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. B

    Abstract Algebra - Cyclic groups

    1. Problem: Suppose a is a group element such that |a^28| = 10 and |a^22| = 20. Determine |a|. I was doing some practice problems for my exam next week and I could not figure this out. (This is my first post on PF btw) 2. Homework Equations : Let a be element of order n in group and let k...
  2. L

    Determining groups not sure how to prove it.

    I'm going through a abstract algebra book I found and am trying to learn more about group theory by going through some of the proofs and practice sets, but am having trouble with the following: Prove that G={a+b*sqrt(2) | a,b E R; a,b not both 0} is a group under ordinary multiplication...
  3. L

    Orders of Quotient Groups (Abstract Algebra)

    Homework Statement Let H be a subgroup of K and K be a subgroup of G. Prove that |G:H|=|G:K||K:H|. Do not assume that G is finite Homework Equations |G:H|=|G/H|, the order of the quotient group of H in G. This is the number of left cosets of H in G. The Attempt at a Solution I...
  4. J

    Can You Help with These Abstract Algebra Proofs?

    Abstract Algebra Proof: Groups... A few classmates and I need help with some proofs. Our test is in a few days, and we can't seem to figure out these proofs. Problem 1: Show that if G is a finite group, then every element of G is of finite order. Problem 2: Show that Q+ under...
  5. L

    Groups and Inner Automorphisms

    Homework Statement Let G be a group. Show that G/Z(G) \cong Inn(G) The Attempt at a Solution G/Z(G) = gnZ(G) for some g ε G and for any n ε N choose some g-1 such that g(g-1h) = g(hg-1) and the same can be done switching the g and g-1 This doesn't feel right at all...
  6. E

    Infinite groups with elements of finite order

    Can anyone think of an example of an infinite group that has elements with a finite order?
  7. A

    Proving the nonisomorphic groups of order 16 are indeed, not isomorphic.

    Homework Statement Find at least 7 pairwise non-isomorphic groups of order 16, and prove that no two among them are isomorphic. Homework Equations I found 7 nonisomorphic groups, but I just am having trouble how to precisely prove they are not isomorphic... The Attempt at a Solution...
  8. L

    Why are General Linear Groups Non Abelian?

    Homework Statement Show that if n>1 and F is an arbitrary field, the general linear group defined by n and F is non-abelian Homework Equations A general linear group is the group of invertible matrices with entries from F A non abelian group is a group where the binary operation isn't...
  9. T

    Exploring Lie Groups: Questions and Concepts

    Hi everybody! Ok, so from a few days I've begun a group theory class, and i have to say i love the subject. In particular i happened to like Lie groups, but there are things that are not cristal clear to me, hope you'll help to figure'em out!First of all, Lie groups are continuous group, so...
  10. M

    Fundamental Theorem of Abelian Groups

    Show that there are two abelian groups of order 108 that have exactly one subgroup of order 3. 108 = 2^ 2 X 3 ^ 3 Using the fundamental theorem of finite abelian groups, we have Possible abelian groups of order 108 can be : Z108, Z4 + Z27, Z2+Z2+Z27, Z4+Z9+Z3, Z2+Z2+Z9+Z3, Z4+Z3+Z3+Z3...
  11. C

    Searching for quantum loop gravity groups

    I'm searching for research groups of quantum loop gravity, Does anyone can help me to find at least one of them?. I want to do my PhD on that topic and I have found kind of hard to find those groups and I am looking for a good supervisor, any reference would be very very helpful. Thanks...
  12. B

    What is the subgroup and order of a matrix group generated by A and B?

    Homework Statement A= \left( \begin{matrix} i & 0 \\ 0 &-i \end{matrix} \right) , B= \left( \begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix} \right) \\ Show that \langle A, B \rangle is subgroup of GL_2(\mathbb{C}). And Show that \langle A, B \rangle generated by A and B, and order of...
  13. T

    Are Study Groups for Everyone? Benefits & Downsides Examined

    Hey guys, would you say study groups benefit everyone? I've always studied by myself and so far I'm an A+ student. Are study groups something that help everyone, or are they mostly aimed at certain mindsets? For example, I could see the "slowest" persons of the group getting a lot out of study...
  14. B

    Are the phosphate groups of ATP protonated at pH = 7?

    Homework Statement The problem states: "Draw the chemical structure of ATP at a pH of 7. Homework Equations The Attempt at a Solution The textbook diagrams the phophate groups as unprotonated, but since H3PO4 has a pKa of <7, I was thinking that maybe each phosphate group would have lost one...
  15. O

    Horrible problem about abelian groups

    G is a group and for all elements a,b in G, (ab)^i = (a^i)(b^i) holds for 3 consecutive positive integers. Show that G is abelian. I know how to prove that if (ab)^2 = (a^2)(b^2) then G is abelian. I was thinking that you could reduce the given equality integer by integer till 2 or...
  16. K

    2 questions on coholomogy groups

    Question 1: $$0 \to A\mathop \to \limits^f B\mathop \to \limits^g C \to 0$$ is an exact short sequence,in order to prove $$\cdots \to H^q (A)\mathop \to \limits^{f^* } H^q (B)\mathop \to \limits^{g^* } H^q (C)\mathop \to \limits^{d^* } H^{q + 1} (A) \to \cdots$$ is an exact long...
  17. A

    Help in understanding groups (undergraduate level).

    I have studied a fair portion of groups, but couldn't imagine what they are all about. Please help me in this regards.
  18. M

    Finding Elements of Order 6 in Aut(Z720)

    I have to find the number of elements in Aut(Z720) with order 6. Please suggest how to go about it. 1) Aut(Z720) isomorphic to U(720) (multiplicative group of units). 2 ) I am using the fundamental theorem of abelian group that a finite abelian group is isomorphic to the direct products of...
  19. S

    Groups of order 21 (Need help understanding an inference)

    Sylow's theorem tells us that there is one 7-Sylow subgroup and either one of seven 3-Sylow subgroups. Call these subgroups H and K respectively. Sylow's theorem also tells that H is normal in G. I'm not going to write it all out as I don't think it's necessary but in the case when we have...
  20. G

    Proving Isomorphic Groups U(5) and U(10)

    Homework Statement For any positive integern, let U(n) be the group of all positive integers less than n and relatively prime to n, under multiplication modulo n. Show the the Groups U(5) and u(10) are isomorphic Homework Equations The Attempt at a Solution any 2 cyclic groups of...
  21. G

    Proving U(n) is a Multiplication Group Modulo n: Homework Solution

    Homework Statement Let U(n) be the set of all positive integers less than n and relatively prime to n. Prove that U(n) is agroup under multiplication modulo n Homework Equations The Attempt at a Solution n | a-a' implies n | b(a-a') =ab-a'b and n |b-b' implies n |a'(b-b')...
  22. B

    Corollaries of the Theorem of Finite Abelian Groups

    Homework Statement If G is a finite abelian group and p is a prime such that p^n divides order of G, then prove that G has a subgroup of order p^n Homework Equations Theorem of Finite Abelian Groups: Every finite abelian group G is a direct sum of cyclic groups, each of prime power...
  23. M

    Group Isomorphism: Proving f: Us(st)->U(t) is Onto

    To prove that : f : U_{s} (st) \rightarrow U(t) is an onto map. Note that Us(st)= {x \in U(st): x= 1 (mod s)} Let x \inU(t) then (x, t)=1 and 1<x< t How to proceed beyond point ?
  24. N

    Lie groups and non-vanishing vector fields

    I'm trying to understand why a Lie group always has a non-vanishing vector field. I know that one can somehow generate one by taking a vector from the Lie algebra and "moving it around" using the group operations as a mapping, but the nature of this map eludes me.
  25. R

    Director product expansion of Lie groups.

    For discrete groups, we can easily find the decomposition of the direct product of irreducible representations with the help of the character table. All we need to do is multiply the characters of the irreducible representations to get the characters of the direct product representation and then...
  26. W

    How many mutually nonisomorphic Abelian groups of order 50

    Homework Statement My question is about the rules behind the method of finding the solution not necessarily the method itself. (I am prepping for the GRE subject) How many mutually nonisomorphic Abelian groups of order 50? The Attempt at a Solution so, if understand this...
  27. B

    Finite-Index Subgroups of Infinite Groups.

    Hi, All: Please forgive my ignorance here: let G be an infinite group, and let H be a subgroup of G of finite index . Does H necessarily have torsion? I can see if , e.g., G was Abelian with G=Z^n (+) Z/m , then , say, would have subgroups of finite index, but I can't tell if...
  28. O

    Semi-Direct Product & Non-Abelian Groups

    Homework Statement Let p, q be distinct primes s.t. p \equiv 1 (mod q). Prove that there exists a non-Abelian group of order pq and calculate the character table. Homework Equations Semi-Direct Product: Let H = < Y | S > and N = < X | R > be groups and let \phi : H \rightarrow Aut...
  29. C

    Are Cyclic Groups with the Same Order Isomorphic?

    Homework Statement Just want to make something clear. Are all cyclic groups that have the same number of elements isomorphic to each other. The Attempt at a Solution I think yes because theirs is a one-to-one correspondence and the groups are cyclic which means they have generators.
  30. P

    Fundamental Forces and Lie Groups

    Hi all, Sorry, I'm not quite sure that I've posted this question in the proper place, but I figured field theory matches best with lie groups in this context. Anyway, my question has to do with the relationship between the fundamental forces (electromagnetism, weak, and strong) and their...
  31. Shackleford

    Groups and Subgroups: Clarifying Questions

    Well, apparently, I'm not too clear on a few things. For #4, what is <[8]> in the group Z18? What does the <> mean around the congruence class? Is my work correct? http://i111.photobucket.com/albums/n149/camarolt4z28/IMG_20110723_161645.jpg?t=1311456158...
  32. C

    Do Euler's Totient Function and Relative Primality Determine Group Generators?

    Homework Statement Does Euler's totient function tell me how many elements are in my group? And once I know how many elements are in my group. the generators are the ones that are relatively prime with the number of element in my group. Are my statements correct.
  33. C

    Good Books on Groups for Algebra Learners: A Book of Abstract Algebra

    Does anyone know of a good book to read about groups for algebra? I've head that a good book was "A Book of Abstract Algebra" by Charles R. Pinter. And I am just learning about groups so it should be basic.
  34. I

    What is the identity element in abstract algebra groups?

    The .pdf can be ignored. Let A + B = (A - B) U (B - A) also known as the symmetric difference. 1. Look for the identity and let e be the identity element A + e = A (A - e) U (e - A) = A Now there are two cases: 1. (A - e) = A This equation can be interpreted as removing from A all elements...
  35. S

    Find subgroups of finitely generated abelian groups

    Is there an "easy" method to finding subgroups of finitely generated abelian groups using the First Isomorphism Theorem? I seem to remember something like this but I can't quite get it. For example, the subgroups of G=Z_2\oplus Z are easy...you only have 0\oplus nZ and Z_2\oplus nZ for n\geq...
  36. Greg Bernhardt

    Facebook Hate Groups Emerge as Casey Anthony Released from Jail

    Cyber-bullying is a hot topic these days. *Many social networking sites have policies directed toward preventing cyber-bullying. *Facebook is one of those sites, having been the venue for several high profile bullying incidents recently*. *Facebook’s policy*directs victims of cyber-bullying to …...
  37. P

    Proving the Subgroup Property of Even Permutations in Permutation Groups

    Homework Statement Show that if G is any group of permutations, then the set of all even permutations in G forms a subgroup of G. I am not sure where to start - I know there is a proposition that states this to be true, but I know that is not enough to prove this statement.
  38. Jim Kata

    Structure theorem for abelian groups

    So say you have a presentation matrix A for a module, and you diagonalise it and you get something like diag[1,5] well you can interpret that as A can be broken down as the direct sum of 1+Z/5Z. That is the trivial group of just the identity plus cyclic 5. What if your module is defined over...
  39. K

    Intermediate subgroups between symmetric groups

    Homework Statement For n>1, show that the subgroup H of S_n (the symmetric group on n-letters) consisting of permutations that fix 1 is isomorphic to S_{n-1} . Prove that there are no proper subgroups of S_n that properly contain H.The Attempt at a Solution The first part is fairly...
  40. K

    Composition length of cyclic groups.

    Homework Statement Let G be a finite cyclic group and \ell(G) be the composition length of G (that is, the length of a maximal composition series for G). Compute \ell(G) in terms of |G|. Extend this to all finite solvable groups. The Attempt at a Solution Decompose |G| into its prime...
  41. K

    Extending automorphism groups to inner automorphism groups.

    I wanted to make clear just a quick technical thing. If G is a group, N is a normal subgroup, and \phi_g \in \text{Inn}(G), \phi_g(h) = g h g^{-1} then \phi_g is an automorphism of N, right? However, is it the case that we cannot say that \phi_g is an inner automorphism, since we are not...
  42. K

    Elements of odd-order abelian groups are squares.

    Homework Statement Let G be a finite abelian group, and assume that |G| is odd. Show that every element of G is a square. The Attempt at a Solution So we want to show that \forall g \in G, \exists h \in G, g = h^2 . Let g \in G be arbitary, and consider the subgroup generated by g, denoted...
  43. Matterwave

    Understanding Lie Groups: A Simple Definition

    Hi, so I didn't see exactly where group theory stuff goes...but since Lie groups are also manifolds, then I guess I can ask this here? If there's a better section, please move it. I just have a simple question regarding the definition of a Lie group. My book defines it as a group which is...
  44. B

    Why are only non-commutative groups called non-abelean or is this wrong?

    According to wiki: "a non-abelian group, also sometimes called a non-commutative group, is a group (G, * ) in which there are at least two elements a and b of G such that a * b ≠ b * a." I thought in order to be an abelean group, 5 axioms must be satisfied. If one of them is not satisfied it...
  45. A

    Expanding knowledge on the Classical Groups

    I need some material on the properties and relationships between classical groups. I was using Robert Gilmore's "Lie Groups, Lie Algebras and Some of their Aplications", but it barely covers it (section 2.iv). Does someone know about a book or any lecture notes that could be used to...
  46. L

    Group Theory: Showing the Order of Element 2 is 2k

    Homework Statement Let p=2^(k)+1 , in which k is a positive integer, be a prime number. Let G be the group of integers 1, 2, ... , p-1 under multiplication defined modulo p. By first considering the elements 2^1, 2^2, ... , 2^k and then the elements 2^(k+1), 2^(k+2), ... show that the...
  47. S

    Groups in Physics: What are they Used For?

    Hi there PF What are groups, and what are they used for in physics? For example if you look at QED, http://en.wikipedia.org/wiki/Quantum...cs#Mathematics , it is said here that QED is a abelian gauge theory with symmetry group U(1). What is this symmetry group, and what is it used for when...
  48. G

    Non-Isomorphic Abelian Groups of Order 54 and the Isomorphism of Factor Groups

    Homework Statement Determine how many non-isomorphic (and which) abelian groups there are of order 54. Determine which of these groups the factor group Z6 x Z18 / <(3,0)> is isomorphic to. Homework Equations The Attempt at a Solution Fundamental theorem for abelian groups...
  49. V

    Groups and orthogonal matrices question

    Let A and B be nxn matrices which generate a group under matrix multiplication. Assume A and B are not orthogonal. How can I determine an nxn matrix X such that X-1AX and X-1BX are both orthogonal matrices? Is it possible to do this without any special knowledge of the group in question?
  50. PainterGuy

    Mixing 2nd and 6th groups' elements to semiconductor silicon

    Hi everyone, :smile: I was wondering why they only use 5th and 3rd groups elements to create n-type and p-type semiconductors respectively. Couldn't we mix 6th or 2nd groups' elements instead to make the semiconductors? What would happen if we do this? Perhaps elements of those groups will...
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