Group Definition and 1000 Threads

I'm a little bit confused. Matrices \begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{bmatrix} ##\theta \in [0,2\pi]## form a group. This is special orthogonal group ##SO(2)##. However it is possible to diagonalize this matrices and get \begin{bmatrix} e^{i\theta} & 0...

I started by inserting ##ds=\sqrt{dx'^{\mu} dx'_{\mu}}## and ##p'^{\mu}=mc \frac{dx'^{\mu}}{ds}##. So we have: $$\frac{dp'^{\mu}}{ds}=mc \frac{d}{dx'^{\mu}} \frac{d}{dx'_{\mu}} (x'^{\mu})$$ Now I know that ##dx'^{\mu}=C_\beta \ ^\mu dx^\beta## and ##dx'_{\mu}=C^\gamma \ _\mu dx_\gamma## where...

Every group needs to have that every element appear only once at each row and each column. But in the case of unfaithful representations of ##Z_2## sometimes we have ##D(e)=1##, ##D(g)=1##. When we write the Caley table we will have that one appears twice in both rows and in both columns. How is...

Summary:: need help with solution group of Homogeneous system Is the solution group of the system A^3X = 0 , Is equal to the solution group of the system AX = 0 If this is true you will prove it, if not give a counterexample. thank you.

Summary:: Good introductory book about Group Theory? Hi, I am looking for a good introductory book about Group Theory for physicists.

The doubt is about B and C. b) n = 4, $C = {I,e^{2\pi/4}} n = 5, $C = {I,e^{2\pi/5}} n = 6, $C = {I,e^{2\pi/6}} Is this right? c) I am not sure what does he wants...

Hello everyone, I am new here, so please let me know if I am doing something wrong regarding the formatting or the way I am asking for help. I did not really know how to start off, so first I tried to just write out all the ##\mu \nu \rho \sigma## combinations for which ##\epsilon \neq 0## and...

This is going to be controversial and might even be taken down, but I think what I will say is absolutely true, and I'm sorry if it offends people. I'm applying for the second time to condensed matter PhDs. I was in a group that did a lot of device fabrication as part of their experiments and...

I see that the first four equations are definitions. The problem is about the dimensions of the quotient. Why does the set Kx forms a six dimensional Lie algebra?

I am probably missing a crucial point here, but what does it means that (12)(34) squares to the identity? How do we prove it? ((12)(34))² = (12)(34)(12)(34) = (12)(12)(34)(34) = (12)(34) ##\neq I ## Is not this the algorithm?

Maybe my problem is misunderstand the concept of " a modulo n ". I would appreciate any help to get this concept and understand the grou´p

I'd like to better understand spinors on curved spacetime, but started wandering along the following tangent. I've looked at but not particularly understood the sections on spinors in the texts by Penrose and (Misner, Thorne and Wheeler). Let ##g_{ij}## be a spacetime metric (a symmetric...

If group ##(G,\cdot)## is defined with two generators ##a## and ##b##. And ##a^n=e##, ##b^{m}=e##. Is there any Theorem to tell us what is the largest group they can form?

If you have a U(1) generator, can it just be normalized to SU(1)?

Hello there.Questions I have: what is the value of group theory?I am not trying to say that it is not important I want to know what made mathematicians study these objects and we still study them today.I know there are very interesting for me at least examples of groups like the Lie group but...

I would like to investigate a function that sends ##f(x)## to ##f(x) - \frac{1}{c}f(x^{c})##, or a function ##g## such that ##g(f(x)) = f(x) - \frac{1}{c}f(x^{c}).## Since symmetries produced by groups are used in physics, I thought there might be someone here who could help explain what the...

Reorder the statements below to give a proof for $$G/G\cong \{e\}$$, where $$\{e\}$$ is the trivial group. The 3 sentences are: For the subgroup G of G, G is the unique left coset of G in G. Therefore we have $$G/G=\{G\}$$ and, since $$G\lhd G$$, the quotient group has order |G/G|=1. Let...

"The group given by ## H = \left\{ \left( \begin{array} { c c } { e ^ { 2 \pi i \theta } } & { 0 } \\ { 0 } & { e ^ { 2 \pi i a \theta } } \end{array} \right) | \theta \in \mathbb { R } \right\} \subset \mathbb { T } ^ { 2 } = \left\{ \left( \begin{array} { c c } { e ^ { 2 \pi i \theta } } & { 0...

Hi, The textbook asks me to use subgroup axioms to prove why a set of permutations that interchange two specific symbols in S4 is or isn't a subgroup of the symmetric group, and the same for a set of permutations that fix two elements. My guess is that the set of permutations that interchange...

hi guys i saw this problem : if G is a group and a,b belongs to G and O(a) = e , b.a =a.b^2 then find O(b) , but i want to tackle this problem using Cayley diagrams , so my attempt is as following : $$ba =ab^{2}$$ then i might assume b as flipping , a as rotation : $$ fr = rf^{2}$$ then...

Hi everyone, I'm working through some group theory questions online. But unfortunately they don't have answers to go with them. So, I'm hoping you can say if I'm on the right track. If this is a binary operation on ℝ, am I right in thinking it satisfies the closure and associativity axioms...

I have failed a course on group theory for physicists in my university, and i need a good book to learn group theory from because anthony zee's book is simply too hard to read. His book is verbose, glosses over many concepts, and is not very rigorous. Then the exercises in the book are very...

Consider the group . "The map defined by for all is a group homomorphism." Is this true or false? I'm guessing it's true because φ (j) = | j |, which means φ (j * k) = | j * k | =| j | * | k | = φ ( j ) * φ ( k ).

Proof: We note ##60 = 2^2\cdot3\cdot5##. By Sylow's theorem, ##n_5 = 1## or ##6##. Since ##G## is simple, we have ##n_5 = 6##. By Sylow's theorem, ##n_3 = 1, 4, ## or ##10##. Since ##G## is simple, ##n_3 \neq 1##. Let ##H## be a Sylow ##3## subgroup and suppose ##n_3 = 4##. Then ##[N_G(H) : G] =...

I learned that the Lorentz group is the set of rotations and boosts that satisfy the Lorentz condition ##\Lambda^T g \Lambda = g## I recently learned that a representation is the embedding of the group element(s) in operators (usually being matrices). Examples of Lorentz transformations are...

Summary:: Looking for best literature or online courses on projective unitary representations of the Poincare Group. I'm watching an online course on relativistic QFT. I understand that because this theory deals with both QM and SR, there is a need to represent Lorentz transformations with...

I am very confused about how to find all the automorphisms of a group. The book I am using is very vague and the exercises don't show any solutions. I get how to do it for cyclic groups but not the general case. I will outline what I know of the procedure and insert my questions into it. To...

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...

The alkyl group of acetate ion in acetic acid pushes more negative charge inductively toward already negative COO- end destabilize it.In this context I wish to know what actually is destabilization?What happen when acetate ion is destabilized?How the anion is stabilized ultimately? Could you...

Is is true that the dihedral group ##D_n## does not have an irreducible representation with a dimension higher than two?

This is note about O(3,3) space-time. The related article is: https://doi.org/10.3390/sym12050817 Here's some background: In O(3,1) space-time (Minkowski), the six generators of rotations and boosts can form an SU(2) x SU(2) Lie algebra. This algebra is then used generically by all the...

I'm trying to 'see' what the generators of the Poincare Group are. From what I understand, it has 10 generators. 6 are the Lorentz generators for rotations/boosts, and 4 correspond to translations in ℝ1,3 since PoincareGroup = ℝ1,3 ⋊ SO(1,3). The 6 Lorentz generators are easy enough to find in...

I am reading Group Theory in a Nutshell for Physicists by A. Zee. I have big problems when learning chapter IV.1 Tensors and Representations of the Rotation Groups SO(N). It reads I can understand why ##D\left ( R \right )## is a representation of SO(3), but I hardly can see why the tensor T...

Given that the galaxies within a cluster of galaxies are generally gravitationally bound, and not affected by the expanding universe, would it not also be expected that after some large number of billions of years, all of the individual galaxies would merge together to become one single very...

Hey! 😊 Let $(G, \#), \ (H, \square )$ be groups. Show: For $(g,h), (g',h')\in G\times H$ we define the operation $\star$ on $G\star H$ as follows: \begin{equation*}\star: (G\times H)\times (G\times H,\star), \ \left ((g,h), (g',h')\right )\mapsto (g\# g', h\square h')\end{equation*}...

In Griffith's Introduction to Elementary Particles, he provides a very cursory introduction to group theory at the start of chapter four, which discusses symmetries. He introduces SO(n) as "the group of real, orthogonal, n x n matrices of determinant 1 is SO(n); SO(n) may be thought of as the...

Hey! 😊 Let $G$ be a permutation group of a set $X\neq \emptyset$ and let $x,y\in X$. We define: \begin{align*}&G_x:=\{g\in G\mid g(x)=x\} \\ &G_{x\rightarrow y}:=\{g\in G\mid g(x)=y\} \\ &B:=\{y\in X\mid \exists g\in G: g(x)=y\}\end{align*} Show the following: $G_x$ is a subgroup of $G$. The...

The first ionization energy decreases between group 5 and group 6 due to the repulsion between the electrons in the p orbital. Although I understand that the effective nuclear charge increases between group 1 and group 2 elements, why isn't this the case between group 1 and group 2 elements...

Hello, I'm trying to make sense of some of the group theoretic discussion found in Griffith's Introduction to Elementary Particles. I have had a fair amount of exposure to elementary group theory, but no representation theory, and have some specific questions related to this which refer to the...

How to prove that direct product of two rep of Lorentz group ##(m,n)⊗(a,b)=(m⊗a,n⊗b)## ? Let ##J\in {{J_1,J_2,J_3}}## Then we have : ##[(m,n)⊗(a,b)](J)=(m,n)(J)I_{(a,b)}+I_{(m,n)}⊗(a,b)(J)=## ##=I_m⊗J_n⊗I_a⊗I_b+J_m⊗I_n⊗I_a⊗I_b+I_m⊗I_n⊗J_a⊗I_b+I_m⊗I_n⊗I_a⊗J_b## and...

I am just starting a QM course. I hope these are reasonable questions. I have been given my first assignment. I can answer the questions so far but I do not really understand what's going on. These questions are all about so(N) groups, Pauli matrices, Lie brackets, generators and their Lie...

Hi, I am the leader of a University of the Third Age discussion group which has just been suspended due to the ongoing Coronavirus situation. I am hoping to be able to continue our discussions online in some way. All group members are tech savvy enough to use email, and search the internet...

three sub algebra of Unitary group (6) as 1. U(5) . 2. SU(3) 3. O(6) here the three chains in attachment is attached. I want to know how these chains are understands in group theory.

When it comes to completing homework all they seem to care about is getting the right answer and being done with it. I mean I get it, physics is time intensive and we're all doing the same work for the same classes (me with some extra actually) . But if you don't want to actually put the work...

After noting w=vk and differentiating with respect to k, and lots of simplifying, I get: Vg = c/n +(2*pi*0.6)/(k*n) This doesn't correspond to any numerical value though...

I'm not really sure where to begin with this problem. Any insight as to where to begin or what to look out for would be much appreciated! Orders of ##S_3## ##|e|=1## ##|f|=3## ##|f^2|=2## ##|g|=2## ##|gf|=2## ##|gf^2|=3## Orders of ##Z_2## ##|0|=1## ##|1|=2## Orders of ##S_3 x Z_2##...

I am not sure if this is the right forum to post this question. The title says it all: are there examples of Lie groups that cannot be represented as matrix groups? Thanks in advance.

For a specific wave vector, ##k##, the group of wave vector is defined as all the space group operations that leave ##k## invariant or turn it into ##k+K_m## where ##K_m## is a reciprocal vector. How the translation parts of the space group, ##\tau##, can act on wave vector? Better to say, the...

I was studying mathematical logic and came across this statement of group theory I'm having a hard time in understanding it. I have concluded that ##G## is any set but not an empty one, ##\circ## is a function having input as two variables (both variables are from set...