Group theory Definition and 365 Threads

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##...

Homework Statement Is it true that for each ##n\geq 2## there are two primes ##p, q \neq 1## that divide every ##\binom{n}{k}## for ##1\leq k\leq n-1##?Examples: For ##n=6: \binom{6}{1}=6; \binom{6}{2}=15; \binom{6}{3}=20; \binom{6}{4}=15; \binom{6}{5}=6.## So we can have ##p=2## and...

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?

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...

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

Homework Statement Determine all the subgroups of (A,x_85) justify. where A = {1, 2, 4, 8, 16, 32, 43, 64}.The Attempt at a Solution To determine all of the subgroups of A, we find the distinct subgroups of A. <1> = {1} <2> = {1,2,4..} and so on? <4> = ... ... is this true? are there any other...

Homework Statement Exercises: https://mega.co.nz/#!YdIgjA7T!WmgIpFjCoO-elDyPtUkDNarm21sZ_xet6OTJndPGiRY Text: https://mega.co.nz/#!pVRxVKIC!RfFZiW2atRNj9ycGa4Xx_7Nu5FO4a1e6wmyQVLCcGlQ 2. Homework Equations The Attempt at a Solution This is what I made, obviously all help would be appreciated...

I know this post is in the topology thread of this forum, for group theory, this seemed like the reasonable choice to post it in. I realize group theory is of great importance in physics and I'm trying to eventually understand Emmy Noether's theorem. I'm learning group theory on my own, and...

[SIZE="4"]Definition/Summary A group is a set S with a binary operation S*S -> S that is associative, that has an identity element, and that has an inverse for every element, thus making it a monoid with inverses, or a semigroup with an identity and inverses. The number of elements of a...

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...

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

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...

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...

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

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...

I just though of this and though "it's abstract math meeting physics, so probably not". After looking up fields in several abstract algebra books I thought that maybe fields in physics were called as such in physics because they share something with the mathematical structure of fields in group...

Homework Statement (a) For SU(N), we have: N ⊗ N = A_A + S_S where A corresponds to a field with two antisymetric fundamental SU(N) in- dices φij = −φji, and S corresponds to a field with two symmetric fundamental SU(N) indices φij = φji. By considering an SU(2) subgroup of SU(N), compute...

Homework Statement How do I prove that the inner automorphisms is isomorphic to ##S_3##? The attempt at a solution I know ##S_3 = \{f: \{ 1,2,3 \}\to\{ 1,2,3 \}\mid f\text{ is a permutation}\}## and I know for every group there is a map whose center is its kernel so the center of of...

My prof has been throwing around some group theory terms when talking about spin and isospin (product representations, irreducible representations, SU(3), etc.) I'm looking for a brief intro to group theory, the kind you might find in a first chapter of a physics textbook, so I can get familiar...

Hello, I'm following the proof for this theorem in my textbook, and there is one part of it that I can't understand. Hopefully you can help me. Here is the part of the theorem and proof up to where I'm stuck: Let ##N## be a normal subgroup of a group ##G##. Then every subgroup of the...

Hi! I keep hearing that in the large N limit (so I am talking in specific AdS/CFT but more general too I guess) U(N) and SU(N) are isomorphic. So if I construct, say, the ## \mathcal{N}=1 ## SYM Lagrangian in the large N limit, I can take as gauge group both of the ones mentioned above...

Homework Statement Prove or disprove the following assertion. Let G, H, and K be groups. If G × K \cong H × K, then G \cong H.Homework Equations G × H = \left\{ (g,h): g \in G, h \in H \right\} The Attempt at a Solution I don't even know whether the statement is true or false... I tried...

I am reading QFT from Srednicki's book. In the 2nd chapter of this book and in the spin half part of this book, group theory and group representation theory is used. Can you suggest me a book from where I can learn this?

So I'm intending to teach myself some Particle Physics and Standard Model type stuff, I was wondering if someone who's already covered this could give me some advice. I did some Group Theory a few years back and looking over content pages of lecture notes I occasionally spot references to...

Homework Statement Show that there exists a group of order 21 having two generators s and t for which s^3 = I and sts^{-1} = t^2. Do this exercise by constructing the graph of the group.Homework Equations Based on the given relations, we have t^7 = I.The Attempt at a Solution Since ##s## and...

Interesting question I've happened upon: If there is an epimorphism (i.e. onto homomorphism) $\phi:G\times G \to H\times H$, is there necessarily an epimorphism $\psi:G\to H$? If not, under what conditions can we ascertain such an epimorphism given the existence of $\phi$? I would think that...

Please show me some group theory books that considering the combination of quantum mechanics and relativity theory that leads to the needing of notion of fields.I have heard that the irreducible representation of Poicare group leading to the infinite dimensions representation(meaning field...

I already know about generators, rotations, angular momentum, etc. When I see questions about SO(3), SU(3), and lie groups as it pertains to quantum mechanics, I always hold off on getting into the discussion because I think maybe I don't know what that means. It all seems really familiar...

Hi, I'm interested in doing some self-study this summer and learning some group theory. This has come up a lot as I'm getting into graduate level physics courses, so I'd like a good solid introduction to it. Any recommendations on a book? Preferably one that's at the level of an introductory...

I recall visiting a website that was a wiki for group theory and had many articles on specific groups, but I don't find it today doing a simple-simon search on keywords like "group theory". Anyone know the website that I'm talking about?

Does anyone know what this guy is on about? I understand some of the basics of group theory and I know there's a connection between Galois theory and the solving of a Rubik's cube, but I'm not sure what law he is even trying to disprove here. I'm assuming something with regards to symmetry or...

Homework Statement I am wondering how for determine the central atom's orbitals from the point group character tables described by group theory. For example CO3^-2 (D3h) Carbon's (central atom) p-orbitals are described by a1''+e'. The s-orbital is a1' Homework Equations The...

Hi all, Here i ask the fisrt serie of questions i couldn't solve; A basic knowledge of group theory is supposed for solving them! ------------------------------------------------------------ 1- Can you find 3 subgroups H, k and L of a group G such that H U k U L = G ;and no one of the 3...

I am reading about group theory in particle physics and I'm slightly confused about the word "representation". Namely, it is sometimes said that the three lightest quarks form a representation of SU(3), or that the three colors do. But at the same time, it is said that a group can be...

I just studied group theory. Its all nice with all the definitions and rules that are supposed to be followed for a set with a given operation to be called a group. But I fail to see the importance of defining such an algebraic structure. What are its uses?

Homework Statement Let G be a finite group, a)Prove that if ##g\,\in\,G,## then ##\langle g \rangle## is a subgroup of ##G##. b)Prove that if ##|G| > 1## is not prime, then ##G## has a subgroup other than itself and the identity. The Attempt at a Solution a) This one I would just like...

I'm looking for a text that covers group theory and its applications for QM and QFT, targeted towards an audience that knows their QM but is ignorant of everything quantum fieldy. Any recommendations?

Let G be a group, |G|=n and m an integer such that gcd(m,n)=1. (i) show that $x^m=y^m$ implies $x=y$ (ii)Hence show that for all g in G there is a unique x such that $x^m=g$ (i) there exist a, b such that am+bn=1 so that $m^{-1}=a (mod n)$. Hence $x^m=y^m ->x=y$ ok? (ii) (i) shows...

[FONT=CMMI12]Let G [FONT=CMR12]be a finite group and [FONT=CMMI12]N [FONT=CMR12]a normal subgroup of [FONT=CMMI12]G[FONT=CMR12]. Assume further that [FONT=CMMI12]N [FONT=CMR12]is a p [FONT=CMR12]-group for some prime [FONT=CMMI12]p[FONT=CMR12]. 1) By considering G/N, show that there is a...

I'm pursuing a degree in nuclear physics. However, I have a huge interest in particle physics (i know they are closely related). I am wondering how much a math course in group theory will help me understand particle physics. I want to minor in math, so I'm going to take some extra math...

[FONT=CMR12]Let [FONT=CMMI12]G [FONT=CMR12]be a group with normal subgroups [FONT=CMMI12]H[FONT=CMR8][FONT=CMR8]1 [FONT=CMR12]and [FONT=CMMI12]H[FONT=CMR8][FONT=CMR8]2 [FONT=CMR12]with H[FONT=CMR8][FONT=CMR8]2 not a subset of H[FONT=CMR8][FONT=CMR8]1. [FONT=CMR12][FONT=CMR12]Let...

Homework Statement I have read the following text in a textbook(look the attaxhement) ,and i have a simple question .WHY every 2x2 hermitian matrix would have to satisfy this Equation.It is not obvious to me why.Does anyone know the answer? The textbook stops there without giving any...

Homework Statement Let (G,.) be an non-abelian group. Choose distinct x and y such that xy≠yx. Show that if x2≠1 then x2\notin{e,x,y,xy,yx} The Attempt at a Solution If x2=x would imply x.x.x-1=x.x-1 and x=e which cannot be. If x2= xy or x2=yx would imply x=y which also cannot be...

Homework Statement Let H be a subgroup of G Prove xH=yH ⇔ x-1.y\inH Homework Equations The Attempt at a Solution If x.H = y.H then x,y\inH since H is a subgroup x-1,y-1\inH and the closure of H means x-1.y\inH Proving the reverse is my problem despite the fact that I'm sure...

I'm working on a problem where I have to find the little co-group and star of two wave vectors for a diamond structure (space group 227). I know I have to act on the vector by the symmetry operations in the group (perhaps only the ones in the isogonal point group, Oh?) and see if it remains the...

You know that the current theories in particle physics are expressed in the language of group theory and the symmetries of the theory describe its properties I don't know how is that but my question is,can we do that to classical physics too? I mean,can we use maxwell's equations and derive a...

What is the relationship between topology, functional analysis, and group theory? All three seem to overlap, and I can't quite see how to distinguish them / what they're each for.

Thanks to those who participated in last week's POTW! Here's this week's problem (I'm going to give group theory another shot). ----- Problem: (i) Prove, by induction on $k\geq 1$, that \[\begin{bmatrix}\cos\theta & -\sin\theta\\ \sin\theta & \cos\theta\end{bmatrix}^k =...

Homework Statement Prove that for a finite group A, the order of any element in A divides the order of A. Homework Equations The order of an element a of a group A is the smallest positive interger n such that an = 1. The Attempt at a Solution Well, I know that the order of a...