Group Definition and 1000 Threads

Group action on ##2##x##2## complex matrices of group ##C_{3v}## for all matrices from ##C^{22}##, for all ##g## from ##C_{3v}## is given by: ##D(g)A=E(g)AE(g^{-1}), A=\begin{bmatrix}...

What is difference between subgroup and closed subgroup of the group? It is confusing to me because every group is closed. In a book Lie groups, Lie algebras and representations by Brian C. Hall is written "The condition that ##G## is closed subgroup, as opposed to merely a subgroup, should be...

Being myself a chemist, rather than a physicist or mathematician (and after consulting numerous sources which appear to me to skip over the detail): 1) It’s not clear to me how one can go generally from a choice of basis vectors in real space to a representation matrix for a spatial symmetry...

In Quantum Mechanics, by Wigner's theorem, a symmetry can be represented either by a unitary linear or antiunitary antilinear operator on the Hilbert space of states ##\cal H##. If ##G## is then a Lie group of symmetries, for each ##T\in G## we have some ##U(T)## acting on the Hilbert space and...

Proof: Let ##B = \lbrace a \rbrace \subseteq A## and ##\rho \in S_4##. We have two cases, ##\rho(a) = a## in which case ##\rho(B) = B##, or ##\rho(a) \neq a## in which case ##\rho(B) \cap B = \emptyset##. Its clear that ##\rho(A) = A##. So these sets are indeed blocks. Now let ##C## be any...

Let ##S = \lbrace a, b \rbrace## and define ##F_S## to be the free group, i.e. the set of reduced words of ##\lbrace a, b \rbrace## with the operation concatenation. We then have the universal mapping property: Let ##\phi : S \rightarrow F_S## defined as ##s \mapsto s## and suppose ##\theta : S...

I know R4N+ is a quaternary ammonium, R2C=N+R2 is an iminium, and R-C≡N+-R is a nitrilium, but what is an R2C=N+=CR2 cation called?

Progress:𝜙:𝑂(3)→ℤ2𝜓:𝑂(3)→𝑆𝑂(3)𝜃:𝑂(3)/𝑆𝑂(3)→ℤ2 𝜙(𝑂)=det(𝑂) with 𝑂∈𝑂(3), that way 𝜙(𝑂)↦{−1,1}≅ℤ2, where 1 is the identity element.Ker(𝜙) = {𝑂∈𝑆𝑂(3)|𝜙(𝑂)=1}=𝑆𝑂(3), since det(𝑂)=1 for 𝑂∈𝑆𝑂(3).By the multiplicative property of the determinant function, 𝜙 = homomorphism. ***What is the form of the...

can anyone explain me how O negative and COO negative acts as electron releasing group,I understood how alkyl groups acts as electron releasing group but I can't understand this

Hi all, Group theory show us that irreducible representation of SO(3) have dimension 2j+1. So we expect to see state with 2j+1 degeneracy. But does group theory help to understand the principle quantum number n ? And in the case of problems with SO(3) symmetry does it explain its strange link...

1) How do we determine a Lie group's global properties when the manifold that it represents is not immediately obvious? Allow me to give the definitions I am working with. A Lie group G is a differentiable manifold G which is also a group, such that the group...

1) How do we determine a Lie group's global properties when the manifold that it represents is not immediately obvious? Allow me to give the definitions I am working with. A Lie group G is connected iff \forall g_1, g_2 \in G there exists a continuous curve connecting the two, i.e. there...

Okay, having devised the antimatter bomb, I'm moving along to the concept of using Apollo group asteroids as freight trains for the inner solar system, but my knowledge of orbital mechanics is zilch. For example, (343158) 2009 HC82 appears to have a max speed of about 56 km/s at perihelion...

Hey guys, Sorry that it's been a decent amount of time since my last posting on here. Just want to say upfront that I am extremely appreciative of all the support that you all have given me over my last three or four posts. Words cannot express it and I am more than grateful for it all. But, in...

Hey guys, Sorry that it's been a decent amount of time since my last posting on here. Just want to say upfront that I am extremely appreciative of all the support that you all have given me over my last three or four posts. Words cannot express it and I am more than grateful for it all. But, in...

I am trying to learn group theory on my own from Schaum's Outline of Group Theory. I chose this book because there are a lot of exercises with solutions, but I have several problems with it. 1) In many cases the author just makes some handwavey statement and I have to spend hours or days trying...

Dear Every one, I am having some difficulties with computing an element in the Integral dihedral group with order 6. Some background information for what is a group ring: A group ring defined as the following from Dummit and Foote: Fix a commutative ring $R$ with identity $1\ne0$ and let...

It is often said that the renormalization group (RG) is not a true group but only a semi-group, because the RG transformation is not invertible. But for renormalizable theories, the renormalized Hamiltonian has the same form as the original Hamiltonian, only with some different values of the...

Weinberg QFT book aside, what are good sources for the representation of the Poincare group used in physics?

There are two related Lemmas in Schaum's Outline of Group Theory, Chapter 4 that seem excessively convoluted. Either I am missing something or they can be made much simpler and clearer. Lemma 4.2: If H is a subgroup of G and {\rm{X}} \subseteq {\rm{H}} then {\rm{H}} \supseteq \left\{...

Hi, let ##G## be a Lie group, ##\varrho## its Lie algebra, and consider the adjoint operatores, ##Ad : G \times \varrho \to \varrho##, ##ad: \varrho \times \varrho \to \varrho##. In a paper (https://aip.scitation.org/doi/full/10.1063/1.4893357) the following formula is used. Let ##g(t)## be a...

Hello, I am an undergraduate who has taken basic linear algebra and ODE. As for physics, I have taken an online edX quantum mechanics course. I am looking at studying some of the necessary math and physics needed for QFT and particle physics. It looks like I need tensors and group theory...

An electromagnetic wave has a phase speed and a group speed. Or velocities, for that matter. In a medium, the phase speed of a wave is generally determined by the medium's permeability μ and permittivity ε. What are the general parameters that determine the group speed of a wave in a medium?

Schaum's Outline of Group Theory, Section 3.6e defines {{\rm{L}}_n}\left( {V,F} \right) as the set of all one to one linear transformations of V, the vector space of dimension n over field F. It then says "{{\rm{L}}_n}\left( {V,F} \right) \subseteq {S_V}, clearly". ({S_V} here means the set...

Is it correct saying that the Exponential limit is an exact solution for passing from a Lie Algebra to a Lie group because a differential manifold with Lie group structure is such that for any point of the transformation the tangent space is by definition the Lie algebra: is that the underlying...

Hi all. Awesome site! Just wondering if anyone can answer my question: If the Sextans galaxies are inside the group's zero velocity surface, why is there uncertainty over whether they're part of the group?

In general, the textbooks says that, if the set ##G## is a group, so to every element ##g \in G## there is other element ##g^{-1} \in G## such that ##g g^{-1} = g^{-1}g = e##, where ##e## is the identity of the group. But I am reading a book where this propriete is write only as ##g^{-1} g =...

Hello, I am newish in group theory so sorry if anything in the following is not entirely correct. In general, one can anticipate if a matrix element <i|O|j> is zero or not by seeing if O|j> shares any irreducible representation with |i>. I know how to reduce to IRs the former product but I...

I'll have to think a bit about what you've written, but just to note this is not true due to Haag's theorem.

I'm trying to wrap my head around the dispersion relation ##\omega(k)##. I understand how you can construct a wavepacket by combining multiple traveling waves of different wavelengths. I can then calculate the phase and group velocities of this wavepacket: \begin{align*} v_p &=...

Hello, I am currently struggling to understand how one can write a Hamiltonian using group theory and change its form according to the symmetry of the system that is considered. The main issue is of course that I have no real experience in using group theory. So to make my question a bit less...

Given a representation of a Lie Group, is there a equivalence between possible electric charges and projections of the roots? For instance, in the standard model Q is a sum of hypercharge Y plus SU(2) charge T, but both Y and T are projectors in root space, and so a linear combination is. But I...

Over in the thread The eight-queens chess puzzle and variations of it | Physics Forums I discovered that with a toroidal board, one with periodic boundary conditions, the amount of symmetries becomes surprisingly large (A group-based search for solutions of the n-queens problem - ScienceDirect)...

Dear Everyone, I want to show that a subset of a group is still a group by using the subgroup criterion which states that a subset $H$ of a group $G$ is a subgroup if and only if $H \ne \emptyset$ and for all $x,y \in H, xy^{-1}\in H$. I am having trouble how to show that criterion in the...

It seems that there is a difference between Galilean transformations and (the transformations of the) Galilean group, for one thing: rotations. The former is usually defined as the transformations ##\{\vec{x'} = \vec x - \vec v t, \ t' = t \}##, where ##\vec v## is the primed frame velocity...

please suggest me a good book on the high energy physics where group theory is discussed for the beginner.

Dear Everyone, I am having some troubles with the problem. The problem states: Let $(G,\star)$ be a group with ${a}_{1},{a}_{2},\dots, {a}_{n}$ in $G$. Prove using induction that the value of ${a}_{1}\star {a}_{2} \star \dots \star {a}_{n}$ is independent of how the expression is bracketed...

Dear Everyone, $\newcommand{\Z}{\mathbb{Z}}$Suppose the set is defined as: $\begin{equation*} {(\Z/n\Z)}^{\times}=\left\{\bar{a}\in \Z/n\Z|\ \text{there exists a}\ \bar{c}\in \Z/n\Z\ \text{with}\ \bar{a}\cdot\bar{c}=1\right\} \end{equation*}$ for $n>1$ I am having some trouble Proving that...

Dear Everyone, Here is the problem that I am attempting to prove: "Prove that a group $(G,\star)$ is abelian if and only if ${(a\star b)}^{-1}={a}^{-1}\star {b}^{-1}$ for all a and b in $G$." My attempt: Let $(G,\star)$ be a group $G$ under the binary operation $\star$. Then suppose $G$ is...

I just confused about it.Why can't we discribe a particle just one wave function instead of wave packet(group of waves with different phase velocities)?

What do we mean when we are talking about something that transforms under a representation of a group? Take for example a spinor. What is meant by: this two component spinor transforms under the left handed representation of the Lorentz group? When we talk about something that transforms...

Hey all I previously asked about some math structure fulfilling some requirements and didn't get much out of it ( Graph or lattice topology discretization ). It was a vague question, granted. Anyway, I seem to have stumbled upon something interesting called geometric group theory. It looks...

Hi all, I have stumbled upon Artin's book "Algebra" and was wondering if I could use it to do some self-study on Group Theory. Some background: I am a physics undergraduate who has some competence in elementary logic, proofs and linear algebra. It seemed to me that ideas related to Group...

Landau and Lifshitz, second volume - Classical Theory of Fields, page 7 $$e_mu,nu,alpha,beta e^alpha, beta, gamma, sigma = -2 ( delta^gamma_mu * delta^sigma_nu delta delta^sigma_mu * delta^gamma_nu ) $$ If for example I calculate the following: $$ e^0,1_alpha,beta e^alpha,beta_0,1 = e_0123...

Why for the free particle, the group velocity and phase velocity are not the same while we have only one wave? What is the envelope here?

$$\text{ Let } n∈ \mathbb{N} \text{ and } S_{n} \text{ symmetrical group on } \underline n\underline . \text{ Let } π ∈ S_{n} \text{ and z } \text{ the number of disjunctive Cycles of π. Here will be counted 1 - Cycle }. (a) \text{ Prove that } sgn (π) = (-1)^{n-z}. (b) \text{ Prove that...

Homework Statement [G,G] is the commutator group. Let ##H\triangleleft G## such that ##H\cap [G,G]## = {e}. Show that ##H \subseteq Z(G)##. Homework EquationsThe Attempt at a Solution In the previous problem I showed that ##G## is abelian iif ##[G,G] = {e}##. I also showed that...

https://imgur.com/a/FuCPJLe I am trying to attempt this problem, but I am wondering why exactly these are the two cases the problem is split into. I can understand the first case, since that let's us count elements and get a contradiction, but why is the second case there? In other words, why...

Homework Statement Show that there is no simple group of order ##10^4##. Homework EquationsThe Attempt at a Solution By way of contradiction, suppose ##G## is simple and ##|G| = 10000 = 5^42^4##. Sylow theory gives ##|\operatorname{Syl}_2(G)| = 1## or ##16##. If ##|\operatorname{Syl}_2(G)| =...

Homework Statement Let ##G## be a group. Let ##H \triangleleft G## and ##K \leq G## such that ##H\subseteq K##. a) Show that ##K\triangleleft G## iff ##K/H \triangleleft G/H## b) Suppose that ##K/H \triangleleft G/H##. Show that ##(G/H)/(K/H) \simeq G/K## Homework Equations The three...