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. E

    Prove Free Abelian Group \mathbb{Z}_{p^r}[p] is Isomorphic to \mathbb{Z}_p

    Homework Statement Show that \mathbb{Z}_{p^r}[p] is isomorphic to \mathbb{Z}_p for any r \geq 1 and prime p. \mathbb{Z}_{p^r}[p] is defined as the subgroup \{x \in \mathbb{Z}_{p^r} | px = 0 \}Homework Equations The Attempt at a Solution I don't think I should need to use Sylow's Theorems for...
  2. M

    Comparing Books on Lie Groups: Representations & Compact Lie Groups

    I'm taking a course on Lie Groups and the Representations. We are using the book: Representations of compact Lie Groups by Bröcker and Dieck, and I find it very unorganized and sometimes sloppy. Can anybody recommend a very clear and rigorous book, where it is not prove by example, "it is easily...
  3. M

    Classification of groups of order 8

    Hi next one, bit confused with this problem: any hints on any of the parts would be greatly appreciated. QUESTION: --------------------------------------- let G be a group of order 8 and suppose that y \epsilon G has ord(y)=4. Put H = [1,y,y^2,y^3] and let x \epsilon G-H (i) show that H...
  4. H

    Can We Identify Quotient Groups as Subgroups of the Original Group?

    Let G be a group and let N\trianglelefteq G , M\trianglelefteq G be such that N \le M. I would like to know if, in general, we can identify G/M with a subgroup of G/N. Of course the obvious way to proceed is to look for a homomorphism from G to G/N whose kernel is M, but I can't think of...
  5. P

    Proof: Discreteness of Topological Groups

    Homework Statement Prove: a topological group is discrete if the singleton containing the identity is an open set. The statement is in here http://en.wikipedia.org/wiki/Discrete_group The Attempt at a Solution Is that because if you multiply the identity with any element in the group, you get...
  6. F

    Solving Maths Problems: Numbers, Symmetries & Groups

    http://img180.imageshack.us/img180/9589/simplell9.jpg Is 1. c) as simple as i think it is? I have gone through my notes and can't find anything to do with it, the module for it is Numbers, symmetries and groups, any ideas or do i simple just wack in 13/7 on my calculator and write down...
  7. E

    Classifying Z_2(\alpha) and Z_2(\alpha)^* Groups

    [SOLVED] extension field Homework Statement Let E be an extension field of Z_2 and \alpha in E be algebraic of degree 3 over Z_2. Classify the groups <Z_2(\alpha),+> and <Z_2(\alpha)^*,\cdot> according to the fundamental theorem of finitely generated abelian groups. Z_2(\alpha)^* denotes the...
  8. K

    Groups, show GxH is a group (final question)

    Homework Statement Let G and H be groups. We define a binary operation on the cartesian product G x H by: (a,b)*(a',b') := (a*a', b*b') (for a,a' \inG and b,b'\in)H Show that G x H together with this operation is a group. Homework Equations The Attempt at a Solution To...
  9. P

    What Are the Effects of Carboxyl Groups on Electron Distribution?

    what are electron withdrawing groups? please give any important advice to solve questions based on it concept.
  10. M

    Sylow's Theorem and Recognition Criterion for Groups of Order pq

    Let p,q be distinct primes with q < p and let G be a finite group with |G| = pq. (i) Use sylow's theorem to show that G has a normal subgroup K with K \cong G (ii) Use the Recogition Criterion to show G \cong C_p \rtimes_h C_q for some homomorphism h:C_q \rightarrow Aut(C_p) (iii)...
  11. E

    Isomorphic Quotient Groups: A Counterexample

    Homework Statement Let H and K be normal subgroups of a group G. Give an example showing that we may have H isomorphic to K while G/H is not isomorphic to G/K. Homework Equations The Attempt at a Solution I don't want to look in the back of my book just yet. Can someone give me a...
  12. E

    Finite Simple Groups: Exploring Order and Lagrange's Theorem

    [SOLVED] simple groups Homework Statement T or F: All nontrivial finite simple groups have prime order. Homework Equations The Attempt at a Solution I want to say yes with Lagrange's Theorem, but I am not sure that applies...
  13. M

    Is the group (G,dG) isomorphic to the original group G?

    Homework Statement Exercise 1.2:2. (i) If G is a group Define an operation dG on |G| by dG(x, y) = x*y^-1. Does the group given by (G,dG) determine the original group G with * (I.e., if G1 and G2 yield the same pair, (G1,dG1) = (G2,dG2) , must G1 = G2?) There is a part II, but I would...
  14. M

    Proving Infinite Groups: Algebra - Groups

    Homework Statement Prove that the following sets form infinite groups with respect to ordinary multiplication. a){2^k} where k E Z b){(1+2m)/(1+2n)} where m,n E Z Homework Equations The Attempt at a Solution I sort of know about closure associativity identity inverses...
  15. J

    Proving Lie Group \rho Preserves Inner Product/Cross Product

    Let \rho : \mathbb{H} \to \mathbb{H}; q \mapsto u^{-1}q u where u is any unit quaternion. Then \rho is a continuous automorphism of H. I'm asked to show that \rho preserves the inner product and cross product on the subspace \mathbf{i}\mathbb{R} + \mathbf{j}\mathbb{R} +...
  16. J

    Order of elements in finite abelian groups

    prove that if G is a finite and abelian group and m is the least common multiple of the order of it's element, that there is an element of order m. My idea: if ai are the elements of G, the order of a1*a2 is lcm(a1,a2) and the result follows directly when applied to all ai... but why is this...
  17. E

    Free Abelian Groups: Isomorphic to Z x Z...xZ?

    Homework Statement What is the point of giving free abelian groups a special name if they are all isomorphic to Z times Z times Z ... times Z for r factors of Z, where r is the rank of the basis? Homework Equations The Attempt at a Solution
  18. J

    Solve Permutation Group Homework Questions

    Hehe, I'm working through the complete groups books right now, so don't think I ask you all my homework questions... I'm doing a lot myself too =). Homework Statement 1) If H is a subgroup of S_n, and is not contained in A_n, show that precisely half of the elements in H are even permutations...
  19. S

    Exploring Groups: G(n) and its Properties

    ok I've managed to solve the other 2 questions. here is my final one: (1) If G is a group and n \geq 1 , define G(n) = { x E G: ord(x) = n} (2) If G \cong H show that, for all n \geq 1 , |G(n)| = |H(n)|. (3) Deduce that, C_3 X C_3 is not \cong C_9. Is it true that C_3 X C_5...
  20. P

    Normal Subgroups: Why Every Kernel is a Homomorphism

    Homework Statement Expain why every normal subgroup is the kernel of some homomorphism. The Attempt at a Solution Every kernel is a normal subgroup but the reverse I can't show rigorously. It seems possible how to show?
  21. S

    Order of Groups: Proving ord(\theta(x)) = ord(x)

    Hi i have completed the answer to this question. Just need your verification on whether it's completely correct or not: Question: If G is a group and xEG we define the order ord(x) by: ord(x) = min{r \geq 1: x^r = 1} If \theta: G --> H is an injective group homomorphism show that, for...
  22. M

    What groups have exactly 4 subgroups?

    I was wondering about the classification of groups with a certain number of subgroups. I (sort of mostly I think maybe) get the ideas behind classification of groups of a certain (hopefully small) order, but I came across a question about classifying all groups with exactly 4 subgroups, and I...
  23. Shaun Culver

    Exploring the Connection Between Lie Groups & Manifolds

    Why are Lie groups also manifolds?
  24. B

    Which Lie Groups are Riemann Manifolds?

    What Lie groups are also Riemann manifolds? thanks
  25. W

    Which Book on Lie Groups and Lie Algebras is a Classic?

    I'm looking for a solid book on Lie groups and Lie algebras, there is too many choices out there. What is a classic text, if there is one?
  26. J

    Sorting Objects into Percentile Groups

    Hello, Is there a well known method (algorithm, process, etc.) by which objects are sorted into percentile groups based on some aspect of the object such as weight or size?
  27. P

    Lie Groups and Representation theory?

    What is the connection between the two if any? What kind of algebra would Lie groups be best labeled under?
  28. D

    Abelian groups from the definition of a field

    Just a pregrad-level curiosity: I see often repeated (in the Wikipedia page defining "Field", for one) that, from the field's axioms, it can be deduced that F,+ and F\{0},* are both commutative groups. Yet, the closure property of * is only guaranteed on F, not necessarily on F\{0}. If I'm...
  29. M

    COSETS are equal for finite groups

    Homework Statement Prove that if H is a subgroup of a finite group G, then the number of right cosets of H in G equals the number of left cosets of H in G Homework Equations Lagrange's theorem: for any finite group G, the order (number of elements) of every subgroup H of G divides...
  30. M

    Groups masquerading as isomorphic

    Hi, I recall being told in an algebra course in college that there exist groups with matching order tables and that are nonetheless not isomorphic. That is, if you list out the orders of all the elements in one group and all the orders of the elements in the other, the lists are "the same"...
  31. Q

    Defining Group Multiplication in Particle Physics

    Everyone must be familiar with U(1),SU(2) and SU(3) Lie groups in particle physics . But how does one define the multiplication of two groups of different dimensions aka SU(2) X U(1) or SU(3) X SU(2) X U(1).
  32. W

    What is the Role of Lie Groups in Isometry Actions on Spaces?

    Hi, everyone: I am asked to show that a group G acts by isometries on a space X. I am not clear about the languange, does anyone know what this means?. Do I need to show that the action preserves distance, i.e, that d(x,y)=d(gx,gy)?. Thanks.
  33. P

    Soluble Groups in Algebra: What Are They and What Branch Do They Fall Under?

    How does soluble groups fit into algebra? Why is there another name for it called solvable groups? What branch does it fall under?
  34. C

    Why emit alphas, not other nucleon groups?

    [SOLVED] Why emit alphas, not other nucleon groups? Homework Statement (Advanced Physics; Adams and Allday; Spread 8.18, Question Section 8.18, question 3) Why do you think helium-4 nuclei (alpha particles) are often emitted from unstable heavy nuclei whereas bundles of neutrons or protons...
  35. H

    Is Calculus a Prerequisite for Abstract Algebra?

    Hi, I want to take this course next term. One reason is because I think it will help me with mechanics, classical and quantum, which are taken next year at advanced level. The problem is I'm taking calc2 atm, and its a listed prereq for this group course. I got all the other prereq's...
  36. D

    [Algebra] Conjugacy classes of Finite Groups

    So, the question is: Determine all finite groups that have at most three conjugacy classes I'm a little confused by how to start. Right now, we can say for sure that cyclic groups of order 1, 2, and 3 satisfy this criterion. Also, with Lagrange's Theorem and the counting formula(I'm using...
  37. S

    Prime Order Groups: Understanding Lagrange's Theorem and its Corollary

    Ok, well a corollary to Lagrange's theorem is that every group of prime order, call it G, must be cyclic. Consider the cyclic subgroup of G generated by a (a not equal to e), the order of the subgroup must divide the order of p, since the only number less than or equal to p that divides p is p...
  38. B

    Proving Cyclic Property in Factor Groups

    Homework Statement Prove that a factor group of a cyclic group is cyclic Homework Equations The Attempt at a Solution For a group to be cyclic, the cyclic group must contain elements that are generators which prodduced all the elements within that group . A factor group is...
  39. A

    Explained: Decomposing Lie Groups in Theoretical Physics

    It's common in theoretical physics papers/books to talk about the decomposition of Lie groups, such as the adjoint rep of E_8 decomposing as \mathbf{248} = (\mathbf{78},\mathbf{1}) + (\mathbf{1},\mathbf{8})+(\mathbf{27},3) + (\overline{\mathbf{27}},\overline{\mathbf{3}}) How is this...
  40. P

    Understanding Abelian Groups, QLG, and Fiber Bundles in String Theory

    Can anyone tell me what type of string theory QLG is? Or explain fiber bundles and (non)abelian groups? This isn't homework, or anything btw.
  41. T

    Isomorphism and direct product of groups

    Just wondering if there is a general way of showing that (Z, .)n isomorphic to Zm X Zp with the obvious requirement that both groups have the same order?
  42. R

    Is U(t)=exp(-iH/th) a Lie Group in Quantum Mechanics?

    Is U(t)=exp(-iH/th) a Lie group? Is it an infinite dimensional Lie group? To what 'family' of Lie groups does it belong? thank you
  43. K

    Proving U(8) is not Isomorphic to U(10): Insights and Techniques

    Hi. Hoping a could have a little bit of guidance with this question Show that U(8) is not isomorphic to U(10) So far, I've realized that in U(8) each element is it's own inverse while in U(10) 3 and 7 are inverses of each other. I guess that's really all I need to say that they aren't...
  44. L

    Theoretical Groups: Explaining SU(2), E8XE8 & More

    What are these groups like SU(2) and E8XE8 etc. Can anybody explain it with full details and also give pre-requisite knowledge.
  45. H

    What will I learn in a course on Groups and Symmetries?

    Hello. As some of you know I'm a chemistry student, but I plan to take some math for the hell of it next summer. I've come across a course called "Groups and Symmetries" and intend to take it, mainly because it is one of the few upper maths avaialbe in the summer. I've never heard of this...
  46. B

    Finding Generators for Cyclic Groups Z(6), Z(8), and Z(20)

    Homework Statement Find all generators of Z(6), Z(8) , and Z(20) Homework Equations The Attempt at a Solution I should probably list the elements of Z(6), Z(8) and Z(20) first. Z(6)={0,1,2,3,4,5} Z{8}={0,1,2,3,4,5,6,7}...
  47. B

    Symmetry groups and Caley tables

    Homework Statement I have a shape about the origin. It has rotational symmetry but not reflectional symmetry (its an odd star shape!). I have to write down in standard notation the elements of the symmetry group and I have to construct a caley table under composition of symmetries. I...
  48. S

    Biology/Chemistry: pH Level and Ionizable Groups

    Homework Statement Select any molecule [from the assigned chapter] that has more than 2 ionizable groups. Write its structures, showing every atom, at pH levels 1 and 7. Homework Equations See below. The Attempt at a Solution http://en.wikipedia.org/wiki/Geranyl_pyrophosphate...
  49. D

    Groups of order 60 and elements of order 5

    Homework Statement Let G be a group with order \left| G \right| = 60. Assume that G is simple. Now let H be the set of all elements that can be written as a product of elements of order 5 in G. Show that H is a normal subgroup of G. Then conclude that H = G Homework Equations The...
  50. W

    One-Parameter Groups:John Lee's Vector Field Analysis

    Hi, everyone. I am new here, so I hope I am follow the protocols. Please let me know otherwise. Also, I apologize for not knowing Latex yet, tho I hope to learn it soon. am trying to show that the vector field: X^2(del/delx)+del/dely Is not a complete vector field. I think this is...
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