Group Definition and 1000 Threads

Hello! Can someone recommend me some good reading about Lorentz and Poincare groups. I would like something that starts from introductory notions but treats the matter both from math (proofs and all that) and physics point of view. Thank you

I constantly read physics topics that are generally more QM, and i always find descriptions of SU groups. I have no idea what they mean? this is not a discussion topic and i don't mind if it's taken down but i really would like a simple, yet informative answer! Thanks!

In the framework of Einstein-Yang-Mills (EYM) theory, suppose the following action: \begin{equation}S=\int\left({\kappa R + \alpha tr(F_{\mu \nu}F^{\mu \nu})d^4 x}\right)\,,\end{equation} where F is the gauge curvature associated with a non-abelian Lie group G and a gauge connection A. Then...

My textbook says the following: "Let ##G## be a group and ##G## act on itself by left conjugation, so each ##g \in G## maps ##G## to ##G## by ##x \mapsto gxg^{-1}##". I am confused by the wording of this. ##g## itself is not a function, so how does it map anything at all? I am assuming this is...

By commutative, we know that ##ab = ba## for all a,b in G. Thus, why do we need to prove separately that ##a^n b^m = b^ma^n##? Isn't it the case that ##a^n## and ##b^m## are in fact elements of the group? So shouldn't the fact that they commute automatically be implied?

Hi everybody, I work currently with permutation group, and with the good advice of this forum I discover GAP software (https://www.gap-system.org/) which is an excellent tools for working with group. My question is about something that is too strange for me: I have a permutation group G...

The textbook proves that ##x^a x^b = x^{a+b}## by an induction argument on b. However, is an induction argument really necessary here? Can't we just look at the LHS and note that there are a ##a## x's multiplied by ##b## x's, so there must be ##a+b## x's?

Hi everybody, I have a question: is an abelian group can be isomorphic to a non-abelian group? Thank you everybody.

In group theory, what is the direct product of a symmetry group with itself? Say T*T or O*O?

Hi everybody, My post today is about Molecular Symmetry group (MS) for non-rigid molecules. I read from this excellent work (Longuet-Higgins), that MS is obtained by selecting only feasible operation from Complete Nuclear Permutation Inversion Group (CNPI). My question is, As I have a quite...

If general relativity in the formal sense constrains all velocities to the speed of light as a maximum, how would superluminal group velocities exceeding speeds of light (at their superpositions) be evaluated in mainstream physics? Would this be a case of General Relativity and Physics...

Homework Statement Given that G is an abelian group of order pq, I need to show that G is isomorphic to ##\mathbb{Z}_{pq}## Homework EquationsThe Attempt at a Solution I am trying to do this by showing that G is always cyclic, and hence that isomorphism holds. If there is an element of order...

Hello! Starting from a gaussian waveform propagating in a dispersive medium, is it possible to obtain an expression for the waveform at a generic time t, when the dispersion is not negligible? I know that a generic gaussian pulse (considered as an envelope of a carrier at frequency k_c) can be...

Homework Statement Consider G = {1, 8, 12, 14, 18, 21, 27, 31, 34, 38, 44, 47, 51, 53, 57, 64} with the operation being multiplication mod 65. By the classification of finite abelian groups, this is isomorphic to a direct product of cyclic groups. Which direct product? Homework EquationsThe...

Homework Statement Let ##G## be any group. Recall that the center of ##G##, or ##Z(G)## is ##\{ x \in G ~ | ~ xg = gx, ~ \forall g \in G\}##. Show that ##G / Z(G)## is isomorphic to ##Inn(G)##, the group of inner automorphisms of ##G## by ##g##. Homework EquationsThe Attempt at a Solution I am...

I am told that ##\varphi_g (x) = g x g^{-1}## is a group action of G on itself, called conjugacy. However, I am a little confused. I thought that a group action was defined as a binary operation ##\phi : G \times X \rightarrow X##, where ##G## is a group and ##X## is any set. However, this...

I am looking for an accessible textbook in group theory. The idea here is to use it to learn basic group theory in order to take up Galois Theory. My background includes Calculus I-IV, P/Differential Equations, Discrete Mathematics including some graph theory, Linear algebra, and am currently...

Homework Statement Find the group velocity for a shallow water wave: ##\nu = \sqrt{\frac{2\pi\gamma}{\rho\lambda^3}}## Homework Equations Phase velocity: ##v_p = \nu\lambda## group velocity: ##v_g = \frac{d\omega}{dk}## ##k=\frac{2\pi}{\lambda}## ##\omega = 2\pi \nu##The Attempt at a Solution...

Hello everybody, I have read some very interesting book (Molecular symmetry and Spectroscopy - Bunker and Jensen) that talk about how to find the Molecular Symmetry group (MS) of a molecule by using the concept of "feasible" operation from the Complete Nuclear Permutation Inversion (CNPI)...

Hey everyone, I'm currently studying A Course in Algebra by E.B. Vinberg, and I was wondering if anyone is studying the same book. I think it will be a less lonely journey for all of us self learners if we can form a book study group to discuss ideas and exercises.

Homework Statement Prove that if ##1 \leq d \leq n##, then ##S_n## contains elements of order d. Homework EquationsThe Attempt at a Solution Here is my idea. The order of the identity permutation is 1. Written in cycle notation, the order of (1,2) is 2, the order of (1,2,3) is 3, the order of...

Homework Statement Show that a group with no proper nontrivial subgroups is cyclic. Homework EquationsThe Attempt at a Solution If a group G has no proper nontrivial subgroups, then its only subgroups are ##\{e \}## and ##G##. Assume that G has at least two elements, and let ##a## be any...

Homework Statement Prove that a group has exactly one idempotent element. Homework EquationsThe Attempt at a Solution So we need to show that the identity element is the unique idempotent element in a group. First, we know that by definition of a group there is at least one element, e, such...

This is a simple question but i keep finding conflicting answers and don't understand scientific and mathamatical language well enough to consult reliable data.. 'Cause I'm a dunce. So, is EVERYTHING outside the local group leaving us, or are some things in the virgo supercluster contracting...

Homework Statement Suppose a cyclic group, G, has only three distinct subgroups: e, G itself, and a subgroup of order 5. What is |G|? What if you replace 5 by p where p is prime? Homework EquationsThe Attempt at a Solution So, G has three distinct subgroups. By Lagrange's theorem, the order of...

Homework Statement Is the group of positive rational numbers under multiplication a cyclic group. Homework EquationsThe Attempt at a Solution So a group is cyclic if and only if there exists a element in G that generates all of the elements in G. So the set of positive rational numbers would...

Homework Statement Consider the binary structure given by multiplication mod 20 on {4, 8, 12, 16}. Is this a group? If not, why not? Homework EquationsThe Attempt at a Solution I started by constructing a Cayley table, and working things out. It turns out that 16 acts as an identity element, 4...

Homework Statement Show that the set GL(n, R) of invertible matrices forms a group under matrix multiplication. Show the same for the orthogonal group O(n, R) and the special orthogonal group SO(n, R). Homework EquationsThe Attempt at a Solution So I know the properties that define a group are...

Recently, I have applied for a Ph.D. position in a research group abroad. I have received an invitation as a respond to my request for a visit for a one week to their research lab. Actually, I am very interested in joining their research group and I have accepted the invitation. Could anyone...

Compare this with the definition of the inverse transformation Λ-1: Λ-1Λ = I or (Λ−1)ανΛνβ = δαβ,...(1.33) where I is the 4×4 indentity matrix. The indexes of Λ−1 are superscript for the first and subscript for the second as before, and the matrix product is formed as usual by summing over...

I am a Computer Science major - I am unfortunately in a group where the other two members are lazy, and I end up doing most of the work. It's too late to change groups. The class itself consists of a research paper/presentations, as well as a programming project/app. Every time I clearly tell...

Why group of symmetry of rectangle does not have more reflections but only two. Why does not have reflections over diagonal as in case of square? Thanks for the answer. http://mathonline.wikidot.com/the-group-of-symmetries-of-a-rectangle

Homework Statement The group ##G = \{ a\in M_n (C) | aSa^{\dagger} =S\}## is a Lie group where ##S\in M_n (C)##. Find the corresponding Lie algebra. Homework EquationsThe Attempt at a Solution As far as I've been told the way to find these things is to set ##a = exp(tA)##, so...

Is it mathematically correct to call any group operation in ##(G,\cdot)## composition?

I need the ionic radius of the cation in the following anhydrous salts: FeCl2 and CoCl2 Looking at this database: http://www.knowledgedoor.com/2/elements_handbook/shannon-prewitt_effective_ionic_radius_part_2.html Knowing that the coordination number of both Fe2+ and Co2+ cations is 6, I am...

Homework Statement Let ##R(\theta) = \left( \begin{array}{cc} \cos(\theta) & -\sin(\theta)\\ \sin(\theta)& \cos(\theta)\\ \end{array} \right) \in O(2)## represent a rotation through angle ##\theta##, and ##X(\theta) = \left( \begin{array}{cc} \cos(\theta) & \sin(\theta)\\ \sin(\theta)&...

The lagrangian of a non interacting quark is made to be invariant under local SU(3) transformations by introduction of a new field, the gauge field, giving rise to the gluon. This gives us a locally gauge invariant lagrangian for the quark field and together with the construction of a locally...

It seems that for folks of my generation (Baby Boomer, Generation X), the canonical arrangement of the Periodic Table was for the f-orbital-filling elements to be in a separate section at the bottom. OK, I can see why this makes sense as otherwise the table would be really wide with a lot of...

This is a companion question to https://www.physicsforums.com/threads/why-su-3-xsu-2-xu-1.884004/ Of course the Higgs mechanism over the standard model produces this low-energy group, SU(3)xU(1), which acts on Dirac fermions (this is, no Left-Right asymmetry anymore). Is there some reason...

Are there any good examples of how group theory can be applied to solve multivariate algebraic equations? The type of equations I have in mind are those that set a "multilinear" polynomial (e.g. ## xyz + 3xy + z##) equal to a monomial (e.g. ##x^3##). However, I'd like to hear about any sort...

The de Sitter group is often used as an extension of the Poincaré group, because its a simple group and preserves, in addition to a velocity c, a length L. A natural candidate for this length scale is the Planck length. Thus it seems to make sense to think about the invariant Planck length as...

Hey! :o For each group $G$, $\text{exp}(G)$ is the exponent of the group $G$, i.e., the smallest positive integer $k$, such that $g^k=e$ for each $g\in G$. Let $G$ be a finite group. I have shown that $\text{exp}(G)$ divides $|G|$, and if $G$ is cyclic, then $\text{exp}(G)=|G|$, as follows...

Homework Statement Prove that a finite group is the union of proper subgroups if and only if the group is not cyclic. Homework Equations None The Attempt at a Solution [/B] " => " If the group, call it G, is a union of proper subgroups, then, for every subgroup, there is at least one...

Homework Statement 2 questions here: 1) Let g(x) = x^6 - 10 be a polynomial in Q(c) where c is a primitive 6th root of unity. Find a splitting field for this polynomial and determine it's Galois group 2) let f(x) = x^3 + x^2 + 2 with coefficients in ##F_3##. Find a splitting field K for this...

Homework Statement Find the Galois group of f(x) = x^7-x^6-2x+2 over ##F_7##. Homework EquationsThe Attempt at a Solution 1 is a root of f(x) so dividing f(x) / (x-1) we get the quotient x^6-2. Now all elements of ##F_7## satisfy a^6 = 1 since it's multiplicative group is of order 6, and thus...

Homework Statement Let c be a primitive 3rd root of unity and b be the third real root of four. Now consider the extension Q(c,b):Q. Find the degree of this extension, show that it is Galois, and calculate Gal(Q(c,b):Q) and then use the Galois group to calculate all intermediate fields...

Hey! :o I want to make the diagram for the dihedral group $D_6$: Subroups of order $2$ : $\langle \tau \rangle$, $\langle \sigma\tau\rangle$, $\langle\sigma^2\tau\rangle$, $\langle\sigma^3\tau\rangle$, $\langle\sigma^4\tau\rangle$, $\langle\sigma^5\tau\rangle$, $\langle\sigma^3\rangle$...

Hey! :o Let $\rho=\sqrt[3]{\frac{1+\sqrt{5}}{2}}$. We have that $\rho$ is a root of $f(x)=x^6-x^3-1\in \mathbb{Q}[x]$, that is irreducible over $\mathbb{Q}$. We have that all the roots of $f(x)$ are $\rho, \omega\rho, \omega^2\rho, -\frac{1}{\rho}, -\frac{\omega}{\rho}...

Homework Statement Let f(x) = x^4 - 6x^2 - 2. Let K be a splitting field for this polynomial over Q, show that Gal(K:Q) is non-abelian of order 8. Homework EquationsThe Attempt at a Solution So I calculated the roots of this polynomial, one root was r = (3+(11^1/2))^1/2, and the others were...

The left-handed Weyl operator is defined by the ##2\times 2## matrix $$p_{\mu}\bar{\sigma}_{\dot{\beta}\alpha}^{\mu} = \begin{pmatrix} p^0 +p^3 & p^1 - i p^2\\ p^1 + ip^2 & p^0 - p^3 \end{pmatrix},$$ where ##\bar{\sigma}^{\mu}=(1,-\vec{\sigma})## are sigma matrices.One can use the sigma...