Groups Definition and 867 Threads

Homework Statement Let U(n) be the set of all positive integers less than n and relatively prime to n. Prove that U(n) is agroup under multiplication modulo n Homework Equations The Attempt at a Solution n | a-a' implies n | b(a-a') =ab-a'b and n |b-b' implies n |a'(b-b')...

Homework Statement If G is a finite abelian group and p is a prime such that p^n divides order of G, then prove that G has a subgroup of order p^n Homework Equations Theorem of Finite Abelian Groups: Every finite abelian group G is a direct sum of cyclic groups, each of prime power...

To prove that : f : U_{s} (st) \rightarrow U(t) is an onto map. Note that Us(st)= {x \in U(st): x= 1 (mod s)} Let x \inU(t) then (x, t)=1 and 1<x< t How to proceed beyond point ?

I'm trying to understand why a Lie group always has a non-vanishing vector field. I know that one can somehow generate one by taking a vector from the Lie algebra and "moving it around" using the group operations as a mapping, but the nature of this map eludes me.

For discrete groups, we can easily find the decomposition of the direct product of irreducible representations with the help of the character table. All we need to do is multiply the characters of the irreducible representations to get the characters of the direct product representation and then...

Homework Statement My question is about the rules behind the method of finding the solution not necessarily the method itself. (I am prepping for the GRE subject) How many mutually nonisomorphic Abelian groups of order 50? The Attempt at a Solution so, if understand this...

Hi, All: Please forgive my ignorance here: let G be an infinite group, and let H be a subgroup of G of finite index . Does H necessarily have torsion? I can see if , e.g., G was Abelian with G=Z^n (+) Z/m , then , say, would have subgroups of finite index, but I can't tell if...

Homework Statement Let p, q be distinct primes s.t. p \equiv 1 (mod q). Prove that there exists a non-Abelian group of order pq and calculate the character table. Homework Equations Semi-Direct Product: Let H = < Y | S > and N = < X | R > be groups and let \phi : H \rightarrow Aut...

Homework Statement Just want to make something clear. Are all cyclic groups that have the same number of elements isomorphic to each other. The Attempt at a Solution I think yes because theirs is a one-to-one correspondence and the groups are cyclic which means they have generators.

Hi all, Sorry, I'm not quite sure that I've posted this question in the proper place, but I figured field theory matches best with lie groups in this context. Anyway, my question has to do with the relationship between the fundamental forces (electromagnetism, weak, and strong) and their...

Well, apparently, I'm not too clear on a few things. For #4, what is <[8]> in the group Z18? What does the <> mean around the congruence class? Is my work correct? http://i111.photobucket.com/albums/n149/camarolt4z28/IMG_20110723_161645.jpg?t=1311456158...

Homework Statement Does Euler's totient function tell me how many elements are in my group? And once I know how many elements are in my group. the generators are the ones that are relatively prime with the number of element in my group. Are my statements correct.

Does anyone know of a good book to read about groups for algebra? I've head that a good book was "A Book of Abstract Algebra" by Charles R. Pinter. And I am just learning about groups so it should be basic.

The .pdf can be ignored. Let A + B = (A - B) U (B - A) also known as the symmetric difference. 1. Look for the identity and let e be the identity element A + e = A (A - e) U (e - A) = A Now there are two cases: 1. (A - e) = A This equation can be interpreted as removing from A all elements...

Is there an "easy" method to finding subgroups of finitely generated abelian groups using the First Isomorphism Theorem? I seem to remember something like this but I can't quite get it. For example, the subgroups of G=Z_2\oplus Z are easy...you only have 0\oplus nZ and Z_2\oplus nZ for n\geq...

Cyber-bullying is a hot topic these days. *Many social networking sites have policies directed toward preventing cyber-bullying. *Facebook is one of those sites, having been the venue for several high profile bullying incidents recently*. *Facebook’s policy*directs victims of cyber-bullying to …...

Homework Statement Show that if G is any group of permutations, then the set of all even permutations in G forms a subgroup of G. I am not sure where to start - I know there is a proposition that states this to be true, but I know that is not enough to prove this statement.

So say you have a presentation matrix A for a module, and you diagonalise it and you get something like diag[1,5] well you can interpret that as A can be broken down as the direct sum of 1+Z/5Z. That is the trivial group of just the identity plus cyclic 5. What if your module is defined over...

Homework Statement For n>1, show that the subgroup H of S_n (the symmetric group on n-letters) consisting of permutations that fix 1 is isomorphic to S_{n-1} . Prove that there are no proper subgroups of S_n that properly contain H.The Attempt at a Solution The first part is fairly...

Homework Statement Let G be a finite cyclic group and \ell(G) be the composition length of G (that is, the length of a maximal composition series for G). Compute \ell(G) in terms of |G|. Extend this to all finite solvable groups. The Attempt at a Solution Decompose |G| into its prime...

I wanted to make clear just a quick technical thing. If G is a group, N is a normal subgroup, and \phi_g \in \text{Inn}(G), \phi_g(h) = g h g^{-1} then \phi_g is an automorphism of N, right? However, is it the case that we cannot say that \phi_g is an inner automorphism, since we are not...

Homework Statement Let G be a finite abelian group, and assume that |G| is odd. Show that every element of G is a square. The Attempt at a Solution So we want to show that \forall g \in G, \exists h \in G, g = h^2 . Let g \in G be arbitary, and consider the subgroup generated by g, denoted...

Hi, so I didn't see exactly where group theory stuff goes...but since Lie groups are also manifolds, then I guess I can ask this here? If there's a better section, please move it. I just have a simple question regarding the definition of a Lie group. My book defines it as a group which is...

According to wiki: "a non-abelian group, also sometimes called a non-commutative group, is a group (G, * ) in which there are at least two elements a and b of G such that a * b ≠ b * a." I thought in order to be an abelean group, 5 axioms must be satisfied. If one of them is not satisfied it...

I need some material on the properties and relationships between classical groups. I was using Robert Gilmore's "Lie Groups, Lie Algebras and Some of their Aplications", but it barely covers it (section 2.iv). Does someone know about a book or any lecture notes that could be used to...

Homework Statement Let p=2^(k)+1 , in which k is a positive integer, be a prime number. Let G be the group of integers 1, 2, ... , p-1 under multiplication defined modulo p. By first considering the elements 2^1, 2^2, ... , 2^k and then the elements 2^(k+1), 2^(k+2), ... show that the...

Hi there PF What are groups, and what are they used for in physics? For example if you look at QED, http://en.wikipedia.org/wiki/Quantum...cs#Mathematics , it is said here that QED is a abelian gauge theory with symmetry group U(1). What is this symmetry group, and what is it used for when...

Homework Statement Determine how many non-isomorphic (and which) abelian groups there are of order 54. Determine which of these groups the factor group Z6 x Z18 / <(3,0)> is isomorphic to. Homework Equations The Attempt at a Solution Fundamental theorem for abelian groups...

Let A and B be nxn matrices which generate a group under matrix multiplication. Assume A and B are not orthogonal. How can I determine an nxn matrix X such that X-1AX and X-1BX are both orthogonal matrices? Is it possible to do this without any special knowledge of the group in question?

Hi everyone, :smile: I was wondering why they only use 5th and 3rd groups elements to create n-type and p-type semiconductors respectively. Couldn't we mix 6th or 2nd groups' elements instead to make the semiconductors? What would happen if we do this? Perhaps elements of those groups will...

Hi, i want to determine the homology groups of S^2 using a Morse function with at least 3 critical points. Is there anyone to helpm me in this way. I know how i can describe the homology of sphere in usual way. That is by using a Morse function with 2 critical points( index 0 and 2)...

*edit* problem solved.

http://arxiv.org/abs/1104.1106 Lecture Notes in Lie Groups Vladimir G. Ivancevic, Tijana T. Ivancevic (Submitted on 6 Apr 2011) These lecture notes in Lie Groups are designed for a 1--semester third year or graduate course in mathematics, physics, engineering, chemistry or biology. This...

If I had a nucleophile, say the conjugate base of benzene, and a molecule with a bromide, and a primary alcohol. Would the nucleophile attack the carbon bound to the bromide, or the proton on the alcohol? If it's hard to follow I can try to make a picture. Basically I want to add Isobutanol...

Integrating Floor Function Homework Statement If \lfloor{x}\rfloor denotes the greatest integer not exceeding x, then \int_{0}^{\infty}\lfloor{x}\rfloor e^{-x}dx= Homework Equations none The Attempt at a Solution I don't know how to start this problem. At first, I tried bringing...

Hi, All: I am given two groups G,G', and their respective presentations: G=<g1,..,gn| R1,..,Rm> ; G'=<g1,..,gn| R1,..,Rm, R_(m+1),...,Rj > i.e., every relation in G is a relation in G', and they both have the same generating set. Does this relation (as a...

Let G group and N subgroup normal from G if b \in{G} and p is prime number then (Nb)^p=Nb^p, Please help me with steps to this proof.

So I know that every smooth manifold can be endowed with a Riemannian structure. In particular though, I'm wondering if there is a natural structure for the unitary and special unitary groups. I often see people using the "trace/Hilbert-Schmidt" inner product on these spaces, where \langle X...

1. Suppose that Ø:Z(50)→Z(50) is an automorphism with Ø(11)=13. Determine a formula for Ø(x). this is the problem I am getting, its chapter 6 problem 20 in Gallian's Abstract Algebra latest edition (you can find it on googlebooks) Am i wrong in thinking there's something wrong with the problem...

What are some simple groups that have non-normal subgroups? The only example I can think of is the alternating group for n > 4.

Homework Statement prove that R under addition is not isomorphic to R^*, the group of non zero real numbers under multiplication. Homework Equations The Attempt at a Solution \varphi:(R,+) \rightarrow (R^* , .) let \varphi(x) = -x then \varphi(x+y) = -(x+y) = -x-y \neq...

When a group has a prime order, does that mean that it is always isomorphic to the cyclic group of the same order? I just am a little confused and need some clarification on this matter. Thanks

Is there a way to "remove" functional groups? I see a lot of pages online that show how you can change them, but not how to completely remove one from a molecule. Is it even possible?

Homework Statement [PLAIN]http://img689.imageshack.us/img689/3047/directproduct.png < denotes a subgroup. \triangleleft denotes a normal subgroup. The Attempt at a Solution Have I done (a) correctly? 0 \in A so A \neq \emptyset If a=x+ix and b=y+iy then ab^{-1} = x-y + ix...

Hello! Is someone aware if there are lecture notes about Lie Groups from a physics course? I would to study an exposition of this subject made by a physicist. Thank you in advance!

I'm new here and I have checked the FAQs. I'm not sure if this question has been posted before. This may actually be a silly question. Why do we study Lorentz and Poincare Groups? I have studied a bit of the theory but was wondering what exactly are we talking about when we study the...

Hi everyone, I am studying the feasibility of using the collision of two beams to obtain nuclear fusion energy. I would like you to recommend me some serious and intersting articles about this. Is there any experiment that does fusion by beam collitions already? Im open to hear about any...

Hello, I want to improve my posture. I tend to slouch and walk with my head jutted forward. I hear that muscle strength affects posture. Is this true? If so, which group of muscles should I strengthen to improve my posture? If possible, can you also supply some reliable links for correct...

According to wikipedia: "A crystallographic point group is a set of symmetry operations, like rotations or reflections, that leave a point fixed while moving each atom of the crystal to the position of an atom of the same kind." "The space groups in three dimensions are made from combinations...