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. S

    Lie Groups and Canonical Coordinates

    Hello. I have a question that has been on my mind for some time. I always see in mathematical physics books that they identify elements of the Lie algebra with group elements "sufficiently close" to the identity. I have never seen a real good proof of this so went on an gave a proof. Let Xi be...
  2. Math Amateur

    Braid Groups at undergraduate level

    In M.A. Armstrongs book "Groups and Symmetry" in Chapter 12 he introduces the reader to the fascinating topic of Braid Groups. Does anyone know of a book at undergraduate level (or even a popular book) that deals with Braid Groups Can you progress with Braid Groups if you lack a...
  3. Math Amateur

    Representation Theory of Finite Groups - CH 18 Dummit and Foote

    I am reading Dummit and Foote on Representation Theory CH 18 I am struggling with the following text on page 843 - see attachment and need some help. The text I am referring to reads as follows - see attachment page 843 for details \phi ( g ) ( \alpha v + \beta w ) = g \cdot ( \alpha v +...
  4. S

    Are Abelian Groups of Relatively Prime Orders Isomorphic?

    Homework Statement Let m and n be relatively prime positive integers. Show that if there are, up to isomorphism, r abelian groups of order m and s of order n, then there are rs abelian groups of order mn. Homework Equations The Attempt at a Solution I'm not sure how to go about...
  5. P

    Isomorphism of relatively prime groups

    Homework Statement Allow m,n to be two relatively prime integers. You must prove that Z(sub mn) ≈ Z(sub m) x Z(sub n) Homework Equations if two groups form an isomorphism they must be onto, 1-1, and preserve the operation. The Attempt at a...
  6. S

    Combinations-different way to form groups of people

    Hello everyone, I am new to this forum. Need help with this problem How many ways you can select 3-person groups from a group of 8 students? My solution: ---------------- Number of ways to make one group of 3 persons = 8C3 How do I proceed from here? Thank you.
  7. F

    Symmetry Groups Algebras Commutators Conserved Quantities

    Symmetry, Groups, Algebras, Commutators, Conserved Quantities OK, maybe this is asking too much, hopefully not. I'm trying to understand the connection between all of these constructions. I wonder if a summary about these interrelationship can be given. If I understand what I'm reading, there...
  8. C

    Finitely Generated Abelian Groups

    Homework Statement Let p be prime and let b_1 ,...,b_k be non-negative integers. Show that if : G \simeq (\mathbb{Z} / p )^{b_1} \oplus ... \oplus (\mathbb{Z} / p^k ) ^ {b_k} then the integers b_i are uniquely determined by G . (Hint: consider the kernel of the homomorphism f_i :G \to...
  9. Math Amateur

    Linear Algebra Preliminaries in Finite Reflection Groups

    Linear Algebra Preliminaries in "Finite Reflection Groups In the Preliminaries to Grove and Benson "Finite Reflection Groups' On page 1 (see attachment) we find the following: "If \{ x_1 , x_2, ... x_n \} is a basis for V, let V_i be the subspace spanned by \{ x_1, ... , x_{i-1} ...
  10. Math Amateur

    Orthogonal Transformations _ Benson and Grove on Finite Reflection Groups

    I am reading Grove and Benson's book on Finite Reflection Groups and am struggling with some of the basic linear algebra. Some terminology from Grove and Benson: V is a real Euclidean vector space A transformation of V is understood to be a linear transformation The group...
  11. D

    Which Is the Best Leaving Group: Cl-, Enolate, or NR3?

    which is the best leaving group? Cl- enolate NR3 I couldn't figure out how to draw the enolate here, but it is a basic 3 carbon one. I believe the answer is Cl because resonance on the enolate will disperse the neg. charge. Am I on the right track or is there a factor I am missing?
  12. M

    Are These Group Statements True or False?

    Homework Statement Which of following statements are TRUE or FALSE. Why? In any group G with identity element e a) for any x in G, if x2 = e then x = e. b) for any x in G, if x2 = x then x = e. c) for any x in G there exists y in G such that x = y2. d) for any x, y in G there exists z...
  13. B

    Is Every Group of Order 15 Cyclic?

    I have a question where it says prove that G \cong C_3 \times C_5 when G has order 15. And I assumed that as 3 and 5 are co-prime then C_{15} \cong C_3 \times C_5 , which would mean that G \cong C_{15} ? So every group of order 15 is isomorohic to a cyclic group of order 15...
  14. J

    Algorithm for dividing a set of numbers into groups

    I'm looking for an algorithm for dividing a set of numbers into groups, and then doing it again in such a way that no numbers are in a group together more than once. For instance if you have 18 numbers and divide them into groups of three you should be able to do this 8 times without any...
  15. T

    Abstract Algebra: Groups and Subgroups

    Homework Statement The problem says: Suppose that * is an associative binary operation on a set S. Let H= {a ε S l a * x = x * a for all x ε s}. Show that H is closed under *. ( We think of H as consisting of all elements of S that commute with every element in S) My teacher is horrible so...
  16. T

    Abstract Algebra: Groups and Subgroups

    The problem says: Suppose that * is an associative binary operation on a set S. Let H= {a ε S l a * x = x * a for all x ε s}. Show that H is closed under *. ( We think of H as consisting of all elements of S that commute with every element in S) My teacher is horrible so I am pretty lost in...
  17. B

    How to show an isomorphism between groups?

    Is the only way to show an isomorphism between groups is to just define a map which has the isomorphism properties? So for example for a group G with order 15 to show that G \cong C_3 \times C_5 would I just have to define all the possible transformations to define the isomorphism...
  18. M

    Using Group Axioms to Solve for x in a Group Equation

    Homework Statement Let G be a group with identity e, and suppose that a, b, c, x in G. Determine x, given that x2a=bxc-1 and acx = xac. Homework Equations The Attempt at a Solution I know the three axioms for group. G1. Associativity. For all a, b, c in G, (a * b) * c = a...
  19. StevieTNZ

    Two Groups of Entangled Photons

    If we have two groups of photons; each group consisting of two entangled photons. We allow one of the photons from each group to interact with another object. If we perform a polarisation measurement on each photon of one group, will the photons in the other group, independent of whether...
  20. C

    Question about groups and limit points?

    Homework Statement We are supposed to say how many limit points the set A={sin(n)} where n is a positive integer. My teacher said to use a theorem by Kronecker to help with it. His theorem says from wiki, that an infinite cyclic subgroup of the unit circle group is a dense subset...
  21. Math Amateur

    How Do Permutations Act on Vector Spaces in S3?

    I am reading Dummit and Foote Ch 18, trying to understand the basics of Representation Theory. I need help with clarifying Example 3 on page 844 in the particular case of S_3 . (see the attahment and see page 844 - example 3) Giving the case for S_3 in the example we have the...
  22. V

    Formula to compute number of groups from given points with overlap

    The problem is kind of easy to understand. Given is some points, say 10 points. (I am using numbering for understanding) 0 1 2 3 4 5 6 7 8 9 Now group these such that the group size is 5 and there is no overlap so, there can be 2 groups. the groups are (0 1 2 3 4) & (5 6 7 8 9) Now...
  23. B

    Isomorphic direct product cyclic groups

    Help! For p prime I need to show that C_{p^2} \ncong C_p \times C_p where C_p is the cyclic group of order p. But I've realized I don't actually understand how a group with single elements can be isomorphic to a group with ordered pairs! Any hints to get me started?
  24. B

    Can Every Element in a Finite Cyclic Group Be a Generator?

    Regarding finite cyclic groups, if a group G, has generator g, then every element h \in G can be written as h = g^k for some k. But surely every element in G is a generator as for any k , (g^k)^n eventually equals all the elements of G as n in takes each integer in turn. Thanks...
  25. M

    Explaining Finite Solvable Groups: Understanding Burnside's Theorem

    HI, I was reading an article and it says that a finite group of order p^aq^b, where p, q are primes, is solvable and therefore not simple. But I can't quite understand why this is so. I do recall a theorem called Burnside's theorem which says that a group of such order is solvable. But then I...
  26. F

    Proving the Surjectivity of Maps in Cyclic Groups with Relatively Prime Integers

    Homework Statement Let G be a cyclic group of order n and let k be an integer relatively prime to n. Prove that the map x\mapsto x^k is sujective. Homework Equations The Attempt at a Solution I am trying to prove the contrapositon but I am not sure about one thing: If the map is...
  27. B

    Isomorphism between groups of real numbers

    Apparently there is an isomorphism between the additive group (ℝ,+) of real numbers and the multiplicative group (ℝ_{>0},×) of positive real numbers. But I thought that the reals were uncountably infinite and so don't understand how you could define a bijection between them?! Thanks for...
  28. M

    Comparing two dependent? groups

    Hi, I have a mathematics/Matlab question. Suppose I have a speaker that serves as a sound source, and two IDENTICAL microphones to the left and right of this speaker. Suppose that each microphone collects data regarding the sound level of the speaker, and that there are over 3,000 data values...
  29. F

    What Are the Elements of the Quotient Group D4/N?

    Homework Statement Let D4 = { (1)(2)(3)(4) , (13)(24) , (1234) , (1432) , (14)(23) , (12)(34) , (13), (24) } and N=<(13)(24)> which is a normal subgroup of d4 . List the elements of d4/N . Homework Equations The Attempt at a Solution I computed the left and right cosets to...
  30. H

    Direct product of two groups with different n-spaces

    how does one evaluate the direct product between a group G with components that are say 2-tuple and a group H with components that are just 1-tuple?
  31. J

    Galois Groups for a system of Linear equations?

    If I were to solve a system of multiple equations in the form αx+βy+ζz=p_{1} Where α,β,ζ are constants x,y,z are variables, and p is a prime, how would I use Galois theory and/or number theory to find the number of solutions if the other equations could all be written in the form...
  32. A

    SU(2)L, SU(2)R, other symmetric groups and SSB

    Hello everyone, When we speak about the SU(2)L group (in electroweak interactions for example), about what group do we talk ? What is the difference with the SU(2) group ? And with the SU(2)R ? Why is the label so important ? I ask this because I see that a Lagrangien can be invariant...
  33. M

    Maximal subgroup of a product of groups?

    Let G and H be finite groups. The maximal subgroups of GxH are of the form GxM where M is maximal subgroup of H or NxG where N is a maximal subgroup of G. Is this true?
  34. D

    Can 3-fields be consistently defined and constructed?

    I was thinking about some similarities in the definitions of group and field, and if it would be possible to generalize in some sense, like follows. A field is basically a set F, such that (F,+) is a commutative group with identity 0, and (F-{0}, .) is a commutative group with identity 1, and...
  35. J

    Free product of non-trivial groups is non-abelian

    Hello I have to show that the free product of a collection of more than one non-trivial group is non-abelian. But doesn't this just follow from the definition of the free product? Or how would you tackle this question?
  36. X

    Abstract Algebra Problem involving the order of groups

    Homework Statement Let G be a group with identity e. Let a and b be elements of G with a≠e, b≠e, (a^5)=e, and (aba^-1)=b^2. If b≠e, find the order of b. Homework Equations Maybe the statement if |a|=n and (a^m)=e, then n|m. Other ways of writing (aba^-1)=b^2: ab=(b^2)a...
  37. G

    Can anyone advise on the strength of modelling/theory groups in the UK

    I am in the fourth year of an MPhys and feel a PhD is the best way to further myself. I want to apply to groups that specialize in the theory and implementation of computational modelling. I would prefer this to be a group with a wide range across several branches of Physics as opposed to a...
  38. S

    Abstract Algebra Question: order, stabilizer, and general linear groups

    The question asks: 3) Let X be the set of 2-dimensional subspaces of F_{p}^{n}, where n >= 2. (a) Compute the order of X. (b) Compute the stabilizer S in GL_{n}(F_{p}) of the 2-dimensional subspace U = {(x1, x2, 0, . . . , 0) ε F_{p}^{n} | x1, x2 ε F_{p}}. (3) Compute the order of S. (4)...
  39. B

    Multiplicative groups of nonzero reals and pos. reals

    WTS is that \mathbb R^*/N \ \cong \ \mathbb R^{**} where N = (-1, 1) then prove that \mathbb R^*/\mathbb R^{**} \ is \ \cong \ to \mathbb Z/2\mathbb Z So the best answer in my opinion is to construct a surjection and use the first iso thm. f:\mathbb R^*\rightarrow\mathbb R^{**}...
  40. B

    What do they mean by classify all groups of a certain order

    what does it mean. I'm thinking list all groups of such order for instance. 115 = 5* 23 hence Z5⊕ Z23 ≈ Z115 ?
  41. R

    Order of Automorphism and Abelian Groups

    If |Aut H| = 1 then how can I show H is Abelian? I've shown a mapping is an element of Aut H previously but didn't think that would help. I have been looking through properties and theorems linked to Abelian groups but so far have had no luck finding anything that would help. The closest I...
  42. B

    Are All Groups Nonempty and How Does the Group Identity Factor In?

    Are all groups nonempty? If so, is it because all groups have an identity (element)?
  43. L

    External direct products of cyclic groups

    I'm wondering if anyone can help me with learning how to write groups as an external direct product of cyclic groups. The example I'm looking at is for the subset {1, -1, i, -i} of complex numbers which is a group under complex multiplication. How do I express it as an external direct...
  44. ArcanaNoir

    Isomorphic groups G and H, G has subgroup order n implies H has subgroup order n

    Homework Statement G is isomorphic to H. Prove that if G has a subgroup of order n, H has a subgroup of order n. Homework Equations G is isomorphic to H means there is an operation preserving bijection from G to H. The Attempt at a Solution I don't know if this is the right...
  45. E

    Prove product of infinite cyclic groups not an infinite cyclic group

    Homework Statement Show that the product of two infinite cyclic groups is not an infinite cyclic? Homework Equations Prop 2.11.4: Let H and K be subgroups of a group G, and let f:HXK→G be the multiplication map, defined by f(h,k)=hk. then f is an isomorphism iff H intersect K is...
  46. S

    Need Ideas on a project on Groups

    So this isn't really a homework question and I'm sorry if I posted in the wrong section. Please feel free to move it if needed. I'm doing a project on groups. I will be working on groups of order 12 through 16. I have no questions to follow or really have no idea on what to do with them...
  47. I

    Programs Graphene Research Groups (PhDs, UK)

    I am currently looking at what research groups to apply to for starting a PhD next year, is there anywhere particularly good that I am missing out? I did my Masters in Theoretical Physics but I am actually leaning more towards a mixture of both experimental and theory at this stage, I did do...
  48. H

    Is S4 a Subset of S5?

    This is not a homework question, just a question that popped into my head over the weekend. My apologies if this is silly, but would you say that the symmetric group S4 is a subset of S5? My friends and I are having a debate about this. One argument by analogy is that we consider the set...
  49. A

    Factor Groups: Conjugation & 2 Conjugates

    Factor groups! Please I just want to ask about factor groups.. how could a factor group G/A acts on A by conjugation, knowing that A is a normal & abelian subgroup of G.. and what do we mean when we say that an element in a group has jus 2 conjugates?? thanks in advance :)
  50. B

    Abstract Algebra - Cyclic groups

    (This is my first post on PF btw - I posted on this another thread, but I'm not sure if I was supposed to) I was doing some practice problems for my exam next week and I could not figure this out. Homework Statement Suppose a is a group element such that |a^28| = 10 and |a^22| = 20...
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