Groups Definition and 906 Threads

Homework Statement Hi all, i have to identify 5 samples (1,2,3 were solids, 4,5 were liquids) by classifying them as 1) Aliphatic or aromatic and 2) Carboxylic acid, amine (primary, secondary, tertiary) or ammonium carboxylate We did a burn test on the solids, tested solubility in water...

Hi! I'm looking for a complete derivation of space groups as Schoenflies did over 100 years ago... Does anybody know where I can find this paper (in English or in German at least): A. Schoenflies Kristallsysteme und Kristallstruktur, Leipzig, 1891 or maybe a book where the whole process of...

Homework Statement Show that the set of all 2x2 matrices of rank 1 is a submanifold of R^4 Homework Equations The Attempt at a Solution The hint in the book was to show that the determinant function is a submersion on the manifold of nonzero 2x2 matrix M(2) - 0. This is easy to...

Z(G) = { x in G : xg=gx for all g in G } (center of a group G) C(g) = { x in G : xg=gx } (centralizer of g in G) I have to show both are subgroups, but what's the difference in the methods? To me the first set is saying all the elements x1, x2,... in G when composed with every element in g...

D(g) is a representaiton of a group denoted by g. The representaion is recucible if it has an invariant subspace, which means that the action of any D(g) on any vector in the subspace is still in the subspace. In terms of a projection operator P onto the subspace this condition can be written...

Homework Statement I'll be delighted to receive some guidance in the following questions: 1. Let G1,G2 be simple groups. Prove that every normal non-trivial subgroup of G= G1 x G2 is isomorphic to G1 or to G2... 2. Prove that every group of order p^2 * q where p,q are primes is...

Hi there, I 'm currently reading topics relating to type IIB superstring theory. One of the things I am always confused with is Groups. I looked on various websites including Wikipedia but I still haven't quite got it. Could anyone please give me a nice introduction about Groups? What are...

Homework Statement Show that every element of the quotient group \mathbb{Q}/\mathbb{Z} has finite order but that only the identity element of \mathbb{R}/\mathbb{Q} has finite order. The Attempt at a Solution The first part of the question I solved. Since each element of...

As i understand it, the commutation rules for the quantum angular momentum operator in x, y, and z (e.g. Lz = x dy - ydx and all cyclic permutations) are the same as the lie algebras for O3 and SU2. I'm not entirely clear on what the implications of this are. So I can think of Lz as generating...

Hey guys, I've been doing a lot of reading on quantum mechanics lately and realized immediately that i am not going to get far without first understanding the meanings of lie groups, SU groups etc. Now I've loked at wiki but unfortunately wiki is not a very good tool for learning math, it's more...

I am trying to show show that there is no homomorphism from Zp1 to Zp2. if p1 and p2 are different prime numbers. (Zp1 and Zp2 represent cyclic groups with addition mod p1 and p2 respectively). I am not sure how to do this but here are some thoughts; For there to be a homomorphism we...

Hi, I am trying to work through some problems I have found in preparation for a test. I am running out of time, and getting somewhat confused, however... Homework Statement a) Show that every irreducible representation of SO(2) has the form \Gamma\ \left( \begin{array}{ccc} cos(\theta) &...

Homework Statement How can irreducible representations of O(3) and SO(3) be determined from the irreducible representations of SU(2)? The Attempt at a Solution I believe there is a two-one homomorphic mapping from SU(2) to SO(3); is that enough for some shared representations? If I had...

Homework Statement Let Cn be a cyclic group of order n. A. How many sub-groups of order 4 there are in C2xC4... explain. B. How many sub-groups of order p there are in CpxCpxC(p^2) when p is a prime? explain. C. Prove that if H is cyclic of order 8 then Aut(H) is a non-cyclic group. WHAT is...

I am looking for results which provides the homology and homotopy groups from some property of the space. For instance, if a space X is contractible then H_0(X)=\mathbb{Z} and H_n(X)=0 if n\neq 0. Another example is the Eilenberg MacLane spaces K(\pi,n) where \pi_n(K(\pi,n))=\pi and...

Homework Statement This question is about sylow-p groups of Sp. I've proved these parts of the question: A. Each sylow p-sbgrp is from order p and there are (p-2)! p-sylow sbgrps of Sp. B. (p-1)! = -1 (mod p ) [Wilson Theorem] I need your help in these two : C. 1) Let G be a group...

Homework Statement Let P be a p-sylow sbgrp of a finite group G. N(P) will be the normalizer of P in G. The quotient group N(P)/P is cyclic from order n. PROVE that there is an element a in N(P) from order n and that every element such as a represnts a generator of the quotient group...

Homework Statement a) Classify all groups of order 51. b) Classify all groups of order 39. Homework Equations Sylow theorems. The Attempt at a Solution a) C51 b) Z3 X Z13 and Z13 x Z3, the semi-direct product with presentation <a,b|a13=b3=1, ab=a3 > Are these all of the...

Homework Statement What functional groups are present based on the compound's names? A. Methyl Hydroxybenzoate B. 2-Hydroxypropanoic acidHomework Equations The Attempt at a Solution We've learned about the basic Hydrocarbon derivatives in class, but only dealing with problems like...

Homework Statement The problem is: Let G be a group of order 12 ( o(G)=12). Let's assume that G has a normal sub-group of order 3 and let a be her generator ( <a>=G ). In the previous parts of the questions I've proved that: 1. a has 2 different conjucates in G and o(N(a))=6 or...

Homework Statement If the order of G is p^2 and p is prime, then show that G is either cyclic or isomorphic to ZpXZp... Homework Equations The Attempt at a Solution Any hints here will help!

Benjamin Crowell writes here, "The discontinuous transformations of spatial reflection and time reversal are not included in the definition of the Poincaré group, although they do preserve inner products." http://www.lightandmatter.com/html_books/genrel/ch02/ch02.html So, if I've...

Homework Statement let p be a prime number and let G be a group with order p^2. the task is to show that G is either cyclic or isomorphic to Zp X Zp. a. let a, not equal to the identity,be an element in G and A=<a>. What's the order of A. b. consider the cosets of A: G/A={A,g2A,...gnA}...

Hello Lately, I have been studying some group theory. On my own, I should add, so I don't really have any professor (or other knowledgeable person for that matter) to ask when a problem arises; which is why I am here. I had set out to find all small groups (up to order 30 or something), up to...

Hi guys just a quick question on how I would go about showing the units of Z21 is isomorphic toC2*C6(cyclic groups).I have done out the multiplicative table but they seem to be different to me. What else can I do?

H,K are normal subgroups of a (finite) group G, and K is also normal in H. If G/K and G/H are simple, does it follow that H=K? I'm almost convinced it does, but I'm having trouble proving it. I mean, the cosets of H partition G and the cosets of K partition G in the same way and on top of that...

HELP! Find all abelian groups (up to isomorphism)! I am really confused on this topic. can you give me an example and explain how you found, pleaseee! for example, when i find abelian group of order 20; |G|=20 i will find all factors and write all of them, Z_20 (Z_10) * (Z_2) (Z_5)*...

Homework Statement Write the multiplication table of C_{6}/C_{3} and identify it as a familiar group. Homework Equations The Attempt at a Solution C_{6}={1,\omega,\omega^2,\omega^3,\omega^4,\omega^5} C3={1,\omega,\omega^2} The cosets are C3 and \omega^3C3 I just need help...

Homework Statement Show that every abelian group of order 70 is cyclic.Homework Equations Cannot use the Fundamental Theorem of Finite Abelian Groups.The Attempt at a Solution I've tried to prove the contrapositive and suppose that it is not cyclic then it cannot be abelian. But that has lead...

Homework Statement Let G be a group, and let H be a normal subgroup of G. Must show that every subgroup K' of the factor group G/H has the form K'=K/H, where K is a subgroup of G that contains H. Homework Equations I don't see how to get started. The Attempt at a Solution I wrote...

Homework Statement Prove that the group of order 175 is abelian. Homework Equations The Attempt at a Solution |G|=175=527. Using the Sylow theorems it can be determined that G has only one Sylow 2-subgroup of order 25 called it H and only one Sylow 7-subgroups called it K. Thus...

Let G be a Lie group. Show that there exists a unique affine connection such that \nabla X=0 for all left invariant vector fields. Show that this connection is torsion free iff the Lie algebra is Abelian.

Hello everyone. I need someone to explain a concept to me. I'm confused about how a type of particle can be a representation of a lie group. For example, I read that particles with half-integer spin j are a representation of the group SU(2), or that particles with charge q are a...

Homework Statement In the group <Z\stackrel{X}{13}> of nonzero classes modulo 13 under multiplication, find the subgroup generated by \overline{3} and \overline{10}Homework Equations The Attempt at a Solution Doesnt 3 generate {3,6,9,12} and 10 generate {2,5,10}?

I've been asked to match some galois groups with structures like: Z_2, Z_3, Z_2 X Z_2 ...etc. And I'm very much lost. I know the galois group for a field extension L:K is the set of isomorphism that fix F. These form a group with function composition. OK. But how do I find these isomorphism...

V is a vectoric space. W_1,W_2\subseteq V\\ W_1\nsubseteq W_2\\ W_2\nsubseteq W_1\\ prove that W_1 \cup W_2 is not a vectoric subspace of V. i don't ave the shread of idea on how to tackle it i only know to prove that some stuff is subspace but constant mutiplication and by sum of two...

Homework Statement Recall that given a group G, we defined A(G) to be the set of all isomorphisms from G to itself; you proved that A(G) is a group under composition. (a) Prove that A(Zn) is isomorphic to Zn/{0} (b) Prove that A(Z) is isomorphic to Z2 Homework Equations The Attempt at a Solution

Homework Statement (a) How many distinct cyclic subgroups of D6 are there? Write them all down explicitly. (Here, D6 is the dihedral group of order 12, i.e. it is the group of symmetries of the regular hexagon.) (b) Exhibit a proper subgroup of D6 which is not cyclic. Homework...

Hello, I wonder if somebody could point me to a book (preferably), or paper, link, etc. which explores the relations between number theory and group theory. For example, I am (more or less) following Burton's "Elementary Number Theory" and there is no mention of groups. I also have the...

I was trying to prove the statement "If (2^n)-1 is prime then n is prime". I've already seen the proof using factorisation of the difference of integers and getting a contradiction, but I was trying to use groups instead. I was wondering if it's possible, since I keep getting stuck. So far I've...

Homework Statement suppose that G is a group in which every non-identity element has order two. show that G is commutative. Homework Equations The Attempt at a Solution IS THIS CORRECT? ab=a[(ab)^2]b=(a^2)(ba)(b^2)=ba

Homework Statement suppose that G is a group in which every non-identity element has order two. Show that G is commutative. Homework Equations The Attempt at a Solution Is my answer correct? Suppose that a,b and ab all have order two. we will show that a and b commute. By...

Homework Statement Suppose that G is a group in which every non-identity element has order two. show that G is commutative. Homework Equations The Attempt at a Solution DOES THIS ANSWER THE QUESTION?: Notice first that x2 = 1 is equivalent to x = x−1. Since every element of G...

I need help here: Suppose that G is a group in which every non-identity element has order two. Show that G is commutative. Also, Consider Zn = {0,1,...,n-1} a. show that an element k is a generator of Zn if and only if k and n are relatively prime. b. Is every...

I need help here: Suppose that G is a group in which every non-identity element has order two. Show that G is commutative. Also, Consider Zn = {0,1,...,n-1} a. show that an element k is a generator of Zn if and only if k and n are relatively prime. b. Is every...

Given a group G. J = {\phi: G -> G: \phi is an isormophism}. Prove J is a group (not a subgroup!). Well we know the operation is function composition. To demonstrate J a group we need to satisfy four properties: (i) Identity: (I'm not sure what to do with this) (ii) Inverses: Suppose a...

Hello, I am quite new here, as my number of posts might indicate. Thus I am not really sure whether or not this question should be posted here or somewhere else. It is not a homework, but neither is it a question that could not be a homework. However, here we go. I have, during a course in...

Hi, I am trying to compute homology groups of a space while some of the homotopy groups are known..what is the best way to do that. I hope that you can help. Thanks, Sandra

I'm studying the reaction mechanisms for carboxylic acid and its derivatives and here it says whether a compound with a C=O bond undergoes nucleophilic addition (as in aldehydes and ketones) or nucleophilic acyl substitution depends on the relative basicities of the substituent group. For...

I think I understand why a vector field must have a unique set of integral curves, but I don't see why they must define a one-parameter group of diffeomorphisms. Let X be a vector field on a manifold M, and p a point in M. A smooth curve C through p is said to be an integral curve of X if...