Group Definition and 1000 Threads

[SIZE="4"]Definition/Summary The dihedral group D(n) / Dih(n) is a nonabelian group with order 2n that is related to the cyclic group Z(n). The group of symmetries of a regular n-sided polygon under rotation and reflection is a realization of it. The pure rotations form the cyclic group...

[SIZE="4"]Definition/Summary The dicyclic group or generalized quaternion group Dic(n) is a nonabelian group with order 4n that is related to the cyclic group Z(2n). It is closely related to the dihedral group. [SIZE="4"]Equations It has two generators, a and b, which satisfy a^{2n}...

Hi I need a formula that returns a value representative of the amount of ‘assortment’ a group shows. The groups are made up of individuals, all of a binary class (e.g. male or female), are of difference sizes, and can be from different populations (i.e. different ratio of males to females). I...

Long time reader, first time poster. Originally, it was my contention that all Lie groups could be written as the semidirect product of a connected Lie group and a discrete Lie group. However, I no longer believe this is true. The next best thing I could think of was to say that a Lie group is...

On first reading, the description of ‘group velocity [vg]’ appears to be quite straightforward. However, I also found a number of speculative explanations as to ‘how’ and ‘why’ the group velocity may exceed the ‘phase velocity [vp]’. Therefore, in order to get a better intuitive understanding of...

Hi all, Isn't the mapping class group of a contractible space trivial (or, if we consider isotopy, {+/-Id})? Since every map from a contractible space is (homotopically)trivial.

Hello! For those who took (or have been taking) the organic chemistry, which methodology do you prefer to tackle the mechanism and prediction questions? I have been reading Loudon & Wade (functional group-based) and Clayden (mechanism-based), but I feel like the mechanism-based approach is...

So I take <z10, +> this to be the group Z10 = {0,1,2,3,4,5,6,7,8,9} Mod 10 group of additive integers and I worked out the group generators, I won't do all of them but here's an example : <3> gives {3,6,9,2,5,8,1,4,7,0} on the other hand <2> gives {2,4,6,8,0} and that's it! but...

I'm starting to learn about particle physics but I really want to see the whole picture before going deep. Here is what I know: - There are symmetries in quantum physics, which are symmetry operators commute with the Hamiltonian (translation operators, rotation operators...) which act on a...

Homework Statement which functional group is present in aspatame molecule? Homework Equations The Attempt at a Solution why the carbonyl group COO- is not present in the diagram? I can find it in the diagram

I am reading Joseph Rotman's book Advanced Modern Algebra. I need help in fully understanding the proof of Proposition 1.52 on page 36. Proposition 1.52 and its proof reads as follows: The part of the proof on which I need help/clarification is Rotman's argument where he establishes that...

Homework Statement Theorm 1: If M is a monoid, the set of M* of all units in M is a group using the operation of M, called the group of units of M. My question is this always a "real" group? for example, is this 'group' always closed under the binary operation? Homework Equations...

Hi all, I understand the concept of group velocity when applied to superimposed sine waves of the same amplitude, and even when applied to wave packets (in which case you get the well-known expression ∂ω/∂k). My question is what happens when you add two sine waves of different amplitudes? So...

How to prove that energy travels at group velocity and not phase velocity?

In Ryder's Quantum Field Theory it is shown that the Lie Algebra associated with the Lorentz group may be written as \begin{eqnarray} \begin{aligned}\left[ A_x , A_y \right] = iA_z \text{ and cyclic perms,} \\ \left[ B_x , B_y \right] = iB_z \text{ and cyclic perms,} \\ \left[ A_i ,B_j...

Homework Statement For any integer K, the map ø_k: Z → Z given by ø_k(n) = kn is a homomorpism. Verify this Homework Equations if ø(gh) = ø(g)ø(h) for all g,h in G then the map ø: G → H is a group homomorpism The Attempt at a Solution So I have barely any linear algebra so many...

Homework Statement Let (R^2,+) be the set of ordered pairs with addition defined component wise. Verify {(x,2x)|x£R} is a subgroup and that {(x,2x+1)|x£R} is not a subgroup. The Attempt at a Solution So for something to be a subgroup it has to have all it's set items contained in the...

Is there anything wrong with my logic and is there any way to further optimize this potentially long-running function? I've put a lot of comments to explain what's going on. template <typename ObType, typename BinaryFunction> bool isCyclic(const std::set<ObType> & G, BinaryFunction & op...

Dear all, Please read the text in the attachment. Then... 1)Explain what is meant by "fix k" and "fixed point of ρ" ? 1a) What does ρ(k) = k mean? 2) How to make the permutation of αβ? 3) What does "re-arranging α so that its top row coincides with the bottom row of β, and...

In the general-symmetry-space group table over to the right on the page below: https://en.wikipedia.org/wiki/Mimetite It states: Space group: P63/m What does the letter P indicate and also the subscript 3? Thanks

If the group G/[G,G] is abelian then how do we show that xyx^{-1}y^{-1}=1? Thanx

What type of group is the Renormalization Group? All I've seen is people giving a (differential) equation for beta-function when they teach for the RG... Also I haven't been able to find an algebra characterizing the RG... Any clues?

The World Cup is less than three weeks away. The brackets were set long ago. There are three groups of death. Group B (Spain, Netherlands, Chile, Australia). Total FIFA ranking points: 4009. Difference between second and third ranked teams (Chile and Netherlands): 70. Poor Australia. They have...

Let H be a normal subgroup of G. Then factor group G/H is an abelian subgroup. For x, y not in H xHyH=yHxH and xyH=yxH (xyH)(yxH)^{-1}=id xyx^{-1}y^{-1}=id Are these steps correct? thnx

Higher dimensional groups are parametrised by several parameters (e.g the three dimensional rotation group SO(3) is described by the three Euler angles). Consider the following ansatz: $$\rho_1 = \mathbf{1} + i \alpha^a T_a + \frac{1}{2} (i\alpha^a T_a)^2 + O(\alpha^3)$$ $$\rho_2 = \mathbf{1}...

Homework Statement Let G be an abelian group and let x, y be elements in G. Suppose that x and y are of finite order. Show that xy is of finite order and that, in fact, o(xy) divides o(x)o(y). Assume in addition that (o(x),(o(y)) = 1. Prove that o(xy) = o(x)o(y). The Attempt at a...

The vectors \vec{\alpha}=\{\alpha_1,\ldots\alpha_m \} are defined by [H_i,E_\alpha]=\alpha_i E_\alpha they are also known to be the non-zero weights, called the roots, in the adjoint representation. My question is - is this connection (that the vectors \vec{\alpha} defined by the commutation...

Homework Statement I need to determine dad^1 for each element d in the left-coset formed by acting on the elements in C_G(a) with the element c such that c is not an element of the subgroup C_G(a) Homework Equations The Attempt at a Solution I don't really understand what the...

i was given that D4=[e,c,c2,c3,d,cd,c2d,c3d] therfore D4=<c,d> is the subgroup of itself generated by c,d then they defined properties of D4 as follows ord(c)=d, ord(d)=2, dc=c-1d i am strugging to understand how they got that c4=e=d2

Hi, I can't understand why the statement in the title is true. This is what I know so far that is relevant: - A subgroup of a cyclic group G = <g> is cyclic and is <g^k> for some nonnegative integer k. If G is finite (say |G|=n) then k can be chosen so that k divides n, and so order of g^k...

Hello :) That's my 2nd year in Math, and I want to start writing an article on NT or Group Theory. I know most of the basic GT and some NT. I still don't know residues/congruences completely, I face problems about understanding the theorems. There are a lot of theorems in these chapters and...

Dear all, In Marder's Condensed matter physics, it uses matrix operations to explain how to justify two different lattice systems as listed in attachment. However, I cannot understand why the two groups are equivalent if there exists a single matrix S satisfying S-1RS-1+S-1a=R'+a'...

Perimeter conference http://pirsa.org/C14020 Here are links to the talks' videos and slides PDF Recent developments in asymptotic safety: tests and properties Tim Morris http://pirsa.org/14040085/ What you always wanted to know about CDT, but did not have time to...

Hi, let G be any group . Is there a way of embedding G in some other group H so that G is normal in H, _other_ than by using the embedding: G -->G x G' , for some group G'? I assume this is easier if G is Abelian and is embedded in an Abelian group. Is there a way of doing this in...

To define spinors in QM, we consider the projective representations of SO(n) that lift to linear representations of the double cover Spin(n). Why don't we consider projective representations of Spin?

Just by looking at the cayley table of a group and looking at its subgroups, is their a theorem or something which tells you if the right and left cosets are equal? I have question to do and I would love to half the workload by not having to to work out the same thing twice. Thanks

We have that A and B belong to different representations of the same Lie Group. The representations have the same dimension. X and Y are elements of the respective Lie algebra representations. A = e^{tX} B = e^{tY} We want to show, for a specific matrix M B^{-1} M B = AM Does it suffice to...

let us assume G is not cyclic. Let a be an element of G of maximal order. Since G is not cyclic we have <a>≠G. Let b be an element in G, but not in the cyclic subgroup generated by a. O(a) = m and O(b) = n where O() refers tothe orders. . then how can we use this to construct a subgroup of G...

From my understanding, normal and anomalous dispersion are because the phase velocity is a function of k so it is different for different components of a group so the group will spread out over time. So what's group velocity dispersion? Is it the same affect (dispersion/ spreading out)...

In a Lorentz group we say there is a proper orthochronous subspace. How can I prove that the product of two orthchronous Lorentz matrices is orthochronous? Thanks. Would appreciate clear proofs.

How to you get sets of complete basis functions using group theory ? For example , using triangle group for CH3 Cl ?

Hi, I would like to understand the left-invariant vector field of the additive group of real number. The left translation are defined by \begin{equation} L_a : x \mapsto x + a \; , \;\;\; x,a \in G \subseteq \mathbb{R}. \end{equation} The differential map is \begin{equation} L_{a*} =...

What is the use of infinite-dimensional representation of lie group? Now, I know Hilbert space is infinite-dimensional, and physical states must be in Hilbert space. However, for massive fields, the transformation group is SO(3), its unitary representation is finite. For massless fields, the...

I found what might be the worst written book on Lie Groups. Ever. Until I find one I like better, I'm going to see if I can persevere through the sludge. I'll write out the theorem word for word and then explain what I can. Hopefully someone can decipher it. Typically, I use the term "chart"...

Hello! I'm currently reading some QFT and have passed the concept of Weyl spinors 2-4 times but this time it didn't make that much sense.. We can identify the Lorentz algebra as two su(2)'s. Hence from QM I'm convinced that the representation of the Lorentz algebra can be of dimension (2s_1 +...

Homework Statement Consider the following permutation representations of three elements in ##S_3##: $$\Gamma((1,2)) = \begin{pmatrix} 0&1&0\\1&0&0\\0&0&1 \end{pmatrix}\,\,\,\,;\Gamma((1,3)) = \begin{pmatrix} 0&0&1\\0&1&0\\1&0&0 \end{pmatrix}\,\,\,\,\,; \Gamma((1,3,2)) = \begin{pmatrix}...

Homework Statement Let A = A(p)\times A' where A(p) is a finite commutative p-group (i.e the group has order p^a for p prime and a>0) and A' is a finite commutative group whose order is not divisible by p. Prove that all elements of A of orders p^k, k\geq0 belong to A(p) The Attempt...

In Dijkgraaf and Witten's paper "Topological Gauge Theory and Group Cohomology" it is claimed that... Why are either of these statements (the Lie group case or the finite case) true?

Mathematicians have produced a wide variety of long and complex proofs of the existence of free groups, and there appears to be a strong emphasis upon finding better proofs that involve a variety of techniques. (Examples are http://www.jstor.org/stable/2978086 and "www.jstor.org/stable/2317030"...

Homework Statement Let a,b be elements of a group G. Show that the equation ax=b has unique solution. Homework Equations none really The Attempt at a Solution ax = b . Multiply both sides by a^{-1}. (left multiplication). a is guaranteed to have an inverse since it is an element of a...