Group Definition and 1000 Threads

I read somewhere that the Monster Group appears is related to String Theory as 26D String theory on a 24D Leech Lattice gives a vertex algebra whose symmetries are the Monster Group. Just wondering if the size of a big group like that appears in the actual Universe? For example, there are lots...

I am looking for an introductory book on group theory for my son who is a high school freshman. He has a good grasp of the basics of mathematics and is ready to take calculus classes. He has a very strong intuitive grasp of symmetry and transformations, so I thought that he may be ready to be...

Hi all I don't really understand this... How come that if an amino group is attached to the amino acid side chain, like in arginine or lysine, the molecule is basic, but if an hydroxyl group is attached, like threonine, it is not basic? How come the amino group can accept a H+ and a hydroxyl...

Hi all. I have never posted anything in PhysicsForums, but I am a long time follower. I was lucky enough to get admitted to both the MIT's graduate program in Physics and to the DPhil in Theoretical Physics at Oxford in the condensed matter theory group. I am unsure which programme to choose...

Hi, I that <I|M|J>=M_{I}^{J} is just a way to define the elements of a matrix. But what is |I>M_{I}^{J}<J|=M ? I don't know how to calculate that because the normal multiplication for matrices don't seem to work. I'm reading a book where I think this is used to get a coordinate representation of...

Hi I am trying to get a simple grasp of the concept of a group generator and group representations. As ever wherever I look, I get very mathematical speak definitions such as: "A representation is a mapping that takes elements g in G into linear operators F that preserve the composition...

But for oxides reducing nature decreases down a group. We say non metallic oxides are acidic but for hydrides it is opposite, What's the appropriate reason?

Homework Statement If G is a group of even order, show that it has an element g not equal to the identity such that g^2 = 1. Homework Equations None The Attempt at a Solution What I wrote: If |G| = n, then g^n = 1 for some g in G. Thus, (g^(n/2))(g^(n/2)) = 1, so g^(n/2) is the element of...

I am confused as to why thallium is toxic, while the other members of group 13 are safe? (Quotes are from Wikipedia) Boron - "Elemental boron [is] non-toxic to humans and animals" Aluminum - "... [has] extremely low acute toxicity..." Gallium - "...metallic gallium is not considered toxic..."...

Can anybody name some real world applications of group theory? I would be particularly interested to hear any uses of cyclic groups to solve everyday problems one could encounter.

I was wondering, if we take a "group" G (so multiplication is defined among the elements) it forms a group if it has the following properties: Closure Contains the identity element Contains the inverse elements follows associativity. I was wondering if associativity is not a must though... like...

Homework Statement Our lecturer seemed to skip over how to get from the Group Velocity Dispersion to the actual temporal stretch of a pulse sent down an optical fibre, instead we were given just the two formula. I've been trying to work out where the temporal stretch comes from but can't work...

Hello Big Minds, In the following analysis...It is said that D_2 contains three subgroups Z_2...why did he choose a mathematical constuction contains only two of the the three subgroups? shouldn't he use the three in his construction? what will happen if he used the three? [from the book of...

Is there any application of maths in music, which topic is directly used. I've heard group theory is used, but how it is used. Plz help..i m going to work on a project that will show how maths works in music or sound system.

Homework Statement Let G be a non-abelian group with order ##p^3##, p prime. Then show that the order of the center must be p. Homework Equations Theorem in our book says that for any p-group the center is non-trivial and it's order is divisible by p. Class eq. ##|G|= |Z(G)| + \sum{[G ...

Hello, I’m looking for introductory books/notes on group theory and algebra. We are using “Groups” by Jordan and Jordan in class, but I am looking for something a little more in-depth. I wouldn’t mind a book that was entirely focused on the pure maths side but if it had lots of applications to...

I don't understand the geometry of what happens when you give a manifold a metric, in particular how the group structure reduces to the orthogonal group. I've read the wikipedia article http://en.wikipedia.org/wiki/Reduction_of_the_structure_group a dozen times but I get stuck when it says that...

Why density matrix renormalization group theory works only for 1D systems?

Homework Statement If P: G-->C_6 is an onto group homomorphism and |ker(p)| = 3, show that |G| = 18 and G has normal subgroups of orders 3, 6, and 9. C_6 is a cyclic group of order 6. Homework Equations none The Attempt at a Solution I determined that |G| = 18 by taking the factor group...

Given an element a in a group G, class(a) = {all x in G such that there exists a g in G such that gxg^(-1) = a} class(b) = {all x in G such that there exists a g in G such that gxg^(-1) = b} so let's say y is a conjugate of both a and b, so it is in both class(a) and class(b), does that mean...

Is there a way to determine the group from the commutation relations? For example, the commutation relations: [J_x,J_y]=i\sqrt{2} J_z [J_y,J_z]=\frac{i}{\sqrt{2}} J_x [J_z,J_x]=i\sqrt{2} J_y is actually SO(3), as can be seen by redefining J'_x =\frac{1}{\sqrt{2}} J_x : then J'_x , J_y and...

I have this problem on simple group's homomorphism: Let ##G′## be a group and let ##\phi## be a homomorphism from ##G## to ##G′##. Assume that ##G## is simple, that ##|G| \neq 2##, and that ##G′## has a normal subgroup ##N## of index 2. Show that ##\phi (G) \subset N##. And last year somebody...

Wikipedia says that largest order of any element of Rubik's cube group is 1260 [PLAIN]http://upload.wikimedia.org/math/e/1/c/e1cff178a2562422492a4140a38f93ff.png. http://en.wikipedia.org/wiki/Rubik%27s_Cube_group What about element of smallest order (except the identity element)? I'll...

Hello I'm looking for good books on the group theory of quantum mechanics. I have a BS in Physics, MS in Electrical Engineering and decades of work experience in building lasers, and R&D in laser systems, optics & infrared sensing systems. My main goal is to study & understand quantum...

Hello, I've got difficulties in understanding what is the point group a o crystal. I read that it is the subset of symmetry operations leaving at least one point of the lattice fixed. But I do not understand: 1) This point must be the same for all the members of the point group? 2) if it must be...

The homomorphism p:G-->H induces an isomorphism between G/Ker(p) and H (if p is onto). I am trying to understand why this must be true. I understand why these groups have the same magnitude and so a bijection is possible, but there is something that I am not able to understand. What seems to be...

Homework Statement Let p: G-->M be a group homomorphism with ker(p) = K. If a is an element of G, how that Ka = {g in G | p(g) = p(a)} Homework Equations none needed The Attempt at a Solution Okay, I've been struggling with this problem for awhile and I've ran into a problem: -Let g be an...

I am working on a problem on automorphism group of radical of finite group like this one: Here are what I know and what I don't know: ##Aut(R(G))## is an automorphism group, whose elements consist of isomorphic mappings from ##R(G)## to itself. For visualization purpose, I envision the...

Hey Guys! I am working on a math project and I am stumped. I'm not sure weather I should use the chi-squared test or another test with my set of data. I am testing weather there is a relationship between crime rates and the unemployment rate of cities. Could someone please help? I'm not testing...

I am self-studying a class note on finite group and come across a problem like this: PROBLEM: Let ##G## be a dihedral group of order 30. Determine ##O_2(G),O_3(G),O_5(G), E(G),F(G)## and ##R(G).## Where ##O_p(G)## is the subgroup generated by all subnormal p-subgroups of ##G##; ##E(G)## is the...

For group 15 elements the order of basicity given is NH3 > PH3 > AsH3 > SbH3 > BiH3 And order of reducing strength is BiH3 > sbH3 > AsH3 > PH3 > NH3 Why are they in opposite order? Reducing nature means tendency to donate electrons. Basicity means strength of bases and hence as basicity...

Hi, I need help in proving the following statement: An abelian,transitive subgroup of the symmetric group Sn is cyclic,generated by an n-cycle. Thank's in advance

I am working on myself on a problem looks like this: Let ##G'## be a group and let ##\phi## be a homomorphism from ##G## to ##G'.## Assume that ##G## is simple, that ##|G| \neq 2##, and that ##G'## has a normal subgroup ##N## of index 2. Show that ##\phi (G) \subseteq N##. I have been asking...

My daughter needs my help, and I am stumped. Here is the problem: In the first round of the city soccer tournament, the teams in group A finished as follows: Team------>Goals For------>Goals Against----->Points Naranja---> 4--------------->2-------------------->7 Bleu------>...

Hi, let p : E--->B be a covering map. Then we have a result that for every subgroup of ## \pi_1(B) ## we have an associated covering map. Now, going in sort-of the reverse direction, is there a way of figuring out what ## \pi_1(B) ## is, if we know a collection of covering maps for B; what...

I came across this problem in class note but I was stuck: Assume that ##G## be a group of order 21, assume also that ##G'## is a group of order 35, and let ##\phi## be a homomorphism from ##G## to ##G.'## Assume that ##G## does not have a normal subgroup of order 3. Show that ##\phi (g) = 1##...

What is the relationship between transmission of information and group velocity of a wave packet? I always keep hearing things like information always travels at the group velocity, it can't go faster than light etc. While I do understand (to an extent) about information not exceeding the...

In Srednicki's text on quantum field theory, he has a chapter on quantum Lorentz invariance. He presents the commutation relations between the generators of the Lorentz group (equation 2.16) as follows: $$[M^{\mu\nu},M^{\rho\sigma}] =...

Want to learn QFT but often lose courage when seeing such a huge book(M. Srednicki). The author also suggests learn with someone else. Is there any group for this?

Hey! :o We consider the polynomial $f(x)=x^3+x^2-2x-1 \in \mathbb{Q}[x]$ and let $E$ be its splitting field. How can we find the group $Gal(E/\mathbb{Q})$ ?? (Wondering)

Hello, I feel very uncomfortable with some aspects of the theory of valuations, places, and valuation rings. Here is one of my problems : Assume that L/K is a finite Galois extension of fields, and that F is a place from K to its residual field k, whose associated valuation ring is discrete. F...

Hi, I'm trying to understand the process of finding the elements of a given group, such as SE(2). What I do understand is limited to finding elements of very simple symmetry groups, such as those corresponding to rotations/reflections of shapes. My overall knowledge of groups is also pretty...

Homework Statement What is the strongest acid among the following: H2O, H2S, H2Se, H2Te 2. The attempt at a solution I noticed they are in the same group so I think the idea is to pick the compound with the group 6 element that is largest in atomic size...I think it's H2Te because atomic size...

Homework Statement Let ##G## be a group such that its center ##Z(G)## has finite index. Prove that every conjugacy class has finite elements. Homework EquationsThe Attempt at a Solution I know that ##[G:Z(G)]<\infty##. If I consider the action on ##G## on itself by conjugation, each...

1- How can infer from the determinant of the matrix if the latter is real or complex? 2- Can we have tensors in an N-dimensional space with indices bigger than N?

How exactly is the EM field tensor related to the proper orthochronous Lorentz group?

Hi, I am trying to understand the derivation of the Lorentz generators but I am stuck. I am reading this paper at the moment: http://arxiv.org/pdf/1103.0156.pdf I don't understand the following step in equation 15 on page 3: \omega^{\alpha}_{\beta}=g^{\alpha\mu}\omega_{\mu\beta} I don't...

Hi, In chapter 12 of GSW volume 2, the authors remark, "spinors form a representation of SO(n) that does not arise from a representation of GL(2,R)." What do they mean by this? More generally, since SO(n) is a subgroup of GL(2,R) won't every representation of GL(2,R) be a representation of...

I was hoping someone could check the following solutions to these 3 basic questions on cyclic groups and provide theorems to back them up. 1. How many elements of order 8 are there in C_{45}? Solution: \varphi(8)=4 2. How many elements of order 2 are there in C_{20}\times C_{30}? Solution...

Hey folks, I'm trying to dip into group theory and got now some questions about irreducibility. A representation D(G) is reducibel iff there is an invariant subspace. Do this imply now that every representation (which is a matrix (GL(N,K)) is reducibel if it is diagonalizable?Best regards