Group Definition and 1000 Threads

http://i.imgur.com/JgpJp03.png I've done the Cayley table for the group above and can't find it in any of the group encyclopedias online. I can post it too if you want, but I'll tell you this: It is a non abelian group of order 8 with two generators (a,g) such that a^4=Identity and...

My question concerns both quantum theory and relativity. But since I came up with this while studying QFT from Weinberg, I post my question in this sub-forum. As I gather, we first work out the representation of Poincare group (say ##\mathscr{P}##) in ##\mathbb{R}^4## by demanding the Minkowski...

Homework Statement Good day, I need to show that S_n=\mathbb{Z}_2(semi direct product)Alt(n) Where S_n is the symmetric group and Alt(n) is the alternating group (group of even permutations) note: I do not know the latex code for semi direct product Homework Equations none The Attempt at...

Homework Statement Is the following a valid group? The values contained in the set of all real numbers ℝ, under an operation ◊ such that x◊y = x+y-1 Homework Equations Axioms of group theory: Closure Associativity There must exist one identity element 'e' such that ex=x for all x There...

Homework Statement Good day all! (p.s I don't know why every time I type latex [ tex ] ... [ / tex ] a new line is started..sorry for this being so "spread" out) So I was wondering if my understanding of this is correct: The Question asks: "\mathbb{Z}_4 has a subgroup is isomorphic to...

EDIT: I've just realized this is the 'Calculus and beyond' subforum - I saw 'beyond' and thought, "Well I've done all my calculus, and now I'm doing group theory, so this thread must go here!". But now I realize it surely belongs somewhere else. Sorry about that. Mods feel free to shift it to...

When reading about GUTs you often come across the 'Standard Model decomposition' of the representations of a given gauge group. ie. you get the Standard Model gauge quantum numbers arranged between some brackets. For example, here are a few SM decompositions of the SU(5) representations...

I study physics and currently taking a mathematical physics course. One of the topics is group theory and we will see the following topics: Symmetries, discrete groups, homomorphisms, isomorphisms, continuous groups, and linear transformations in phase space. This topic will be covered with...

I'm doing a small research project on group theory and its applications. The topic I wanted to investigate was Noether's theorem. I've only seen the easy proofs regarding translational symmetry, time symmetry and rotational symmetry (I'll post a link to illustrate what I mean by "the easy...

Hi, How do we determine the group types for a particular order? I know the example of using the Cayley table to show there are only two types of groups of order 4 but do not know how to determine this for other groups. For example, suppose I wanted to show there is a group of order 10 in the...

Hi guys, There is a software package called GAP for "Groups, Algorithms, and Programming" with emphasis on Group Theory. You can download it for free. I did. However I'm finding it so intractable to use. I would like to find the "missing group" in the Symmetric groups. That is, the group...

Hello all! If I have a group of order 20 that has three elements of order 4, can this group be cyclic? What if it has two elements? I am new to abstract algebra, so please keep that in mind! Thanks!

Let $G$ be a set and $*$ a binary operation on $G$ that satisfies the following properties: (a) $*$ is associative, (b) There is an element $e\in G$ such that $e*a=a$ for all $a\in G$, (c) For every $a\in G$, there is some $b\in G$ such that $b*a=e$. Prove that $(G, *)$ is a group. My...

Homework Statement Let ##G## be a group and ##\sim## and equivalence relation on ##G##. Prove that if ##\sim## respects multiplication, then ##\sim## is the equivalence relation associated to some normal subgroup ##N\trianglelefteq G##; i.e., prove there is a normal subgroup ##N## such that...

Hi, I'm confused about the discussion on p28 of Nigel Goldenfeld's "Lectures on phase transitions and the renormalization group" (this question can only be answered by people who have access to the book.) The goal is to compute the potential energy of a uniformly charged sphere where the...

Hi, I was wondering if there is code already available to draw group lattice diagrams if I already know what the subgroup structure of the group and its subgroups are. For example, it's easy to determine the subgroup lattice for cyclic groups simply using divisors via Lagrange's Theorem...

Hi, The Quaternion group, ##Q=\{1,-1,i,-i,j,-j,k,-k\}##, can be realized by ##2x2## matricies: ## \begin{align*} 1=\begin{bmatrix} 1,0 \\ 0,1\end{bmatrix} &\hspace{10pt} i=\begin{bmatrix} \omega,0 \\ 0,-\omega\end{bmatrix} & \hspace{10pt}j=\begin{bmatrix} 0,1 \\ -1,0\end{bmatrix} &...

If someone can check this, it would be appreciated. (Maybe it can submitted for a POTW afterwards.) Thank-you. PROBLEM Prove that if $H$ and $K$ are torsion-free groups of finite rank $m$ and $n$ respectively, then $G = H \oplus K$ is of rank $m + n$. SOLUTION Let $h_1, ..., h_m$ and $k_1...

I am studying Group Theory at the moment and i am not sure about a theorem. Is it true that a Lie Group G is compact if and only if every finite complex representation of it is unitary? I know that is true the if, but what about the viceversa? Same question. Is it true that a Lie group is...

I need to study in detail the rappresentations of the Poincare Group, i am interessed in the idea that particles can be wieved as irriducible representations of it. Do you have some references about it?

I would like to start a discussion about group work both in education and in work-life. Group work, especially in smaller groups, tends to be 'active learning'. Active learning tends to be more efficient than passive learning. I would like to know if you as a teacher like to assign group work...

If G is a finite set closed under an associative operation such that ax = ay forces x = y and ua = wa forces u = w, for every a, x, y, u, w ##\in## G, prove that G is a group. What I attempted: If we can prove that for every x ##\in## G, x##^{-1}## is also ##\in## G, then by the closure of the...

Case 2: I get that D = c I means A must also be proportional to I but how does that mean B must be diagonal? Question: Answers:

I'm reading a paper these days, How can I get 2.28? It seems for a D-dim SO(2) gauge field, we have spin2, spin1, as well as spin0 particles?

I am reading Joseph J. Rotman's book, Advanced Modern Algebra and am currently focused on Chapter 1: Groups I. I need some help with the proof of Proposition 1.52. Proposition 1.52 reads as follows: I have several related questions that need clarification ...Question 1 In the above text...

Hi there. Can anybody recommend a good textbook for an undergraduate wanting to study group theory (especially representation theory). I'm thinking of reading "visual group theory" by Carter for conceptual understanding but I also need a book to study alongside this that gives a more formal...

I want to learn how to write down a particle state in some inertial coordinate frame starting from the state ##| j m \rangle ##, in which the particle is in a rest frame. I know how to rotate this state in the rest frame, but how does one write down a Lorentz boost for it? Note that I am not...

Hello all. I am new here. I am in the last quarter of a 3 quarter sequence of undergrad quantum mechanics and I just had some conceptual questions (nothing pertaining to homework). We just recently covered Berry's Phase and the Dynamical Phase. Now I wanted to start with a more basic quantum...

Suppose to have a Lie group that is at the same time also a Riemannian manifold: is there a relation between Christoffel symbols and structure constants? What can i say about the geodesics in a Lie group? Do they have special properties?

Is the following correct? We begin with a set. Then, we specify a certain collection of subsets and thereby create a topology. This endows the set with certain properties, one of which is “nearness” and “boundedness.” Then we specify that the topology be smooth. In so doing, our topology...

Consider the rotation group ##SO(3)##. I know that ##R_{x}(\phi) R_{z}(\theta) - R_{z}(\theta) R_{x} (\phi)## is a commutator? But can this be called a commutator ##R_{z}(\delta \theta) R_{x}(\delta \phi) R_{z}^{-1}(\delta \theta) R_{x}^{-1} (\delta \phi)##?

(Again, I am sorry for the simplicity of these questions. I am a mechanical engineer learning this stuff late in life.) I have read the following terms or phrases: group algebra group algebra the algebra of a group an algebra group an algebraic group a group of algebras So... could someone...

Hi there, I have a question about the rest mass of an electron. As we all know, the charge of an electron is a function of the energy at which the system is probed. When defining the charge, we typically use as our reference scale the charge measured in Thompson scattering at the orders of...

Homework Statement Show that the group of units in Z_10 is a cyclic group of order 4 Homework EquationsThe Attempt at a Solution group of units in Z_10 = {1,3,7,9} 1 generates Z_4 3^0=1, 3^1=3, 3^2=9, 3^3= 7, 3^4= 1, this shows <3> isomorphic with Z_4 7^0=1 7^1= 7, 7^2= 9 7^3=3 7^4=1, this...

According to Maxwell's equations, $$c=\frac 1 {\sqrt{μ_0 μ_r ε_0 ε_r}}$$ in a medium with an electric permittivity of ##ε_r## and magnetic permeability of ##μ_r##. This means that in any medium which has values for these properties which are greater than that of a vacuum, the speed of light...

I've been studying for my final exam, and came across this homework problem (that has already been solved, and graded.): "Show that the Galois group of ##f(x)=x^3-1## over ℚ, is cyclic of order 2." I had a question related to this problem, but not about this problem exactly. What follows is...

If an individual has a genotype##I^oI^o## and ##hh## is he considered to be O or bombay blood group? Also i read that a person with bombay blood group can donate blood to anyone but can accept only from a person of his blood group, why can't he/she accept blood from O-ve?( is there a H antibody?)

I'm thinking of an object or objects. How do I show that the objects form a representation of the Lorentz group in 1+1 D spacetime? Thanks for any help!

In Chapter 7 of John M. Lee's book on topological manifolds, we find the following text on composable paths and the multiplication of path classes, [f] ... ... Lee, writes the following:In the above text, Lee defines composable paths and then defines path multiplication of path classes (not...

Is anyone interested in a GRE Math Subject Test study group in the Milwaukee area that meets up to 3 times or so per week?

Hello everyone. Does anyone know if it is possible to build a gauge theory with a local ISO(3) symmetry (say a Yang-Mills theory)? By ISO(3) I mean the group composed by three-dimensional rotations and translations, i.e. if ##\phi^I## are three scalars, I'm looking for a symmetry under: $$...

Is it possible for a group velocity of any wave disturbance to be greater than its component phase velocities?

Why does the thermal decomposition of group 1 metals, except for lithium, yield a nitrite while that of group 2 metals yield oxides?

Homework Statement Good afternoon, How can you mathematicaly talk about how how group theory compares to electromagnetism. Homework Equations e^iθ=Cosθ+iSinθ The Attempt at a Solution I know that the above formula is because of a sin wave and a cosine wave. Put them together and you get a...

In class we had to show that ${A}_{5}$ is cyclic. So what we did was, ${A}_{5}$ is cyclic iff there is an $\alpha\in{A}_{5}$ with $<\alpha> = {A}_{5}$. So, the $ord(\alpha) = |<\alpha>| = |{A}_{5}| = \frac{5!}{2} = 60$. So, $60 = {2}^{2}*3*5$. After this, we said that we could do a 4-cycle...

What makes a Lie group a real Lie group? I see on the Wikipedia page http://en.wikipedia.org/wiki/Lie_group "A real Lie group is a group that is also a finite-dimensional real smooth manifold" So when is a manifold a real manifold?

Why must the gauge group be in a complex representation so that chirality of the fermions is respected? thanks

By properties I mean the +I or -I effect as compared to other functional groups along with whether it induces a +R or -R effect?

Let N be a normal subgroup of a group G and let f:G→H be a homomorphism of groups such that the restriction of f to N is an isomorphism N≅H. Prove that G≅N×K, where K is the kernel of f. I'm having trouble defining a function to prove this. Could anyone give me a start on this?

The mean velocity of a wavepacket given by the general wavefunction: \Psi(x,t)=\frac{1}{\sqrt{2\pi}}\int dk A(k)e^{i(k x - \omega(k) t)}, can be expressed in two ways. First, we have that it's the time derivative of the mean position (i.e., its mean group velocity): \frac{d \langle...