Groups Definition and 867 Threads

[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...

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 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...

This particular problem has me stumped. I've looked through all the equations given in the book, but none seem to fit this problem correctly. Instead, I tried solving it logically, but I'm stuck on what seems to be the last step of every part. Any help is appreciated! Homework Statement...

Homework Statement Let A be a normal subgroup of a group G, with A cyclic and G/A nonabelian simple. Prove that Z(G)= AHomework Equations Z(G) = A <=> CG(G) = A = {a in G: ag = ga for all g in G} My professor's hint was "what is G/CG(A)?" The Attempt at a Solution A is cyclic => A is...

Homework Statement Does every group whose order is a power of a prime p contain an element of order p? Homework Equations The Attempt at a Solution I know it certainly can contain an element of order p. I also feel that |G|=|H|[G:H] might be useful. Any help is appreciated!

I know full well the proof using Lagrange's thm. But is there a direct way to do this without using the fact that the order of an element divides the order of the group? I was thinking there might be a way to set up an isomorphism directly between G and Z/pZ. Clearly all non-zero elements...

This is what you get if you compile numerous attractive females and then take all their features to come up with archetypal faces that is supposed to best represent the group they belong to.

Homework Statement [PLAIN]http://img541.imageshack.us/img541/9880/34132542.gif The Attempt at a Solution For part (a), I think since the order of an element g is the smallest integer n such that gn=e, we will have: 8n mod 65 = 1 => n=4 64n mod 65 = 1 => n=2 14n mod 65 =1 =>...

I'm currently learning Lie groups/algebras and I am trying to find the infinitesimal generators of the special orthogonal group SO(n). It is the n-dimensions that throws me off. I know that the answer is n(n-1)/2 generators of the form, X_{\rho,\sigma}=-i\left(x_{\rho}\frac{\partial}{\partial...

Homework Statement [PLAIN]http://img641.imageshack.us/img641/8448/63794724.gif The Attempt at a Solution Firstly, how do I list the elements of H? According to Lagrange's theorem if H is a subgroup of G then the number of distinct left (right) cosets of H in G is |G|\|H|. So I...

Homework Statement Let [G,+,0] be a non-abelian group with a binary operation + and a zero element 0 . To prove that if both the zero element and the inverse element act on the same side, then they both act the other way around, that is: If \forall a \in G , a + 0 = a , and a + (-a)...

Homework Statement Suppose (G, \circ) is a group. Define an operation \star on G by a \star b = b \circ a for all a,b \in G. Show that (G,\star) is a group. The Attempt at a Solution So, I have to show that (G,\star) satisfies the associativity, identity, inverse and closure...

Suppose \mathfrak{g} and \mathfrak{h} are some Lie algebras, and G=\exp(\mathfrak{g}) and H=\exp(\mathfrak{h}) are Lie groups. If \phi:\mathfrak{g}\to\mathfrak{h} is a Lie algebra homomorphism, and if \Phi is defined as follows: \Phi:G\to H,\quad \Phi(\exp(A))=\exp(\phi(A))...

Homework Statement I am trying to solve a question from Abstract Algebra by Hernstein. Can anyone give me hint regarding the following: Show that a group can not be written as union of 2 (proper) subgroups although it is possible to express it as union of 3 subgroups? Thanks...

Is there a relationship between the homotopy groups of a pair (X,A) and of the quotient X/A ? It feels like they should be equal under mild hypothesis. More precisely, I am interested in the case where X is a smooth manifold and A a submanifold. Thx

Why are there so few research groups that focus on relativity and gravity in the US? It may be only my perception, but it seems that studying general relativity is rather "standard" in the UK. The schools I've looked at all seem to have a few classes on relativity at the undergraduate level...

Homework Statement This is a problem from a chapter entitled "Permutation Groups" of an abstract algebra text. 1. Let α = ( 1 3 5 7 ) and β = (2 4 8) o (1 3 6) ∈ S8 Find α o β o α-1. 2. Let α = ( 1 3) o (5 8) and β = (2 3 6 7) ∈ S8 Find α o β o α-1. Homework Equations Sn is the set...

Hi, Everyone: LetB: E-->X be a line bundle, with scructure group G and X has a CW -decomposition. I am trying to understand why/how, if the structure group G of B is connected, then any trivialization over the 0-skeleton of X can be extended to a trivialization of the...

If the Cayley table, of a finite set G, is a latin square (that is, any element g appears once and only once in a given row or column) does it follow G is a group? I know the converse is true, and it seems reasonable that this is true. Since the array will be of size |G|x|G|, inverses exist and...

Hi. I'm looking for an introductory book on Lie Groups and Lie Algebras and their applications in physics. Preferably the kind of book that emphasizes understanding, applications and examples, rather than proofs. Any suggestions? Edit: Please move this to Science Book Discussion.

I would like to know how much weightage is taken into account for activities outside class, when applying for a university scholarship... Are sports activities given preference over, for example, science interest groups such as astronomy? What advantage does multiple club memberships hold? Is...

Alright, I understand that there are redundant degrees of freedom in the Lagrangian, and because transformations between these possible "gauges" can be parametrized by a continuous variable, we can form a Lie Group. What I am not so firm upon is how Lie Algebras, specifically, the Lie Algebra...

Hi Everyone, Back in college i informally learned what i would call point group theory. Most of it never touched on continuous transformations. When I learned it back then it was all pretty straight forward. Recently I have been trying to learn about Lee groups (to understand symmetries in...

A while ago I heard the following two facts about semi-simple Lie groups (though I have a feeling they may have to be restricted to connected semi-simple Lie groups): 1. That semi-simple Lie groups are classified by their weight (and co-weight) and root (and co-root) lattices; 2. That all of...

Hi All, I am trying to learn about Linear Algebraic Groups. I am using the book by James Humpreys. I love the subject, but I find it a bit, say, not so beginner-friendly. My goal is not being spoon-fed, but I am very interested in finding a source(s) whereby one is able to go through some...