Groups Definition and 906 Threads

Every once in a while I use my ancient trick of searching something in google with keywords, and found the above article. I don't think there's a free copy of it, because it's from 1989. I guess I need to read the pink book on foundations of Q-Groups by Majid. You know who also has written a...

Hello I plotted the Parton distribution functions in Mathematica. Now I want to compare the graphs drawn with the graphs of other groups(xu and xd). How should I do this?

1) At first my answer was ##n! \begin{pmatrix} n+r-1 \\ r - 1 \end{pmatrix} ## But I think that's not correct because let say first group consists of person A and B, by multiplying with n!, I also consider first group to be B and A which is just the same as A and B so there is double counting...

0 Hint: Show that the isomorphism preserves the order of the element My solution: C4 = {e,r,r^2,r^3} where e-identity element and r is rotation by 90° Z4 = {0,1,2,3} LEMMA: ! Isomorphism preserves the order of the element ! (PROOF OF IT)Now we calcuate the order of the elements of both...

For each positive integer ##m##, let ##C_m## denote a cyclic group of order ##m##. Show that for all positive integers ##m## and ##n##, there is an isomorphism ##C_m \times C_n \simeq C_d \times C_l## where ##d = \operatorname{gcd}(m,n)## and ##l = \operatorname{lcm}[m,n]##.

I want to identify: ##S^n## with the quotient of ##O(n + 1,R)## by ##O(n,R)##. ##S^{2n+1}## with the quotient of ##U(n + 1)## by ##U(n)##. The orbit-stabilizer theorem would give us the result, but my problem is to apply it. My problem is how to find the stabilizer. In 1 how to define the...

Hello :), I am wondering of the right and direct method to calculate the following tangent spaces at ##1##: ##T_ISL_n(R)##, ##T_IU(n)## and ##T_ISU(n)##. Definitions I know: Given a smooth curve ##γ : (− ,) → R^n## with ##γ(0) = x##, a tangent vector ##˙γ(0)## is a vector with components...

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Pin Groups are the double cover of the Orthogonal Group and Spin Groups are the double cover of the Special Orthogonal Group. Both sets of the double cover are considered to be groups, but it seems that only one of the sets of the double cover actually contains the identity element, which means...

So, a friend of mine has attempted a solution. Unfortunately, he's having numbers spawn out of nowhere and a lot of stuff is going on there which I can't make sense of. I'm going to write down the entire attempt. $$ 0 \in X \; \text{otherwise no subgroup since neutral element isn't included}...

hi, i have studied Standard Model for particle physics - at present it is described by three groups - i have studied - these groups but could not establish what particular feature suggest of these group to be used to describe SM. Thanks

Hi, I'd like to clarify the following terminology (Fradkin, Quantum Field Theory an integrated approach) "carry the quantum numbers of the representation of the gauge group": Does the author basically mean that the wilson loop is a charged operator, in a sense that it transforms non-trivially...

Hi, I was attempting the following question and was getting slightly stuck. Question: We have a bag of 18 marbles: 6 red, 6 blue, and 6 yellow. Now I randomly select 5 marbles from the bag without replacement. What is the probability that I have picked out EXACTLY 2 colors? Attempt: I tried...

Hi all! I've been trying to find a reasonable way to solve following problem: N people meet each others in groups of K people such that no one meets more than once. If I'm not mistaken, this requires two things: N must be divisible by K N - 1 must be divisible by K - 1. Let's assume these...

Hey! :giggle: Let $p<q$ different prime numbers. a) If $q\neq 1\pmod p$ show that each group of order $pq$ is abelian. b) If $q\neq \pm 1\pmod p$ show that each group of order $pq^2$ is abelian.I have done the following : a) Let $H$ be a $p$-Sylow subgroup of $G$ and let $K$ be a $q$-Sylow...

Wikipedia says that a general projective representation cannot be lifted to a linear representation and the obstruction to this lifting can be understood via group cohomology. For example, I see that a spin group is a central extension of SO(3) by Z/2. More generally I can follow the reasoning...

I have often been surprised to find music videos from the late 60s and early 70s, of people and groups like David Bowie, and The Guess Who. This surprises me because the first music videos I remember came along with MTV in the 80s. And I have often heard it discussed that MTV rushed in a new era...

Summary:: 10 values are divided into 4 groups and need a combination of these with restrictions placed on group size, ordering and combinations I have a combinations question.. i have 4 group of values A , B, C and D, with A-2 values, B-3 values , C-2 values, D-3 values. 1. From each group...

I understand that on Riemannian manifolds, the transition functions that glue charts together are coordinate transformations (Jacobian matrices). However, I am not quite sure how transition functions work in the context of Lie groups and Fiber bundles. Do we consider the manifolds to be flat...

Of course it must be an infinite group, otherwise |G/N|=|G|/|N| and then {e} is the only ( and trivial) solution. I understand there is a result that for every quotient Q:=G/N there is a subgroup H that is isomorphic to Q. Is that the case?

So I that I need to prove the axioms: associativity, existence of the identity element, and existence of the right inverse. For associativity I know that the binary operations of G and H have to already be associative, and the elements of G X H are made up of these binary operations, so...

What is the difference between groups ##SU(n)## and ##SO(n,\mathbb{C})##? They look completely the same.

Let M be a manifold. The space of cotangent spaces to M is called the cotangent bundle T*M. a function on M can be lifted into the cotangent bundle. On the manifold T*M we can define a 1-form θ which describes the natural lifts and so on. A vector field on M is a section of the tangent bundle...

Hello there. Do you know any examples of manifolds not being groups?Can you talk about some of them developing them as much as you want?Thank you.

I am confused because ##H## is a subgroup of ##G## and ##H^{\perp\perp}## is a set of homomorphisms. Are we trying to show ##f(H) = H^{\perp\perp}## where ##f## is the isomorphism defined in the definitions above? Proof: We want to show ##H = (H^\perp)^\perp##. ##(\subset):## Let ##h \in H##...

https://arxiv.org/abs/2009.14613 A group-theorist's perspective on symmetry groups in physics Robert Arnott Wilson [Submitted on 29 Sep 2020 (v1), last revised 12 Nov 2020 (this version, v3)] R.A. Wilson (physics blog) worked on finite simple groups such as the famous "Monster". He writes...

According to the book I am using, one can decompose a finite abelian group uniquely as a direct sum of cyclic groups with prime power orders. Uniquely meaning that the structures in the group somehow force you to one particular decomposition for any given group. Unfortunately, the book gives no...

I live in the California Bay Area. Wondering if there are meet-up groups for meeting people in this forum.

Define $$\phi(A)$$ a transformation which, acting on a vector x, returns $$AxA^{*}$$, in such way that if A belongs to the group $$SL(2,C)$$, $$||\phi(A)x||^2 = ||x||^2$$, so it conserves the metric and so is a Lorentz transformation. $$\phi(AB)x = (AB)x(AB)^{*} = ABxB^{*}A^{*} = A(BxB^{*})A^{*}...

The options are rank(B)+null(B)=n tr(ABA^{−1})=tr(B) det(AB)=det(A)det(B) I'm thinking that since it's invertible, I would focus on the determinant =/= 0. I believe the first option is out, because null (B) would be 0 which won't be helpful. The second option makes the point that AA^{−1} is I...

In the context of group theory, there's a theorem that states that for a given positive integer \(n\) there exist finitely different types of groups of order \(n\). Notice that the theorem doesn´t say anything of how many groups there are, only states that such groups exist. In the proof of this...

This thread is motivated by samalkhaiat's comment here I know that the Lorentz Group is formed by all matrices that satisfy $$\eta = \Lambda^{T} \eta \Lambda \tag{1.1}$$ Which is equivalent to $$\eta_{\mu\nu}\Lambda^{\mu}{}_{\rho}\Lambda^{\nu}{}_{\sigma} = \eta_{\rho \sigma} \tag{1.2}$$ If...

Posting for my son (who does not have an account here): He's a sophomore math major in college and is looking for a good book on Lie algebra and Lie Groups that he can study over the summer. He wants mathematical rigor, but he is thinking of grad school in theoretical physics, so he also wants...

Real World Application here. I'm creating a virtual meeting involving six people and will have the first round include two groups of three. Then we'll switch a few times. I tried hopelessly to do it such that after three rounds, everyone would see everyone else at least once. Was so close...

I have been following the proof of the Stone's theorem on one-parameter unitary groups. The question is if the current list of self-adjoint operators used in quantum mechanics, including position and momentum operators, is exhaustive or not? Put it another way, can we say that there is no...

I know for a group to be abelian a*b=b*a I tried multiplying the matrix by itself also but I’m not sure what I’m looking for. picture is below of the matrix https://www.physicsforums.com/attachments/255812

Matrix representation of a finite group G is irreducible representation if \sum^n_{i=1}|\chi_i|^2=|G|. Representation is reducible if \sum^n_{i=1}|\chi_i|^2>|G|. What if \sum^n_{i=1}|\chi_i|^2<|G|. Are then multiplication of matrices form a group? If yes what we can say from...

I am confused. Look for instance cyclic ##C_2## group representation where D(e)= \begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix} and D(g)= \begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix} and let's take invertible matrix A= \begin{bmatrix} 1 & 2\\ 3 & 4 \end{bmatrix}. Then A^{-1}=...

I have questions concerning group theory, esprecially Rotation groups. The first is: Are rotations groups f.ex. SO(2) defined for rotations in the actual physical 2 dimensional plane or are general rotations in any 2 dimensional space included? Someone wrote that "the action of an element of...

Anyone reading Lie Groups and Lie Algebras and Some of Their Applications by Robert Gilmore , might be interested in a series of YouTube videos by "XylyXylyX" that follows the book. The first lecture is:

Dear Everyone, Does every theorem that holds for finite group holds for ring groups? Why or Why not?Thanks Cbarker1

Hi. After learning quantum computation for months, it fascinates me. Quantum computation expands my view about computational methods. I believe that many future achievements can be obtained from quantum computing, especially the combination of AI and quantum computation. Meanwhile, I think...

I realize the question is quite broad but what research groups working on statistical physics, stochastic processes, and complex systems are generally considered the best? Would like to know about Europe and America alike.

When we have a Lie group, we want to obtain number of real parameters. In case of orthogonal matrices we have equation R^{\text{T}}R=I, that could be written in form \sum_i R_{i,j}R_{i,k}=\delta_{j,k}. For this real algebra ##SO(N)## there is ##n=\frac{N(N-1)}{2}## real parameters. Why this is...

Homework Statement This is only a step in a proof I am trying to make. Let Dm be the dihedral group. r is the rotation of 2π/m around the origin and s is a reflexion about a line passing trough a vertex and the origin. Let<s> and <r> be two subgroups of Dm. Is there a theorem that states...

Hi all - This is pulled from a past paper - Homework Statement I'm only going to state the components that I find challenging of this problem - The rest will be attached in my solution set. Essentially - given an intrinsic semiconductor comprised of group II-VI elements. Upon doping with group...

Homework Statement I am translating so bear with me. We have two group homomorphisms: α : G → G' β : G' → G Let β(α(x)) = x ∀x ∈ G Show that 1)β is a surjection 2)α an injection 3) ker(β) = ker(α ο β) (Here ο is the composition of functions.) Homework Equations This is from a...

Homework Statement Decide all abelian groups of order 675. Find an element of order 45 in each one of the groups, if it exists. Homework Equations /propositions/definitions[/B] Fundamental Theorem of Finite Abelian Groups Lagrange's Theorem and its corollaries (not sure if helpful for this...

One could argue that this question should be posted on the maths forum, but I see it so frequently in physics that I find it more productive to ask it here. In a symmetry group, do we have freedom of choice of how the group is going to act in the coordinates? Or is the way the group act on the...

\nmh{837} Compute the orders of the following groups: $U(3), U(4), U(12)$ and $U(3), U(5), U(15)$. On the basis of your answers, make a conjecture about the relationship among $|U(r)|, |U(s)|$, and $|U(rs)|$. ok I still don't have a clear idea on how to do this $ax=1$ $U(3)=3$