Groups Definition and 906 Threads

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...

Homework Statement prove that every finite group has exponent that divides the order of the group Homework Equations The Attempt at a Solution Given G is a finite group and x \in G. Suppose x has order m, then <x> = {e,x,x^2...x^(m-1)} and so \left|<x>\right| = m so by...

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...

[b]1. Let G be a group containing subgroups H and K such that we can find an element h e H-K an an element k e K - H. Prove that h o k is not a subgroup of H U K. Deduce that H U K is not a subgroup of G. I have proved that h o k is not in H U K but I don't know how to deduce that H U K is...

Homework Statement List the elements of the cyclic subgroup of S_6 generated by f = \left(\begin{array}{llllll} 1 & 2 & 3 & 4 & 5 & 6\\ 2 & 3 & 4 & 1 & 6 & 5\\ \end{array}\right)Homework Equations The Attempt at a Solution I really do not understand what the elements of a permutation really...

Why do people try against all odds to make SU(2) isometric with SO(3) when it's clear from the definition that it's actually isometric with SO(4). Either way you've got 4 variables and the same constraint between them. It's interesting to see all the dodgy tricks that go into this deception...

Homework Statement The problem asks me to determine if the matrix [p -q ## q p] is a field with addition and multiplication. However, that is not my question. My question is: How is proving a set is a field different from proving a set is a non-abelian group (under addition then separately...

Let W be the mapping cone of the map f: S^{1} \rightarrow S^{1} defined by f(z)= z^{p} How do you compute the homology groups of W? What about the homology groups of the Universal covering of W? I know that the mapping cone C_f of f:X\rightarrow Y, is defined to be the quotient of the...

Homework Statement I need to prove that any isomorphism between two cyclic groups maps every generator to a generator. 2. The attempt at a solution Here what I have so far: Let G be a cyclic group with x as a generator and let G' be isomorphic to G. There is some isomorphism phi: G...

Homework Statement Prove that S_n and D_n for n>=3 are non-cyclic and non-abelian. Homework Equations I get that I need to show that two elements from each group do not commute and that there is not a single generator to produce the groups... I am just unsure of how to do this...

This is not a homework question, just a general question. Let G be a non-abelian finite group, S < G a non-normal proper subgroup of index v >= 2, and G/S the set of v right cosets S_1 = S, S_2, ..., S_v, of S in G. We know there is a naturally defined right-multiplication action G x G/S -->...

What do I do if I have a hard time contributing in labs? Everytime my professor tells us to work together on a lab, I always walk away feeling like it was a wasted learning experience because I couldn't contribute. Additionally, labs are hard for me because there seems to be a huge mentality of...

Homework Statement 1. Let G and H be finite groups and let a: G → H be a group homomorphism. Show that if |G| is a prime, then a is either one-to-one or the trivial homomorphism. 2. Let G and H be finite groups and let a : G → H be a group homomorphism. Show that if |H| is a prime, then a...

Hello all, We're going to be doing a fundraising competition, and I'm not sure as to the fairness. We will determine the winning group by just comparing the average amount raised by each group. Is this mathematically fair, or is their a better way? For some reason I can't help but think...

Prove that if G is a simple group of order 60, then G is isomorphic to A5. So far, I have shown that there is a subgroup of G with index 5. I know that with this information I should be able to show that G is isomorphic to A5, but I'm not sure how...

Homework Statement Let G be a finite cyclic group of order n. If d is a positive divisor of n, prove that the equation x^d=e has d distinct solutions Homework Equations n=dk for some k order(G)=nThe Attempt at a Solution solved it: <g^k>={g^k, g^2k,...,g^dk=e} and for all x in <g^k> x^d=e...

Homework Statement The question is to show that the for symmetric groups, Sn with n>=3, the only permutation that is commutative is the identity permutation. Homework Equations I didn't know if it was necessary but this equates to saying the center is the trivial group. The Attempt at...

Using this reference as an example let's name the double bonded carbon attached to the methyl as C1. In my textbook it says that C1 has two alkyl groups on it. The CH3 is one alkyl group and the carbon to the left of C1 is the other alkyl group (let's call this C2). I'm confused, why is C2...

I was wondering, if we want to define a morphism from \mathbb{Z}2006 to, let's say \mathbb{Z}3008. Obviously, all linear functions like $ x \rightarrow a\cdot x$ will do, but are there any other functions which can result in a homomorphism?

Basic question, but nevertheless. In a non-Abelian gauge theory, the fermions transform in the fundamental representation, i.e. doublets for SU(2), triplets for SU(3), while the gauge fields transform in the adjoint representation, which can be taken straight from the structure constants of...

I'm having trouble understanding the product of groups and their max normal subgroups. What does it mean to be a max subgroup? How do I take the product of two groups? How do I do it for something like S7 X Z7 ?

I've been doing some work with simple Abelian groups and their generators, and I feel like there is a way to classify all of them, is this possible?

Why do groups descibe symmetry? Why does a set which has an identity and inverse element, is closed under an abstract multplication operation and whose member obey the association law, captures symmetry? Why is that? thanks

Homework Statement Find all groups of order 9, order 10, order 11. Homework Equations None The Attempt at a Solution We have already done an example in class of groups of order 4 and of order 2,3,5, or 7. So I'm going to base my proofs on the example of groups of order 4 except...

Homework Statement A cable supports a cylindrical beam at point B. For the given loading conditions, what type of supports, applied at point A, would improperly constrain the beam, would constrain the beam and make the beam statically determinate, and would constrain the beam and make the beam...

Homework Statement Find all of the subgroups of Z3 x Z3 Homework Equations Z3 x Z3 is isomorphic to Z9 The Attempt at a Solution x = (0,1,2,3,4,5,6,7,8) <x0> or just <0> = {0} <1> = {identity} <2> = {0,2,4,6} also wasn't sure if I did this one correctly x o x for x2 <3> =...

Out of curiosity, how does the product rule work in Lie groups? I ended up needing it because I approached a problem incorrectly and then saw that the product rule was unnecessary, but it seems to create a strange scenario. For example: Consider a Lie group G and two smooth curves \gamma_1...

How would I find the amount of energy that is stored in a particular functional group? I know things like Azide, Nitro, Alkynyl, Cyanides, etc. would all store a lot of energy.

Homework Statement If G is a group with operation * and \alpha,\beta\in G, then \beta\ast\alpha\ast\beta^{-1} is called a conjugate of G. Compute the number of conjugates of each 3-cycle in S_{n} (n\geq3). Homework Equations The Attempt at a Solution For any group S_{n} there...

How do we know that the cartesian product of any two groups is also a group using the axioms of group theory?

Homework Statement The pKa values for the three ionizable groups on tyrosine are pKa (--COOH)=2.2, pKa (--NH3+)=9.11, and pKa (R)=10.07. In which pH ranges will this amino acid have the greatest buffering capacity? A) At all pH's between 2.2 and 10.07 B) At pH's near 7.1 C) At pH's between...

Hello, how do symmetry groups in the Euclidean space differ from the symmetry groups in the hyperbolic space (in the Poincaré disk) ? I've been told that in the hyperbolic case one has at disposal a richer "vocabulary" to describe symmetries, but I don't see how, and maybe I misunderstood...