Group Definition and 1000 Threads

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...

Homework Statement let n ≥ 2 Show that Dn = < a,b | a2, b2, (ab)n> Homework EquationsThe Attempt at a Solution I see that a and b are involutions and therefore are two different reflections of Dn. If we set set b = ar where r is a rotation of 2π/n And Dn = <a,r | a2, rn, (ab)2 > I am unsure...

I would like to know if this group http://www.quantumgravityresearch.org/ and its Emergence Theory has any standing in main stream Physics. Thanks Andrew

Hey there, I've suddenly found myself trying to learn about the Lorentz group and its representations, or really the representations of its double-cover. I have now got to the stage where the 'complexified' Lie algebra is being explored, linear combinations of the generators of the rotations...

I teach group theory for physicists, and I like to teach it following some papers. In general my students work with condensed matter, so I discuss group theory following these papers: [1] Group Theory and Normal Modes, American Journal of Physics 36, 529 (1968) [2] Nonsymmorphic Symmetries and...

https://www.cbc.ca/news/technology/hemimastigotes-supra-kingdom-1.4715823 Citation to the paper being discussed: Lax et al. Hemimastigophora is a novel supra-kingdom-level lineage of eukaryotes. Nature. Published online 14 Nov 2018. https://www.nature.com/articles/s41586-018-0708-8

I do have a fair amount of visual/geometric understanding of groups, but when I start solving problems I always wind up relying on my algebraic intuition, i.e. experience with forms of symbolic expression that arise from theorems, definitions, and brute symbolic manipulation. I even came up with...

Homework Statement Let ##p## be a prime integer. Show that ##\operatorname{Aut}(\underbrace{Z_p\times \dots \times Z_p}_{n \text{ factors }})\cong GL_n(\mathbb{F}_p)##, where ##\mathbb{F}_p## is ##Z_p## viewed as a field. Homework EquationsThe Attempt at a Solution First, note that ##Z_p =...

Why does the image of elements in a homomorphism depend on the image of 1? Why not the other generators?

Suppose a set of basis vectors are eigenvectors of some operator. So they will provide a one dimensional representation of that operator in the vector space?

Ffom exercise 27 of Dummite and Foote: Let ##N## be a finite subgroup of ##G##. Show that ##gNg^{-1}\subseteq N## if and only if ##gNg^{-1} = N##. Why must the subgroup ##N## be finite? Isn't this result true for subgroups of any size?

Suppose that we know that ##G=\langle S \mid R\rangle##, that is, ##G## has a presentation. If ##N\trianglelefteq G##, what can be said about ##G/N##? I know that for example, if ##G=\langle x,y \rangle##, then ##G/N = \langle xN, yN \rangle##. But is there anything that can be said about the...

Homework Statement Let ##G## be a finite group and ##m## a positive integer which is relatively prime to ##|G|##. If ##b\in G## and ##a^mb=ba^m## for all ##a\in G##, show that ##b## is in the center of ##G##. Homework EquationsThe Attempt at a Solution Let ##|G| = n## and ##b\in G##. Note that...

This one may seem a bit long but essentially the problem reduces to some matrix calculations. You may skip the background if you're familiar with Lorentz representations. 1. Homework Statement A Lorentz transformation can be represented by the matrix...

Let's say we have a given matrix ##G##. I want to find a set of ##M## matrices so that ##MG = GM## and prove that this is a group. How can I approach this problem?

Homework Statement Let ##S = \{(x_1, \dots, x_p) \mid x_i \in G, x_1 x_2 \cdots x_p = e\}##. Let ##C_p## denote cyclic subgroup of ##S_p## of order ##p## generated by the ##p##-cycle, ##\sigma = (1 \, 2 \, \cdots \, p)##. Show that the following rule gives an action of ##C_p## on ##S## $$...

Dear All short explanation: I am trying to leverage my limited understanding of representation theory to explain (to myself) how many non-vanshing components of, for example, nonlinear optical susceptibility tensor ##\chi^{(2)}_{\alpha\beta\gamma}## can one have in a crystal with known point...

Dear Everyone, I have some feeling some uncertainty proving one of the axioms for a group. Here is the proof to show this is a group: Let the set T be defined as a set of 2x2 square matrices with coefficients of integral values and all the entries are the same. We want to show that T is an...

Homework Statement Prove that for any finite group ##G## there exists a sequence of nested subgroups of ##G##, ##\{e\}=N_0\leq N_1\leq \cdots \leq N_n=G## such that for each integer ##i## with ##1\leq i\leq n## we have ##N_{i-1}\trianglelefteq N_i## and the quotient group ##N_i/N_{i-1}## is...

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...

Homework Statement My question is, how do I show that speed is equal to group velocity? More information at https://imgur.com/a/m6FwNaG Homework Equations v_g = dw/dk The Attempt at a Solution Part a is substitution, part b uses v_g = dw/dk, part c is multiplication by h-bar, but I am stuck...

Homework Statement For each natural number ##n##, let ##V_n## be the subset of the symmetric group ##S_n## defined by $$V_n = \{(i j)(k l) | i,j,k,l \in \{1,\ldots, n\}, i \neq j, \text{ and } k \neq l\},$$ that is, ##V_n## is the set of all products of two 2-cycles. Let ##A_n## be the...

Homework Statement Let ##\mu=\{z\in \mathbb{C} \setminus \{0\} \mid z^n = 1 \text{ for some integer }n \geq 1\}##. Show that ##\mu = \langle z \rangle## for some ##z \in \mu##. Homework EquationsThe Attempt at a Solution My thought would be just to write out all of the elements of ##\mu## in...

aa3.2 Let Q be the group of rational numbers under addition and let $Q^∗$ be the group of nonzero rational numbers under multiplication. In $Q$, list the elements in $\langle\frac{1}{2} \rangle$, In ${Q^∗}$ list elements in $\langle\frac{1}{2}\rangle $ ok just had time to post and clueless

Homework Statement Show that for every subgroup ##H## of cyclic group ##G##, ##H = \langle g^{\frac{|G|}{|H|}}\rangle##. Homework EquationsThe Attempt at a Solution At the moment the most I can see is that ##|H| = |\langle g^{\frac{|G|}{|H|}}\rangle|##. This is because if...

Homework Statement Prove that, for any group ##G##, the set ##\operatorname{Aut} (G)## is a group under composition of functions. Homework EquationsThe Attempt at a Solution 1) associativity: It is a known fact of set theory that composition of functions is an associative binary operation. 2)...

Homework Statement Let ##n## be a natural number and let ##\sigma## be an element of the symmetric group ##S_n##. Show that if ##\sigma## is a product of disjoint cycles of orders ##m_1 , \dots , m_k##, then ##|\sigma|## is the least common multiple of ##m_1 , \dots , m_k##. Homework...

Homework Statement Let ##G## be a group and ##x \in G## any element. Prove that if ##|x| = n##, then ##|x| = |\{x^k : k \in \mathbb{Z} \}|##. Homework EquationsThe Attempt at a Solution Let ##H = \{x^k : k \in \mathbb{Z} \}##. I claim that ##H = \{1,x,x^2, \dots , x^{n-1} \}##. First, we show...

hi everyone, I'm electrical engineer student and i like a lot arnold's book of ordinary differential equations (3rd), but i have a gap about how defines action group for a group and from an element of the group.For example Artin's algebra book get another definition also Vinberg's algebra book...

Hey! :o I want to check the following sets with the corresponding relations if they satisfy the axioms of groups. $M=\mathbb{R}\cup \{\infty\}$ with the relation $\min:M\times M\rightarrow M$. It holds that $\min (a, \infty)=\min (\infty, a)=a$ for all $a\in M$. $M=n\mathbb{Z}=\{n\cdot...

Hey! :o Let $n\in \mathbb{N}$ and $M=\{1, 2, \ldots , n\}\subset \mathbb{N}$. Let $d:M\times M\rightarrow \mathbb{R}$ a map with the property $$\forall x, y\in M : d(x,y)=0\iff x=y$$ Let \begin{equation*}G=\{f: M\rightarrow M \mid \forall x,y\in M : d(x,y)=d\left (f(x), f(y)\right...

I've been trying to understand representations of the Lorentz group. So as far as I understand, when an object is in an (m,n) representation, then it has two indices (let's say the object is ##\phi^{ij}##), where one index ##i## transforms as ##\exp(i(\theta_k-i\beta_k)A_k)## and the other index...

Homework Statement Let ##H = \langle x \rangle##. Assume ##|x| = \infty##. Show that if ##H = \langle x^a \rangle## then ##a = \pm 1## Homework EquationsThe Attempt at a Solution Here is my attempt: Suppose that ##H = \langle x^a \rangle##. Then, for arbitrary ##b \in \mathbb{Z}##, ##x^b =...

Problem: If ##H = \langle x \rangle## and ##|H| = n##, then ##x^n=1## and ##1,x,x^2,\dots, x^{n-1}## are all the distinct elements of ##H##. This is just a proposition in my book with a proof following it. What I don't get is the very beginning of the proof: "Let ##|x| = n##. The elements...

Problem: Let ##G=S_n##, fix ##i \in \{1,2, \dots, n \}## and let ##G_i = \{ \sigma \in G ~|~ \sigma (i) = i \}##. Use group actions to prove that ##G_i## is a subgroup of G. Find ##|G_i|##. So here is what I did. Let ##A = \{1,2, \dots, n \}##. I claim that ##G## acts on ##A## by the group...

Compute the order of each of the elements in the symmetric group ##S_4##. Is the best way to do this just to write out each element's cycle decomposition, or is there a more efficient way?

Dear All, I've been recently reading the very clear text of Burns and Glazer entitled Space Groups for Solid State Scientists in the context of my thesis which requires understanding of symmetries of crystals, more specifically symmetries of (approximate triply periodic minimal surfaces)...

Homework Statement Determine the Lie bracket for 2 elements of SU(3). Homework Equations [X,Y] = JXY - JYX where J are the Jacobean matrices The Attempt at a Solution I exponentiated λ1 and λ2 to get X and Y which are 3 x 3 matrices.. If the group elements are interpreted as vector...

A set of numbers 1,2,...,4N gets randomly divided into two groups with equal amount of numbers. Calculate the probability:7 a) Each group has an equal amount of odd and even numbers, b) All numbers that are divisible by N, to fall in only one of the groups, c) All numbers that are divisible by...

So I have been learning about measuring nuclei radius using electron diffraction, using sin(theta)=1.22lambda/d, doing some research I found out that is the equation for circular aperture diffraction but I don’t really understand how a group of nucleus can act as a circular aperture. Also is...

Hello! I want to make sure I understand the relation between this and rotation (mainly between SU(2) and SO(3), but also in general). Also, I am a physics major, so I apologize if my statements are not very rigorous, but I want to make sure I understand the basic underlying concepts. So SU(2) is...

A very simple question: if given a vector ##v(t_0)## and two group functions ##G(t)## and ##G'(t)##, here ##t## is the parameter of time, the two group functions act on ##v(t_0)## simultaneously, then we can get a vector field ##v(t)##, then how to get ##v(t)##?

Hello! Can someone recommend me some good reading about the Lorentz group and its representations? I want something to go pretty much in all the details (not necessary proofs for all the statements, but most of the properties of the group to be presented). Thank you!

Our standard model breaks the Higgs Su(2) electroweak symmetry via the Higgs mechanism. In official beyond the standard models. May I know the different lists of models where the Higgs field can be part of larger symmetry group like SU(10) and different ways to break it?

Hello! I am reading some notes on Lorentz group and at a point it is said that the irreducible representations (IR) of the proper orthochronous Lorentz group are labeled by 2 numbers (as it has rank 2). They describe the 4-vector representation ##D^{(\frac{1}{2},\frac{1}{2})}## and initially I...

Or in other words: The renormalization group is a systematic theoretical framework and a set of elegant (and often effective) mathematical techniques to build effective field theories, valid at large scales, by smoothing out irrelevant fluctuations at smaller scales. But does the...

Hi, I'm currently stuck on a homework question and I was hoping if I could get some help. Group A has 50% chance of ordering french fries (price: 5),40%chanceoforderingmilkshake(price:6), and 10% chance of ordering a burger (price: 7).GroupBhas30%chanceoforderingfrenchfries(price:5), 30% of...

I am looking at this proof and I am stuck on the logic that $a^{p}$ = 1. For example, consider the group under multiplication without zero, ${Z}_{5}$, wouldn't 2^4 = 1 imply that the order is 4 not 5? We know that if G is a finite abelian group, G is isomorphic to a direct product...

The Brian Hall's book reads: A Lie group is any subgroup G of GL(n,C) with the following property: If Am is a secuence of matrices in G, and Am converges to some matrix A then either A belongs to G, or A is not invertible. Then He concludes G is closed en GL(n,C), ¿How can this be possible, if...

will it be correct to say (Λ00)2 – (Λ11)2 – (Λ22)2 – (Λ33)2 = 1 if Λ - group is neither proper nor orthochronous