What is Groups: Definition and 906 Discussions

Google Groups is a service from Google that provides discussion groups for people sharing common interests. The Groups service also provides a gateway to Usenet newsgroups via a shared user interface.
Google Groups became operational in February 2001, following Google's acquisition of Deja's Usenet archive. Deja News had been operational since March 1995.
Google Groups allows any user to freely conduct and access threaded discussions, via either a web interface or e-mail. There are at least two kinds of discussion group. The first kind are forums specific to Google Groups, which act more like mailing lists. The second kind are Usenet groups, accessible by NNTP, for which Google Groups acts as gateway and unofficial archive. The Google Groups archive of Usenet newsgroup postings dates back to 1981. Through the Google Groups user interface, users can read and post to Usenet groups.In addition to accessing Google and Usenet groups, registered users can also set up mailing list archives for e-mail lists that are hosted elsewhere.

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  1. W

    Organic Chem - Identifying Functional Groups

    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...
  2. P

    Space groups derivation (Schoenflies)

    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...
  3. G

    What is the Submanifold of Rank 1 2x2 Matrices in R^4?

    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...
  4. F

    I can't tell the difference between these groups :/

    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...
  5. B

    Questions about a projection operator in the representation theoy of groups

    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...
  6. T

    Abstract Algebra - Normal groups

    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...
  7. W

    Understanding Groups in Type IIB Superstring Theory

    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...
  8. 3

    Finite Order in Quotient Groups: Q/Z and R/Q

    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...
  9. S

    Lie groups and angular momentum

    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...
  10. T

    Getting started with symmetry groups

    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...
  11. V

    Question on cyclic groups (addition mod n)

    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...
  12. R

    Reducing infinite representations (groups)

    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) &...
  13. R

    The groups O(3), SO(3) and SU(2)

    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...
  14. T

    Abstract Alg.-Abelian groups presentation

    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...
  15. H

    Homology and Homotopy groups from properties

    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...
  16. T

    Verifying Sylow-p Groups of Sp: A Homework Exercise

    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...
  17. T

    What is the significance of p-sylow groups in finite groups?

    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...
  18. V

    Groups of order 51 and 39 (Sylow theorems).

    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...
  19. D

    Determining Functional groups - carboxylic acid, ester etc

    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...
  20. T

    Proving the Isomorphism between Group G and A4: A Scientist's Perspective

    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...
  21. H

    Proving Group of Order p^2 is Cyclic or ZpXZp

    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!
  22. Rasalhague

    Poincaré and Euclidean groups

    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...
  23. H

    Groups of order p^2 where p is prime

    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}...
  24. J

    Finding groups by semidirect products

    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...
  25. R

    Isomorphic Groups: Z21 & C2*C6

    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?
  26. B

    Normal and Simple Subgroups in Finite Group G: Proof of Equality for K and H

    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...
  27. N

    HELP Find all abelian groups (up to isomorphism)

    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)*...
  28. G

    Mutliplication table of quotient groups

    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...
  29. R

    Abelian groups of order 70 are cyclic

    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...
  30. L

    Proving Normal Subgroups in Factor Groups: G and K

    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...
  31. L

    Is the group of order 175 abelian?

    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...
  32. Z

    Lie groups, Affine Connections

    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.
  33. T

    Particles as representations of groups

    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...
  34. H

    Generating Subgroups in <Z\stackrel{X}{13}> Modulo 13 Under Multiplication

    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}?
  35. F

    At a total loss with these Galois groups.

    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...
  36. L

    Proving that the Union of Two Non-Intersecting Subspaces is Not a Subspace

    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...
  37. M

    Isomorphism of A(Zn) and Zn/{0}: A Proof

    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
  38. M

    Distinct Cyclic Subgroups of D6 with Proper Subgroup Example

    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...
  39. D

    Exploring the Connections between Number Theory and Group Theory

    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...
  40. B

    Proving 'If (2^n)-1 is Prime, Then n is Prime' Using Groups

    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...
  41. H

    Groups whose elements have order 2

    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
  42. H

    Groups whose elements have order 2

    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...
  43. H

    Groups whose elements have order 2

    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...
  44. H

    Show Commutativity of Group with All Elements of Order 2 & Consider Zn

    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...
  45. H

    Groups whose order have order two

    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...
  46. C

    Groups and Isomorphisms.

    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...
  47. J

    Problem regarding computation of factor groups.

    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...
  48. W

    Homology groups from Homotopy groups

    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
  49. L

    Nucleophilic acyl substitution and basicity of groups

    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...
  50. Fredrik

    Integral curves and one-parameter groups of diffeomorphisms

    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...
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