Group Definition and 1000 Threads

Hey! :o I want to show that if $|G|=pqr$ where $p,g,r$ are primes, then $G$ is not simple. We have that a group is simple if it doesn't have any non-trivial normal subgroups, right? (Wondering) Could you give me some hints how we could show that the above group is not simple? (Wondering)...

Hello! As far as I know any subgroup can, in principle, be used to divide group into bundle of cosets. Then any group element belongs to one of the cosets (or to the subgroup itself). And such division still is not qualified as a quotient. Yes, the bundle of cosets in this case will be...

i: B to Y is an inclusion, p: X to Y is a covering map. Define $D=p^{-1}(B)$, we assume here B and Y are locally path-connected and semi-locally simply connected. The question 1: if B,Y, X are path-connected in what case D is path-connected (dependent on the fundamental groups)? 2 What's the...

Hey! :o I want to show that $\text{Inn}(G)\trianglelefteq\text{Aut}(G)$ for each group $G$. We have that the inner automorphisms of $G$ is the following set $\text{Inn}(G)=\{\phi_g\mid g\in G\}$ where $\phi_g$ is an automorphism of $G$ and it is defined as follows: $$\phi_g : G\rightarrow G...

Homework Statement The renormalization group equations for the n-point Green’s function ##\Gamma(n) = \langle \psi_{x_1} \dots \psi_{x_n}\rangle ## in a four-dimensional massless field theory is $$\mu \frac{d}{d \mu} \tilde{\Gamma}(n) (g) = 0$$ where the coupling g is defined at mass scale...

Hey! :o I want to show that $\text{Aut}(\mathbb{Z}_n)$ is an abelian group of order $\phi (n)$. We have that $\mathbb{Z}_n$ is cyclic and it is generated by one element. Does it have $\phi (n)$ possible generators? (Wondering) Let $h\in \text{Aut}(\mathbb{Z}_n)$. Then $h(k)=h(1+1+\dots...

Hello, if we consider a group G and two subgroups H,K such that HK \cong H \times K, then it is possible to prove that: G/(H\times K) \cong (G/H)/K Can we generalize the above equation to the case where HK \cong H \rtimes K is the semidirect product of H and K? Clearly, if HK is a semidirect...

Hey! :o The automorphism group of the group $G$, $\text{Aut}(G)$, is the group of isomorphisms from $G$ to $G$, right? (Wondering) How could we show that for example $\text{Aut}(\mathbb{Z})$ is isomorphic to $\mathbb{Z}_2$ ? (Wondering) What function do we use? (Wondering)

Homework Statement Three electrons form an equilateral triangle 0.800 nm on each side. A proton is at the center of the triangle. Homework Equations U = k[(q_1*q_2)/r] The Attempt at a Solution I tried to use the following equation: k*[(3e^2)/(0.8*10^-9) - (3e^2)/(0.4*10^-9)] I plugged in...

Does an amino group or ethyl group have greater steric hindrance? Is there a "rule" that I can follow to determine this?

Homework Statement Let ##\sigma_4## denote the group of permutations of ##\{1,2,3,4\}## and consider the following elements in ##\sigma_4##: ##x=\bigg(\begin{matrix}1&&2&&3&&4\\2&&1&&4&&3\end{matrix}\bigg);~~~~~~~~~y=\bigg(\begin{matrix}1&&2&&3&&4\\3&&4&&1&&2\end{matrix}\bigg)##...

Homework Statement How many distinct permutations are there of the form (abc)(efg)(h) in S7? Homework Equations 3. The Attempt at a Solution [/B] since we have 7 elements I think for the first part it should be 7 choose 3 then 4 choose 3. And then we multiply those together.

Hey, There are some posts about the reps of SO, but I'm confused about some physical understanding of this. We define types of fields depending on how they transform under a Lorentz transformation, i.e. which representation of SO(3,1) they carry. The scalar carries the trivial rep, and lives...

Hi! I am having problem in understanding the difference between phase and group velocity clearly. In my textbook phase velocity is given by ω/κ while group velocity is by dω/dκ. What is the difference between these two terms? Thank you!

Hello, in group theory a regular action on a G-set S is such that for every x,y∈S, there exists exactly one g such that g⋅x = y. I noticed however that in the theory of Lie groups the definition of regular action is quite different (see Definition 1.4.8 at this link). Is there a connection...

Homework Statement Show that the set U(n) = {x < n : gcd(x, n) = 1} under multiplication modulo n is a group. Homework Equations 3. The Attempt at a Solution [/B] I know that it is important to have the gcd=1 other wise you would eventually have an element that under the group operation...

I'm looking for a nice set of basis matrices ##B_{i,j}## that cover the matrices of size ##n \times n## when linear combinations are allowed. The nice property I want them to satisfying is something like ##B_{i,j} \cdot B_{a,b} = B_{i+a, j+b}##, i.e. I want multiplication of two basis matrices...

If $G$ is a finite group, show that there exists a positive integer $N$ such that $a^N = e$ for all $a \in G$. All I understand is that G being finite means $G = \left\{g_1, g_2, g_3, \cdots, g_n\right\}$ for some positive integer $n.$

Problem: In $S_3$ give an example of two elements $x$ and $y$ such that $(xy)^2 \ne x^2y^2$. Attempt: Consider the mapping $\phi: x_1 \mapsto x_2, x_2 \mapsto x_1, x_3 \mapsto x_3$ and the mapping $\psi: x_1 \mapsto x_2, x_2 \mapsto x_3, x_3 \mapsto x_1$. We have that the elements $\phi, \psi...

$4$. If G is a group in which $(a \cdot b)^i = a^i \cdot b^i$ for three consecutive integers $i$ for all $a, b \in G$, show that $G$ is abelian. I've done this one. The next one says: $5$. Show that the conclusion of Problem $4$ does not follow if we assume the relation $(a \cdot b)^i = a^i...

Question: Is the group R^{x} isomorphic to the group R? Why? R^{x} = {x ∈ R | x not equal to 0} is a group with usual multiplication as group composition. R is a group with addition as group composition. Is there any subgroup of R^{x} isomorphic to R? What I Know: Sorry, I would have liked...

Hello all, I am not very sure with this kind of weld check, could you give me some help and suggestion? Details as following: Given: Ft=force in y direction Lt=distance between Ft action point and weld group geometric center(y direction) Fl=force in x direction Ll=distance between Fl action...

Prove that if $G$ is an abelian group, then for $a, b \in G$ and all integers $n$, $(a \cdot b)^n = a^n \cdot b^n.$ Never mind. I figured it out. We proceed by induction on $n$, then use a lemma in the text.

Hello, I have to solve the following problem: Show that a homomorphism from a finite group G to Q, the additive group of rational numbers is trivial, so for every g of G, f(g) = 0. My work so far: f(x+y) = f(x)+f(y) I know that |G| = |ker(f)||Im(f)| I think that somehow I have to find that...

If you are trying to show that two groups, call them H and G, are isomorphic and you know a presentation for H, is it enough to show that G has the same number of generators and that those generators have the same relations?

I can never derive the prolongation formulas correctly when I want to prove the Lie group symmetries of PDEs. (If I'm lucky I get the transformed tangent bundle coordinate right and botch the rest.) I've gone through a number of textbooks and such in the past, but I haven't found any clear...

Hi y'all, This is more of a maths question, however I'm confident there are some hardcore mathematical physicists out there amongst you. It's more of a curiosity, and I'm not sure how to address it to convince myself of an answer. I have a Lie group homomorphism \rho : G \rightarrow GL(n...

If i am given the de broglie wavelength Of any particle then its sure that i can find the velocity of tat particle if its mass is given. λ= h/mv But the velocity which i found is the group velocity or phase velocity?If it's not the group velocity How can i find it?

Hello Everybody, I am searching for a book that introduces the theory of renormalization other then Peskin Schroeder, I found Peskin Schroeder cumbersome regarding this topic. Can anyone help? Thanks in advance!

(a,b)*(c,d)=(ac,bc+d)on the set {(x,y)∈ℝ*ℝ:x≠0} 1.(b,a)*(d,c)=(bd,ad+c) so not commutative 2.[(a,b)*(c,d)]*(e,f)=(ace,(bc+d)e+f)=(ace,bce+de+f) (a,b)*[(c,d)*(e,f)]=(ace,bce+de+f) so associative Is that correct so far? What do I do next?

I was wondering if anyone knows a good PhD thesis writing support group, either online or in London? I couldn't find any and I wanted to set up a meet up group... but I'm not sure what we would actually do to support each other... and when I finish in a few months, what then? Also I'm afraid...

I've heard it said that the commutation relations of the generators of a Lie algebra determine the multiplication laws of the Lie group elements. I would like to prove this statement for ##SO(3)##. I know that the commutation relations are ##[J_{i},J_{j}]=i\epsilon_{ijk}J_{k}##. Can you...

Homework Statement how to visualize group velocity and phase velocity? I tried to to visualize it The velocity of up and down vibration Of the wave as phase velocity. And The velocity of the whole wave in propagating direction as group velocity. AM I CORRECT OR WRONG? Homework EquationsThe...

The commutation relations for the ##\mathfrak{so(n)}## Lie algebra is:##([A_{ij},A_{mn}])_{st} = -i(A_{j[m}\delta_{n]i}-A_{i[m}\delta_{n]j})_{st}##.where the generators ##(A_{ab})_{st}## of the ##\mathfrak{so(n)}## Lie algebra are given by:##(A_{ab})_{st} =...

This is what I have so far: I'm trying to show that the matrix D has to be unitary. It is the matrix that transforms the wavefunction.

In the propagation of non-monochromatic waves, the group velocity is defined as v_g = \displaystyle \frac{d \omega}{d k} It seems here that \omega is considered a function of k and not viceversa. But in the presence of a signal source, like an antenna in the case of electro-magnetic wave or a...

Hi, I have one spot remaining to take a pure math course, and I'm trying to decide between complex analysis and group theory. Although I've touched some of the basic of dealing with complex numbers in my physics/DE courses, they haven't gone in much depth into them beyond applications. On the...

Homework Statement Show that the set of all ##n \times n## unitary matrices with unit determinant forms a group. 2. Homework Equations The Attempt at a Solution For two unitary matrices ##U_{1}## and ##U_{2}## with unit determinant, det(##U_{1}U_{2}##) = det(##U_{1}##)det(##U_{2}##) = 1...

Homework Statement Show that the set of all ##n \times n## unitary matrices forms a group. Homework Equations The Attempt at a Solution For two unitary matrices ##U_{1}## and ##U_{2}##, ##x'^{2} = x'^{\dagger}x' = (U_{1}U_{2}x)^{\dagger}(U_{1}U_{2}x) =...

Homework Statement Show that the set of all ##n \times n## orthogonal matrices forms a group. Homework Equations The Attempt at a Solution For two orthogonal matrices ##O_{1}## and ##O_{2}##, ##x'^{2} = x'^{T}x' = (O_{1}O_{2}x)^{T}(O_{1}O_{2}x) = x^{T}O_{2}^{T}O_{1}^{T}O_{1}O_{2}x =...

Hello I am studying for my exam and there's a question that i don't know how to solve, I have some difficulties with symmetric/permutations groups 1. Homework Statement Consider a finite group of order > 2. We write Aut(G) for the group of automorphisms of G and Sg for the permutations group...

Good morning. I was wondering how do you prove explicitly that the fundamental group of SO(2) is Z?

Hi everyone. So it's apparent that G/N cyclic --> G cyclic. But the converse does not seem to hold; in fact, from what I can discern, given N cyclic, all we need for G/N cyclic is that G is finitely generated. That is, if G=<g1,...,gn>, we can construct: G/N=<(g1 * ... *gn)*k> Where k is the...

Does anybody can help solve point kinetic equation for one group of delayed neutrons in steps. I am looking forward to solve it by analytical methods. dn(t)/dt=ρ-β/l n(t)+ λC(t) dC(t)/dt= βi* n(t)/l- λC I would really appreciate your help as i am have to submit to clear this paper next week

I've been thumbing back through my organic synthesis book to try and remember how to reduce a carboxylic functional group. I know how to do partial reduction to primary alcohols, or create aldehydes and or acyl halides. But isn't there a way to completely reduce it all the way to a hydrocarbon...

I know that quarks can never exist in isolation, and also group up so that they have a net neutral colour charge. But I am wondering at the start of the universe, or under very, very extreme conditions (such as the start of the universe) would quarks have been able to exist by themselves. I have...

The O(N) nonlinear sigma model has topological solitons only when N=3 in the planar geometry. There exists a generalization of the O(3) sigma model so that the new model possesses topological solitons for arbitrary N in the planar geometry. It is the CP^{N-1} sigma model,†whose group manifold is...

The O(N) nonlinear sigma model has topological solitons only when N=3 in the planar geometry. There exists a generalization of the O(3) sigma model so that the new model possesses topological solitons for arbitrary N in the planar geometry. It is the CP^{N-1} sigma model,†whose group manifold is...

Hi, I am looking for textbooks in relativistic quantum mechanics and group theory. I have just finished my undergraduate studies in Physics and am looking to specialise in theoretical high-energy physics. Therefore, textbooks in relativistic quantum mechanics and group theory suited for that...

EDIT: I think i screwed up and posted in the wrong section. Sorry. Should i make a new one to the correct place? Can this one be moved? Hello. I have a couple of problems here, that i will have to translate from another language, so I am not 100% sure if I am using the correct terms. (1) Let...