Groups Definition and 906 Threads

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. K

    Symmetries and Transformation Groups of Equilateral Triangle & Icosahedron

    How many symmetries (and what symmetries) and how many elements do the transformation groups of the equilateral triangle and the icosahedron have? thanks
  2. P

    Show isomorphism between two groups

    Homework Statement Suppose G is a non-abelian group of order 12 in which there are exactly two elements of order 6 and exactly 7 elements of order 2. Show that G is isomorphic to the dihedral group D12. Homework Equations The Attempt at a Solution My attempt (and what is listed...
  3. L

    Theorem concerning free abelian groups

    I spend much time to study a theorem - Let G be a nonzero free abelian group of finite rank n, and let K be a nonzero subgroup of G. Then K is free abelian of rank s smaller or equal to n. Furthermore, there exists a basis (x1,x2,...,xn) for G and positive integers d1,d2,...,ds where di divides...
  4. T

    What this symbols means regarding groups

    i got two groups of numbers "a" and "b" what is the meaning of a\b and b\a ??
  5. S

    Is a Discrete Group of Rotations Cyclic?

    Homework Statement Prove that a discrete group G cosisting of rotations about the origin is cyclic and is generated by \rho_{\theta} where \theta is the smallest angle of rotation in G The Attempt at a Solution since G is by definition a discrete group we know that if \rho is a...
  6. S

    Isomorphisms between cyclic groups

    Ok, here is something i thought i understood, but it turns out i am having difficulties fully grasping/proving it. Let \theta:G->G' be an isomorphism between G and G', where o(G)=m=o(G'), and both G and G' are cyclic, i.e. G=[a] and G'=[b] So my question is, when we want to find the...
  7. S

    Order of groups in relation to the First Isomorphism Theorem.

    Given H,K and general finite subgroups of G, ord(HK) = [(ord(H))(ord(K))] / ord(H intersection K) I know by the first isomorphism theorem that Isomorphic groups have the same order, but the left hand side of the equation is not a group is it? I am struggling to show this.
  8. T

    Why Are Half of the Permutations in Sn Even and the Other Half Odd?

    How do we know that half of permutations are odd and half are even? Why not 1/4, 3/4 or other proportions?
  9. B

    Proof of Finite Order of G in Quotient Group Q/Z

    1. Show that every element of the quotient group G = Q/Z has finite order. Does G have finite order? he problem statement, all variables and given/known data [b]2. This is the proof The cosets that make up Q/Z have the form Z + q, where q belongs to Q. For example, there is a...
  10. T

    Traceless hermitian matrices form groups?

    is the set of nxn traceless hermitian matrices under addition a group? is the set of nxn traceless hermitian matrices under multiplication a group? is the set of nxn traceless non-hermitian matrices under addition a group? question 1-I thought that traceless means trace=0 is this right...
  11. B

    Oreder of groups and their elements

    [b] Show that any group of even order has at least an element of order 2 Homework Equations [b]3. I know that the order of a groups tells you how many elements the group consist, but just randomly assuming that it has at least an order of 2 is what I can't really understand. For example...
  12. Moonbear

    Support groups: preconceptions and nomenclature

    While having a discussion with my students this week, a topic came up that I found interesting, and was wondering what other people's views were on it. We were talking about patient compliance with treatment (for diabetes in this case) and the role and availability of support groups. The...
  13. S

    Groups whose orders have no common factors

    Homework Statement Let G and G' be finite groups whose orders have no common factor. Prove that the homomorphism \varphi G \rightarrow G' is the trivial one \varphi (x) =1 for all x. The Attempt at a Solution My thoughts are that we need to use lagrange's thm. somehow. or maybe...
  14. F

    How to Determine the Order of an Element in a Factor Group?

    I'm currently studying factor groups in abstract algebra and needed some help understanding how to determine the order of an element in a factor group Suppose I have Z (mod 12) / <4>. And I choose some random element from Z (mod 12) such as 5 or 7. How would I go about determining the order...
  15. E

    Functional Groups: Examples of Esters & Ethers

    Could someone please give me an example of an ether and an ester. In words, not drawing them. Thank you!
  16. T

    Any deep results from Hopf Algebras (or Quantum groups)?

    Hi. I'm currently working on a expository paper about quantum groups and Hopf algebras. However, from all the books I've read, they are more about examples (deformations of various groups) than actual interesting results (Or perhaps I just don't understand them enough to draw any interesting...
  17. J

    Some questions on vector fields on Lie groups

    Homework Statement Let G be a Lie group with unit e. For every g in G let Lg: G -> G, h-> gh be left multiplication. a) Prove that the map G x TeG -> TG, (g,v) -> De Lg (v) is a diffeomorphism, and conclude G has a basis of vector fields. b) Prove for every v in TeG the is a unique...
  18. A

    Non abelian groups of order 6 isomorphic to S_3

    How can I show that all Non Abelian Groups of order 6 are isomorphic to S_3 without using Sylow's Theorems? I have shown the following: G has a non-normal group of normal subgroup of order 2 The elements of G look like: 1, a, a^2, b, c, d, where a,a^2 have order 3 and others have order 2...
  19. S

    Finding Smallest Subgroups of A_4 Containing 2 Permutations

    Homework Statement I was curious to know, say we have two even permutations taken out of A_4, say (12)(34) and (123), and we want to find the smallest subgroup of A_4 that contains both these permutations, then how would we go about it. This subgroup in this case will defenitely be A_4...
  20. N

    Groups, Normalizer, Abstract Algebra, Dihedral Groups help?

    [b]1. Let G be a Group, and let H be a subgroup of G. Define the normalizer of H in G to be the set NG(H)= the set of g in G such that gHg-1=H. a) Prove Ng(H) is a subgroup of G b) In each of the part (i) to (ii) show that the specified group G and subgroup H of G, CG(H)=H, and NG(H)=G...
  21. J

    Why are SOnR and SLnR Lie Groups?

    Homework Statement Prove SOnR and SLnR are Lie groups, and determine their dimensions. SOnR = {nxn real hermitian matrices and determinant > 0} SLnR = {nxn real matrices with determinant 1} The Attempt at a Solution We can see that SLnR is level set at zero of the graph of a smooth...
  22. F

    Proving Normal Subgroup of Abelian Groups

    Homework Statement Let G be a group and let H,K be subgroups of G. Assume that H and K are Abelian. Let L=(H-union-K) be the subgroup of G generated by the set H-union-K. Show that H-intersect-K is a normal subgroup of L. The Attempt at a Solution How do i start this?
  23. T

    Isomorphic Groups: Same Info Studying 1 or Both

    If two groups A and B are isomorphic then by studying one of them, we can deduce all algebraic information about the other? Hence studying one is equivalent to studying the other?
  24. P

    Proof: Intersection of Subgroups is a Subgroup of H in G

    Homework Statement If H, K are subgroups of G, show that H intersect K is a subgroup of H Homework Equations I know that H intersect K is a subgroup of G; I proved this already but I'm wondering how H intersect K is a subgroup of H The Attempt at a Solution I'm quite sure this is true...
  25. I

    Lattices in nilpotent Lie groups

    Please, help me with the following questions or recommend some good books. 1) We have a simply-connected nilpotent Lie group G and a lattice H in G. Let L be a Lie algebra of G. There is a one to one correspondence between L and G via exp and log maps. a) Is it true, that to an ideal in...
  26. S

    Groups of quarks and confinement

    hey guys, I am a first year physics student but my physics lecturer invited me to sit in during her third year physics lecture. Of course i didnt fully understand some of it, but i think i at least grasped the concept of confinement (the lecture was on quantum chromodynamics by the way)...
  27. R

    Normalizer of Sylow p-subgroup of simple groups A_n

    what is the order of normalizer of Sylow p-subgroup of simple groups A_n?
  28. T

    Isomorphic groups that have different properties?

    What are some properties apart from the actual names of the elements that differ between isomorphic groups?
  29. radou

    Infinite cyclic groups isomorphic to Z

    I'm currently going through Hungerford's book "Algebra", and the first proof I found a bit confusing is the proof of the theorem which states that every infinite cyclic group is isomorphic to the group of integers (the other part of the theorem states that every finite cyclic group of order m is...
  30. A

    Findng a group between two groups

    we all know that set of rationals i a subgroup of set of reals. my question is whether there exsts a group between these tw groups. f yes what it can be? and if no, how to prve the non-existence?
  31. S

    Large groups of teens hanging out in downtown areas of cities

    I'm a homeless transient. One constant I've noticed is that around the downtown area in all medium sized ( population 100,000 + ) cities or larger, there are large groups of teenagers that hang around at night, goofing off and shooting the bull for hours. Don't they have any work to do? Don't...
  32. B

    Why Are Two Indices Used for the Generators of Lorentz and Poincare Groups?

    Just a quick question here: I was going through my notes and I noticed that the generators of both these groups are labeled two indices. I was wondering if there is any particular reason for this, since it seems to me that one index would work perfectly well. Thanks
  33. N

    Understanding Symmetric Groups: S4 Order & Products

    What is the order of S4, the symmetric group on 4 elements? Compute these products in order of S4: [3124] o [3214], [4321] o [3124], [1432] o [1432]. Can I get help on how to do this. The solution's manual gives the answer on how to do the last two, but I don't understand the process...
  34. V

    Proving G is Cyclic if No Subgroups Other than G and {e} Exist

    Homework Statement Prove that if a group G has no subgroup other than G and {e}, then G is cyclic... Homework Equations The Attempt at a Solution we could say that, let a E G - {e} then we construct <a>...
  35. H

    Different Type of Extension Problem for Groups

    Let H, K be finite groups. Instead of asking what groups G there are such that K can be embedded as a normal subroup and G/K is isomorphic with H (the usual extension problem for groups), I've been thinking about the following: Which groups G exist such that H and K can be embedded as (not...
  36. C

    Connection between polynomials and groups

    Hey Everyone, I'm reading a paper by Claude LeBrun about exotic smoothness on manifolds and he is talking about a connection between polynomials and groups that I am not familiar with (or at least I think that's what he's talking about). He's creating a line bundle (which happens to be...
  37. T

    Can a Finitely Generated Group Contain an Infinitely Generated Subgroup?

    Could a finitely generated group contain a subgroup which is infinitely generated? Why?
  38. T

    What Can We Learn from Finite Presentations of Groups?

    What's so special about finite presentations? Does it indicate some properties about the group?
  39. C

    Questions involving simple groups

    Hi, it's my first time posting in this forum, so I'm sorry if I have done anything against the forum rules and please point it out to me. Currently revising group theory for an exam in a week's time, and these two practise questions I couldn't finish, so if anyone can push me towards the right...
  40. E

    What Structure Must G Have for All Aut(G) Elements to Send N to N

    Here is another problem from Lang. Let G be a finite group. N a normal subgroup. We want to ask what structure must G have in order for all the elements of Aut(G) to send N to N. It is assumed that the order of N is relatively prime to the order of G/N. I have worked on this problem for a...
  41. E

    A External Direct Sum of Groups Problem

    Homework Statement Find a subgroup of Z_4 \oplus Z_2 that is not of the form H \oplus K where H is a subgroup of Z_4 and K is a subgroup of Z_2. The attempt at a solution I'm guessing I need to find an H \oplus K where either H or K is not a subgroup. But this seems impossible. Obviously...
  42. S

    Finding if two groups are isomorphic

    Homework Statement Show that the group {U(7), *} is isomorphic to {Z(6), +} Homework Equations The Attempt at a Solution I drew the tables for each one. I can see that they are the same size and the identity element for U7 is 1, and for Z6 is 0. I don't really see any...
  43. A

    Number of Non-Isomorphic Abelian Groups

    Homework Statement Determine the number of non-isomorphic abelian groups of order 72, and list one group from each isomorphism class. The Attempt at a Solution 72 = 2^3*3^2 3= 1+1+1= 2+1= 3 (3) 2= 1+1= 2 (2) 3*2 = 6 And then I get lost on the...
  44. G

    Manifolds / Lie Groups - confusing notation

    Hi there, I'm reading over my Lie groups notes and in them, in the introductory section on manifolds, I've written that F_{\star} is a commonly used notation for d_{x}F and so the chain rule d_{x}{G \circ F}=d_{F(x)}G \circ d_{x}F can be written (G\circ F)_{\star}=G_{\star}\circ F_{\star} Is...
  45. G

    Manifolds / Lie Groups - confusing notation

    I'm reading over my Lie groups notes and in them, in the introductory section on manifolds, I've written that F_{\star} is a commonly used notation for d_{x}F and so the chain rule d_{x}{G \circ F}=d_{F(x)}G \circ d_{x}F can be written (G\circ F)_{\star}=G_{\star}\circ F_{\star} Is what I've...
  46. A

    Non-Isomorphic Groups of Order 30

    Homework Statement How many different nonisomorphic groups of order 30 are there? Homework Equations The previous parts of the problem dealt with proving that 3-Sylow and 5-Sylow subgroups of G were normal in G when o(G)=30, though I'm not sure how that relates... The Attempt at a...
  47. F

    How Do Elements of Direct Product Groups Commute in Particle Physics?

    From Gauge Theory of particle physics, Cheng and Li I don't understand the flollowing: "Given any two groups G={g1,..} H= {h1,h2,...} if the g's commute with the h's we can define a direct product group G x H={g_ih_j} with the multplication law: g_kh_l . g_mh_n = g_kh_m . h_lh_n Examples...
  48. M

    Exploring Alkyl Groups and their Role in Organic Chemistry

    I'm a bit confused about what alkyl groups are in organic chemistry. I thought functional groups by definition where groups of atoms which contained at least 1 element other than carbon or hydrogen which were connected to the carbon skeleton of the molecule. What are alkyl groups then? I read...
  49. G

    Are O(3) and SO(3) Both Generators of Angular Momentum in Quantum Mechanics?

    I read that the generator of the O(3) group is the angular momentum L and that the generator of the SU(2) group is spin S. Nevertheless I have some questions. 1. In some books they say that the generator of the SO(3) group is angular momentum L. SO(3) is the group of proper rotations...
  50. E

    Finitely generated abelian groups

    [SOLVED] finitely generated abelian groups Homework Statement My book states that (\mathbb{Z} \times\mathbb{Z} \times\cdots \times \mathbb{Z})/(d_1\mathbb{Z} \times d_2\mathbb{Z} \times \cdots d_s\mathbb{Z} \times {0} \times \cdots \times {0}) is isomorphic to \mathbb{Z}_{d_1}...
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