# 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. ### A Quantum Groups as a Generalization of String Theory

Every once in a while I use my ancient trick of searching something in google with keywords, and found the above article. I don't think there's a free copy of it, because it's from 1989. I guess I need to read the pink book on foundations of Q-Groups by Majid. You know who also has written a...
2. ### A How Can I Compare Parton Distribution Functions Without Data from Other Groups?

Hello I plotted the Parton distribution functions in Mathematica. Now I want to compare the graphs drawn with the graphs of other groups(xu and xd). How should I do this?
3. ### Number of ways to partition n persons and probability to form n groups

1) At first my answer was ##n! \begin{pmatrix} n+r-1 \\ r - 1 \end{pmatrix} ## But I think that's not correct because let say first group consists of person A and B, by multiplying with n!, I also consider first group to be B and A which is just the same as A and B so there is double counting...
4. ### I Isomorphisms between C4 & Z4 Groups

0 Hint: Show that the isomorphism preserves the order of the element My solution: C4 = {e,r,r^2,r^3} where e-identity element and r is rotation by 90° Z4 = {0,1,2,3} LEMMA: ! Isomorphism preserves the order of the element ! (PROOF OF IT)Now we calcuate the order of the elements of both...
5. ### POTW Product of Two Finite Cyclic Groups

For each positive integer ##m##, let ##C_m## denote a cyclic group of order ##m##. Show that for all positive integers ##m## and ##n##, there is an isomorphism ##C_m \times C_n \simeq C_d \times C_l## where ##d = \operatorname{gcd}(m,n)## and ##l = \operatorname{lcm}[m,n]##.
6. ### I Using the orbit-stabilizer theorem to identify groups

I want to identify: ##S^n## with the quotient of ##O(n + 1,R)## by ##O(n,R)##. ##S^{2n+1}## with the quotient of ##U(n + 1)## by ##U(n)##. The orbit-stabilizer theorem would give us the result, but my problem is to apply it. My problem is how to find the stabilizer. In 1 how to define the...
7. ### A About computing the tangent space at 1 of certain lie groups

Hello :), I am wondering of the right and direct method to calculate the following tangent spaces at ##1##: ##T_ISL_n(R)##, ##T_IU(n)## and ##T_ISU(n)##. Definitions I know: Given a smooth curve ##γ : (− ,) → R^n## with ##γ(0) = x##, a tangent vector ##˙γ(0)## is a vector with components...

9. ### I Pin & Spin Groups: Double Covers of Orthogonal & SO Groups

Pin Groups are the double cover of the Orthogonal Group and Spin Groups are the double cover of the Special Orthogonal Group. Both sets of the double cover are considered to be groups, but it seems that only one of the sets of the double cover actually contains the identity element, which means...

30. ### MHB Connecting linear algebra concepts to groups

The options are rank(B)+null(B)=n tr(ABA^{−1})=tr(B) det(AB)=det(A)det(B) I'm thinking that since it's invertible, I would focus on the determinant =/= 0. I believe the first option is out, because null (B) would be 0 which won't be helpful. The second option makes the point that AA^{−1} is I...
31. ### MHB Exploring Finite Group Theory: Finding the Upper Bound of Groups of Order

In the context of group theory, there's a theorem that states that for a given positive integer $$n$$ there exist finitely different types of groups of order $$n$$. Notice that the theorem doesn´t say anything of how many groups there are, only states that such groups exist. In the proof of this...
32. ### I Understanding Lorentz Groups and some key subgroups

This thread is motivated by samalkhaiat's comment here I know that the Lorentz Group is formed by all matrices that satisfy $$\eta = \Lambda^{T} \eta \Lambda \tag{1.1}$$ Which is equivalent to $$\eta_{\mu\nu}\Lambda^{\mu}{}_{\rho}\Lambda^{\nu}{}_{\sigma} = \eta_{\rho \sigma} \tag{1.2}$$ If...
33. ### Algebra Book on Lie algebra & Lie groups for advanced math undergrad

Posting for my son (who does not have an account here): He's a sophomore math major in college and is looking for a good book on Lie algebra and Lie Groups that he can study over the summer. He wants mathematical rigor, but he is thinking of grad school in theoretical physics, so he also wants...
34. ### I Trying to Split Six People into Two Groups and Have Each Person Meet

Real World Application here. I'm creating a virtual meeting involving six people and will have the first round include two groups of three. Then we'll switch a few times. I tried hopelessly to do it such that after three rounds, everyone would see everyone else at least once. Was so close...
35. ### A Stone's theorem on one-parameter unitary groups and new observables?

I have been following the proof of the Stone's theorem on one-parameter unitary groups. The question is if the current list of self-adjoint operators used in quantum mechanics, including position and momentum operators, is exhaustive or not? Put it another way, can we say that there is no...
36. ### Is this matrix a non-abelian group?

I know for a group to be abelian a*b=b*a I tried multiplying the matrix by itself also but I’m not sure what I’m looking for. picture is below of the matrix https://www.physicsforums.com/attachments/255812
37. ### A Representations of finite groups: Irreducible and reducible

Matrix representation of a finite group G is irreducible representation if \sum^n_{i=1}|\chi_i|^2=|G|. Representation is reducible if \sum^n_{i=1}|\chi_i|^2>|G|. What if \sum^n_{i=1}|\chi_i|^2<|G|. Are then multiplication of matrices form a group? If yes what we can say from...
38. ### A Representations of finite groups -- Equivalent representations

I am confused. Look for instance cyclic ##C_2## group representation where D(e)= \begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix} and D(g)= \begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix} and let's take invertible matrix A= \begin{bmatrix} 1 & 2\\ 3 & 4 \end{bmatrix}. Then A^{-1}=...
39. ### I Questions about Rotation Groups in Physics

I have questions concerning group theory, esprecially Rotation groups. The first is: Are rotations groups f.ex. SO(2) defined for rotations in the actual physical 2 dimensional plane or are general rotations in any 2 dimensional space included? Someone wrote that "the action of an element of...
40. ### Algebra Lie Groups and Lie Algebras by Robert Gilmore

Anyone reading Lie Groups and Lie Algebras and Some of Their Applications by Robert Gilmore , might be interested in a series of YouTube videos by "XylyXylyX" that follows the book. The first lecture is:
41. ### MHB Finite vs Ring Groups: Examining Theorems

Dear Everyone, Does every theorem that holds for finite group holds for ring groups? Why or Why not?Thanks Cbarker1
42. ### Schools Are there any good groups about quantum computation?

Hi. After learning quantum computation for months, it fascinates me. Quantum computation expands my view about computational methods. I believe that many future achievements can be obtained from quantum computing, especially the combination of AI and quantum computation. Meanwhile, I think...
43. ### Schools Who are the Top Non-Equilibrium Research Groups in North America and Europe?

I realize the question is quite broad but what research groups working on statistical physics, stochastic processes, and complex systems are generally considered the best? Would like to know about Europe and America alike.
44. ### A Why does the Lie group ##SO(N)## have ##n=\frac{N(N-1)}{2}## real parameters?

When we have a Lie group, we want to obtain number of real parameters. In case of orthogonal matrices we have equation R^{\text{T}}R=I, that could be written in form \sum_i R_{i,j}R_{i,k}=\delta_{j,k}. For this real algebra ##SO(N)## there is ##n=\frac{N(N-1)}{2}## real parameters. Why this is...
45. ### Sub groups of the dihedral group

Homework Statement This is only a step in a proof I am trying to make. Let Dm be the dihedral group. r is the rotation of 2π/m around the origin and s is a reflexion about a line passing trough a vertex and the origin. Let<s> and <r> be two subgroups of Dm. Is there a theorem that states...
46. ### Doping semiconductors compounded from various element groups

Hi all - This is pulled from a past paper - Homework Statement I'm only going to state the components that I find challenging of this problem - The rest will be attached in my solution set. Essentially - given an intrinsic semiconductor comprised of group II-VI elements. Upon doping with group...
47. ### Show injectivity, surjectivity and kernel of groups

Homework Statement I am translating so bear with me. We have two group homomorphisms: α : G → G' β : G' → G Let β(α(x)) = x ∀x ∈ G Show that 1)β is a surjection 2)α an injection 3) ker(β) = ker(α ο β) (Here ο is the composition of functions.) Homework Equations This is from a...
48. ### Group Theory: Finite Abelian Groups - An element of order

Homework Statement Decide all abelian groups of order 675. Find an element of order 45 in each one of the groups, if it exists. Homework Equations /propositions/definitions[/B] Fundamental Theorem of Finite Abelian Groups Lagrange's Theorem and its corollaries (not sure if helpful for this...
49. ### I Symmetry Group Freedom: Choosing How Groups Act on Coordinates

One could argue that this question should be posted on the maths forum, but I see it so frequently in physics that I find it more productive to ask it here. In a symmetry group, do we have freedom of choice of how the group is going to act in the coordinates? Or is the way the group act on the...
50. ### MHB 3.13 Compute the orders of the following groups:

\nmh{837} Compute the orders of the following groups: $U(3), U(4), U(12)$ and $U(3), U(5), U(15)$. On the basis of your answers, make a conjecture about the relationship among $|U(r)|, |U(s)|$, and $|U(rs)|$. ok I still don't have a clear idea on how to do this $ax=1$ $U(3)=3$