Group Definition and 1000 Threads

The following statements are from the paper with the above title, recommended in another thread, are from here: http://fds.oup.com/www.oup.co.uk/pdf/0-19-922719-5.pdf An interpretion of these statements would be appreciated: 1. [first paragraph, page 3] What is 'conservation of...

If you consider the permutation representation of Sn in ℂ^n, i.e the transformation which takes a permutation into the operator which uses it to permute the coordinates of a vector, then of course the subspace such that every coordinate of the vector is the same is invariant under the...

Homework Statement Let G=S_6 acting in the natural way on the set X = \{1,2,3,4,5,6\}. (a)(i) By fixing 2 points in X, or otherwise, identify a copy of S_4 inside G. (ii) Using the fact that S_4 contains a subgroup of order 8, find a subgroup of order 16 in G. (b) Find a copy of S_3...

The group in question is U100, the group of units modulo 100, which, correct me if I'm wrong, is equal to {3, 7, 9, 11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 37, 39, 41, 43, 47, 49, 51, 53, 57, 59, 61, 63, 67, 69, 71, 73, 77, 79, 81, 83, 87, 89, 91, 93, 97, 99}. How many elements are there of...

I'm trying to prove that every nxn matrix can be written as a linear combination of matrices in GL(n,F). I know all matrices in GL(n,F) are invertible and hence have linearly independent columns and rows. I was thinking perhaps there is something about the joint bases for the n-dimensional...

I don't understand how to show that the reflections and rotations are associative. Thanks for any help.

The words Poincare and Lorentz sound pretty elegant. I think they are French words like Loreal Or Laurent. I know Poincare has to do with spacetime translation and Lorentz with rotations symmetry. But how come one commonly heard about Lorentz symmetry in Special Relativity and General...

For some reason, diffeomorphism invariance seems to be treated like a second-class citizen in the land of symmetries. In nonrelativistic quantum mechanics, we consider Galilean invariance so important that we form our Hilbert space operators from irreducible representations of the Galilei...

Hi everyone, I was just wondering if anyone had any suggestions of more-mathematically-rigorous textbooks on Lie groups and Lie algebras for (high-energy) physicists than, say, Howard Georgi's book. I have been eying books such as "Symmetries, Lie Algebras And Representations: A Graduate...

Lets say i have elements a,b,c,d,e in some group. is abcde always = ab(d^-1c^-1)e. My question is and you change elements in the middle of an operation by using the inverse?

Proof showing group is abelian? Homework Statement Show that every group G with identity e such that x*x=e for all x in G is abelian. The Attempt at a Solution I know that Ii have to show that it's commutative. I start by taking x,y in G and then xy is in G, so x*x=e...

Hello! I´m currently reading 'Groups, Representations and Physics' by H.F. Jones and I have drawn some conclusions that I would like to have confirmed + I have some questions. :) Conclusions: 1. An albelian group has always only one irrep. 2. The direct sum of two representations...

Homework Statement If f(x) is an irreducible cubic polynomial over a field F, is it ever possible that C_2 may occur as the \operatorname{Aut}(K/F) where K is the splitting field of f? The Attempt at a Solution It seems that this should be theoretically possible. In particular, if f is an...

In an example it says that, if |G| = 15 and G has subgroups A,B of G with |A| = 5 and |B| = 3 , then A \cap B must equal \{e_G\} and the smallest subgroup of G containing both A and B is G itself. Could anyone explain why? Thanks!

Homework Statement Let G=D_4 (the group of symmetries (reflections/rotations) of a square) and let X=\{ \text{colourings of the edges of a square using the colours red or blue} \} so a typical element of X is: What is the size of X? Let G act on X in the obvious way. You are given...

Can a group, G, have repeating elements? And if so does the order of G include these repeated elements? Thanks!

Hello everyone, I am currently studying the renorm. group in Stat. physics, more precisely how a rescaling (of space) leaves the partition function unchanged, at the price of having an infinite space of parameters due to the interaction proliferation at each rescaling. Let K be our...

I search for an 'elementary' proof of this, where results about structure of abelian groups are not used. I've tried a standard way of proving this, but hit a wall. I'm mainly interested if my work on a proof can be expanded to a full solution. Homework Statement Let G be an abelian group...

Homework Statement What is the normalizer of the Sylow p-subgroup in the symmetric group Sym(p) generated by the element (1,2,...,p) where p is a prime number? Thanks Homework Equations na The Attempt at a Solution I know that the normalizer has order p(p-1). And I know that it has...

Hello, I am reading Naive Lie Theory by John Stillwell, and he gives the definition of a simple Lie group as a Lie group which has no non-trivial normal subgroups. Wikipedia, on the other hand, defines it as a Lie group which has no connected normal subgroups. I was wondering, which...

I have seen proofs that the alternating group A_n cannot have subgroups with index less than n. Ok, but what is the subgroup with index equal to n?

Homework Statement Find all the automorphisms of a cyclic group of order 10. Homework Equations ψ(a)ψ(b)=ψ(ab) For G= { 1, x, x^2,..., x^9}, and some function ψ(a) = x^(a/10) The Attempt at a Solution I know that a homomorphism takes the form Phi(a)*phi(b) = phi (ab)...

I am seeking to gain a good understanding of group presentations Currently I have the following general question: "Does a group presentation completely determine a particular group?" The textbooks I am reading seem to indicate that a group presentation does actually determine/specify...

I am reading James and Liebeck's book on Representations and Characters of Groups. Exercise 1 of Chapter 3 reads as follows: Let G be the cyclic group of order m, say G = < a : a^m = 1 >. Suppose that A \in GL(n \mathbb{C} ) , and define \rho : G \rightarrow GL(n \mathbb{C} ) by...

I am doing exercises from Hungerford's text 'Algebra', and would appreciate if someone took the time to verify my write-up for me, and possibly provide me with tips how this could be done more efficiently (using less mathematical machinery) Homework Statement If a normal subgroup N of order p...

We're doing isomorphisms and I was just wondering, is the dihedral group D_{12} isomorphic to the group of even permutations A_4?

Alright, so excuse my ignorance, but I have no idea why the choice he uses for boosts is "convenient" Just to make sure everyone is on the same metric etc etc. Weinberg uses (-,+,+,+) with gamma defined traditionally and God-given units He requires that transformations..(oh my,,,how...

How can I see that the generators of the Galilean group correspond to energy, momentum, etc.? References which cover the Galilean group and algebra as well as their realization in phase space are appreciated, especially if they are not too sophisticated. Thanks kith

Is it possible to compute the Galois Group of a polynomial manually (without a computer)? If so, can someone please explain how? I can't seem to find any information (aside from computer algorithms) on how to find a Galois Group or how to factor a polynomial modulo a prime. If it helps to...

Homework Statement Taking the Dih(12) = {α,β :α6 = 1, β2 = 1, βα = α-1β} and a function Nr = {gr: g element of Dih(12)} Homework Equations Taking the above I have to find the elements of N3. And then prove that N3 is not a subgroup of Dih(12). The Attempt at a Solution For N3 I...

I'm working with universal enveloping algebras, specifically U(sl(2)). Does anybody know of a nice way of determining what the group like elements are. Of course, one could go a direct route and compare the coproduct Δ(v), v\in U(sl(2)), directly with the desired outcome v\otimesv, but the...

Please teach me this: What is the adjoint representation in Lie group? Where is the vector space that the ''elements of the group'' act on in this representation(adjoint representation)? Thank you very much for your kind helping.

Infrared radiation, of wavelength λair = 1um in air, travels through a dispersive medium with refractive index n = 1.4505 and with dn/dλair = -0.01 per um at this wavelength. Calculate the speed at which the radiation carries information. So know that c/n = λf radiation carries...

Hi, I have been assuming something I never took the time to prove because it is beyond my scope so to speak. But I am on the path, rest assured. My question is, is it possible to describe correctly say, the color force, using a different gauge group, regardless of the difficulty? That is...

I remember an argument which says that closed to critical points all systems are universal in the sense that their behavior is described by the critical exponents and that these critical exponents depend only on the dimension of the system and the dimension of the order parameter. I remember...

Alright, so I was reading Ryder and he defines the generator corresponding to a^{\alpha} as the following X_{\alpha}=\frac{\partial x'^{\mu}}{\partial a^{\alpha}}\frac{\partial}{\partial x^{\mu}} (\alpha =1,...r) for r-parameter group of transformations Now this makes sense for...

I have two questions about this group that I think I have an idea about but am unsure of. The first question is how many elements in the Sn group can map 1 to any particular elements, say n-2? The second question is how do you find the order of the stabilizer of 5 in Sn?

Im taking a group theory course at the moment in my third year of a theoretical physics degree. In my textbook the author says defines an isomorphism by saying that if two groups are isomorphic then their elements can be put in a one-to-one correspondence that preserves the group combination...

Homework Statement How many elements of order 4 are in S6? (symmetric group with order 6) Homework Equations The Attempt at a Solution So, the different forms of elements with order 4 in S6 are (abcd)(ef), (abcd) from there I am sunk on how to calculate. I know there are...

Hi I don't know how to attack the following question, any hints would be appreciated: If G is a simply connected topological group and H is a discrete subgroup, then \pi_1(G/H, 1) \cong H .Thank you James

Homework Statement Use the class equation to show that greup G of order pq, with p and q prime, has an element of order p. Homework Equations |G| = \sum |C(x)| (class equation) Z = center of the group G = {z\inG: zy = yz for all y in G} |G| = |Z(x)||C(x)| (counting formula) The...

I know of the theorem that says that in any fiber bundle with structure group G having finitely many connected component, the structure group can be reduced to a maximal compact subgroup of G. But here I am reading a thesis in which the author says: "[Since U(n,n)] is homotopic to its maximal...

Q: Given a fraction A/B when does there exist a finite group G and an automorphism f s.t. exactly A/B elements of G are mapped to their own inverses (f(a) = a-1? If so how can we find the group? Does anything change if we allow infinite groups? I have a friend who was preparing for an intro...

Ok, so to form the factor groups, let's say G(+)H/J, you take every element of G(+)H and left multiply by elements of J. Well elements of G(+)H look like (g, h) and elements of J look like (g, e_2)...so elements of the factor group look like (g_i, h_i)(g_n, e_2) = (g_i*g_n, h_i) Am I...

Homework Statement It seems to me that every element of Z_45 has order 1, 3, 5, 9, 15, or 45. It seems impossible to have an element of order 2 by Lagrange's Theorem. Is there another way of looking at this problem?

Well it isn't. I'm trying to prove it. So I assume there is more than one sylow-3 subgroup, each has order 9, we have 4 of them, now their intersection is either e or has order 3. if it is e, then we have 32 elements in these subgroups besides e (33rd) then assume we have more than...

To find all subgroups you use the fact that by Legrange theorem and subgroup will divide the order of the group, so for the dihedral group D4 our subgroups are of order 1,2, and 4. I am unsure how to tell whether or not these groups will be normal or not.

If G is a finite group and M is a maximal subgroup, H is a subgroup of G not contained in M. Then G=HM. Is this true?

This is not homework. Self-study. And I'm really enjoying it. But, as I'm going through this book ("A Book of Abstract Algebra" by Charles C. Pinter) every so often I run into a problem or concept I don't understand. Let G be a finite abelian group, say G = (e,a1, a2, a3,...,an). Prove...

I was thinking this: If I have the set A = \{ \mathbb{R}^n, \mathbb{R}^n \} where for the "first" element I mean the real vector space \mathbb{R}^n , and the "second" element is the additive group \mathbb{R}^n , then does the set A contain one element ( \mathbb{R}^n )? Or it contains two...