ab}(M)## with respect to the subgroup generated by all elements of type ##[x+y]-[x]-[y]##. I don't quite understand why he does this. I know that he is trying to create inverse elements, but I don't see why that factorization necessarily satisfies...

I am reading "Introduction to Ring Theory" by P. M. Cohn (Springer Undergraduate Mathematics Series) In Chapter 2: Linear Algebras and Artinian Rings we read the following on page 55:https://www.physicsforums.com/attachments/3142I am trying to get an idea of what Cohn says and means by a group...

In the following stackexchange thread, the answerer says that there is a Riemannian metric on \mathbb{R} such that the isometry group is trivial. http://math.stackexchange.com/questions/492892/isometry-group-of-a-manifold This does not seem correct to me, and I cannot follow what he is...

Homework Statement Let S be the subset of group G that contains identity element 1 such that left co sets aS with a in G, partition G .Probe that S is a subgroup of G. Homework Equations {hS : h belongs to G } is a partition of G. The Attempt at a Solution For h in S if I show that hS is S...

From what I understand, the little group for a particle moving at the speed of light, has 3 generators. 2 generators generate gauge transformation, and 1 generator rotates the particle about its axis of motion. I have 3 questions: 1) Do all particles moving at the speed of light (not...

List every generator of each subgroup of order 8 in $$\mathbb{Z}_{32}$$. I was told to use the following theorem: Let $$G$$ be a cyclic group of order $$n$$ and suppose that $$a\in G$$ is a generator of the group. If $$b=a^k$$, then the order of $$b$$ is $$n/d$$, where $$d=\text{gcd}(k,n)$$...

Hey everyone, I've got a question in elementary group theory. Suppose we have a group G, and we want to completely partition it into multiple subgroups, such that the only element each subgroup shares with any other is the identity element. Is this ever possible? I think that such a...

MIT OCW recently posted their introductory quantum class 8.04 at http://ocw.mit.edu/courses/physics/8-04-quantum-physics-i-spring-2013/ I was wondering if anyone would be interested in going through the course. I'm primarily studying it to get ready for the MIT MOOC 8.05x Quantum Physics...

Hi, I'm hoping to clear up a few uncertainties in my mind about proving that the identity element and inverses of elements in a group are unique. Suppose we have a group \left(G, \ast\right). From the group axioms, we know that at least one element b exists in G, such that a \ast b = b \ast...

We fix some N=1,2,3,\ldots, and define the factor group \mathbb{Z}_N as \mathbb{Z}/N\mathbb{Z}, and denote the elements x+N\mathbb{Z} as [x], where x\in\mathbb{Z}. My question is that how do you solve [x_1] and [x_2] out of \left(\begin{array}{c} \lbrack y_1\rbrack \\ \lbrack y_2\rbrack \\...

[The homework format does not appear on mobile] Problem: Show that a finite group of even order has elements of order 2 Attempt: The book gives a suggested approach that lead me to write the most round about, ugly proof I've ever written. Can't I just say: 1.) If G has even order, G/{1} has...

I know this post is in the topology thread of this forum, for group theory, this seemed like the reasonable choice to post it in. I realize group theory is of great importance in physics and I'm trying to eventually understand Emmy Noether's theorem. I'm learning group theory on my own, and...

Good afternoon : I now what I've written here : https://www.physicsforums.com/showthread.php?t=763322 in the first message. I've made the Clebsh Cordon theorem with the components. Which can be represented by the Young tableau. There also the SU(3) and the su(3) representation of dimension...

Hi, Is still there a bird group classified as Palmipedes? I can not find enough information in the internet for it and this is also the first entry in this science forum. Best Regards.

Hello, why one can use a bi-doublet scalar field (2,2) under SU(2)L x SU(2)R ? In terms of group theory, we should have only triplets (3,1) or (1,3) since 2 x 2=3+1 ? But in left right symmetric models, indeed yukawa coupling are formed with bi-doublet scalars. Best regards

Homework Statement if K is normal in G and has index m, show g^m is an element of K for all g in G Work (I haven't done much with proofs so bear with me): |G/K| = |G| / |K| = m |G| = x |K| = y g^m must be an element of G since m|x if g^m is an element of G and K is normal to G then (g^m)K =...

Homework Statement G is a cyclic group generated by a, G = <a>. |a| = 24, let K=<a^12>. Q: In G/K, find the order of the element Ka^5 Work: K=<a^12> = <1,a^12> --> |K| = 2 |G/K| = |G| / |K| = 24 / 2 = 12, so |Ka^5| = 1,2,3,4,6 or 12. now I'm lost ;-/

To represent operations of a point group by matrix we need to choose basis for this representation. What is the criteria for doing that? How to realize that how many bases are necessary for a matrix representation and how to select them? Or could you please give me an elementary reference to...

Homework Statement Groups G and H are both groups in Z_7 (integer modulo), the mapping Is given by ø(g) = 2g is ø: G-->H a homomorphism? The Attempt at a Solution My textbook says yes, I can't understand why. ø(g1g2) = 2(g1g2) does not equal 2g1*2g2 = ø(g1)ø(g2) something...

Homework Statement Let R be the ring of all 2*2 matrices, over Zp, p a prime. Let G be the set of elements x in the ring R such that det x ≠ 0. Prove that G is a group. Homework Equations Matrix is invertible in ring R. The Attempt at a Solution Group properties and ring properties...

Can anyone explain the idea behind Hadlock's proof that there is an Sn for every poly of degree n? Theorem 37 page 217 I can follow how to build up G from F using symmetric functions and the primitive element theorem. A lso I get the idea of constructing a poly of deg n! from one of deg n...

Homework Statement Prove that if G is a cyclic group with more than two elements, then there always exists an isomorphism: ψ: G--> G that is not the identity mapping. Homework Equations The Attempt at a Solution So if G is a cyclic group of prime order with n>2, then by Euler's...

Homework Statement Prove that the center of the factor group G/Z(G) is the trivial subgroup ({e}). Homework Equations Z(G) = {elements a in G|ax=xa for all elements x in G} The Attempt at a Solution I need to prove G is abelian, because G/Z(G) is cyclic, right? Then I can say that...

Hi, While reading "Superspace: One Thousand and One Lessons in Supersymmetry" by Gates et al. I came across the following paragraph: Maybe I haven't understood what exactly they're trying to say here, but 1. Why is the Lorentz Group SL(2, R) instead of SL(2, C)? 2. Why is the two-component...

Homework Statement Two subgroups of G, H and K are conjugate if an element a in G exists such that aHa^-1= {aha^-1|elements h in H}= K Prove that if G is finite, then the number of subgroups conjugate to H equals |G|/|A|. Homework Equations A={elements a in G|aHa^-1=H} The Attempt...

A co-worker recently shared an article with me that demonstrated a sound pulse traveling faster than c. After doing much research, I am still confused as to how this does not send information faster than light. If the leading edge of the pulse arrives before the rest of it, how would that...

Good morning I'me french so excuse my bad language : so in this course : http://lapth.cnrs.fr/pg-nomin/salati/TQC_UJF_13.pdf take a look at page 16. They say that all rotation auround a unitary vector \vec{u} of angle \theta in the conventionnal space could be right like this with the matrix...

[SIZE="4"]Definition/Summary The symmetric group S(n) or Sym(n) is the group of all possible permutations of n symbols. It has order n!. It has an index-2 subgroup, the alternating group A(n) or Alt(n), the group of all possible even permutations of n symbols. That group has order n!/2...

[SIZE="4"]Definition/Summary A quotient group or factor group is a group G/H derived from some group H and normal subgroup H. Its elements are the cosets of H in G, and its group operation is coset multiplication. Its order is the index of H in G, or order(G)/order(H)...

[SIZE="4"]Definition/Summary A Lie group ("Lee") is a continuous group whose group operation on its parameters is differentiable in them. Lie groups appear in a variety of contexts, like space-time and gauge symmetries, and in solutions of certain differential equations. The elements of...

[SIZE="4"]Definition/Summary A group representation is a realization of a group in the form of a set of matrices over some algebraic field, usually the complex numbers. A representation is irreducible if the only sort of matrix that commutes with all its matrices is a sort that is...

[SIZE="4"]Definition/Summary The character of a group representation is the trace of its representation matrices. Group characters are useful for finding the irrep content of a representation without working out the representation matrices in complete detail. Every element in a...

[SIZE="4"]Definition/Summary A group is a set S with a binary operation S*S -> S that is associative, that has an identity element, and that has an inverse for every element, thus making it a monoid with inverses, or a semigroup with an identity and inverses. The number of elements of a...