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. karush

    MHB Orders of Elements in Groups $$\Bbb{Z}_{12}, U(10), U(12), D4$$

    nmh{909} For each group in the following list, $$ \Bbb{Z}_{12}, \qquad U(10)\qquad U(12) \qquad D4 $$ (a) find the order of the group $$|\Bbb{Z}_{12}|=12$$ (b) the order of each element in the group.ok the eq I think we are supposed to use is $$\textit{ if } o(g)=n \textit{ then }...
  2. Math Amateur

    I Product of Categories .... Product of Groups as an example ...

    I am reading Steve Awodey's book: Category Theory (Second Edition) and am focused on Section 1.5 Isomorphisms ... I need some further help in order to fully understand some aspects of the definition of the product of two categories as it applies to the category Groups ... ... The definition of...
  3. Math Amateur

    MHB How can the product of two categories help us understand groups?

    I am reading Steve Awodey's book: Category Theory (Second Edition) and am focused on Section 1.5 Isomorphisms ... I need some further help in order to fully understand some aspects of the definition of the product of two categories as it applies to the category Groups ... ... The definition...
  4. Math Amateur

    MHB Category Theory .... Groups and Isomorphisms .... Awodey, Section 1.5 .... ....

    I am reading Steve Awodey's book: Category Theory (Second Edition) and am focused on Section 1.5 Isomorphisms ... I need some further help in order to fully understand some aspects of Definition 1.4, Page 12 ... ... The start of Section 1.5, including Definition 1.4 ... reads as...
  5. W

    I Exponential map for Lie groups

    I’ve read about the exponential map that for Lie groups the exponential map is actually the exponential function. But the exponential map is based on the geodesic ODE, so you need Christoffel symbols and thus the metric. But usually nobody gives you a metric with a Lie group. So how can I get...
  6. M

    Proof of Subgroup Property for Cyclic Group G: Homework Help

    Homework Statement Let G be a group. Assume a to be an element of the group. Then the set <a> = {ak I k∈ℤ} is a subgroup of G. I am confused as to why the proof makes the assumption that <a> is a subset of the set G. Homework EquationsThe Attempt at a Solution The proof I think is like the...
  7. W

    I Understanding Quotient Groups in Abstract Algebra

    So I'm just beginning to study abstract algebra and I'm not sure I grasp the definition of a quotient group, I believe it probably has to do with the book providing little to no examples. In trying to come up with my own examples, I imagined the following: Consider the Klein four group, if we...
  8. Mr Davis 97

    I Groups as symmetries of objects

    So it's said that every group is a symmetry group of some tangible object. For example, ##S_3## is the symmetry group of ##\{1,2,3 \}##, and ##D_{2n}## is the symmetry group of an n-gon. But what is ##GL_{10} (\mathbb{R})## the symmetry group of? What about ##\mathbb{Z}##? I have found two...
  9. Mr Davis 97

    Showing that cyclic groups of the same order are isomorphic

    Homework Statement Prove that any two cyclic groups of the same finite order are isomorphic Homework EquationsThe Attempt at a Solution So I began by looking at the map ##\phi : \langle x \rangle \to \langle y \rangle##, where ##\phi (x^k) = y^k##. So, I went through and showed that this is...
  10. Mr Davis 97

    I Showing that two groups are not isomorphic question

    I am trying to show that ##\mathbb{R} - \{ 0\}## is not isomorphic to ##\mathbb{C} - \{0 \}##. If we simply look at ##x^3 = 1##, it's clear that ##\mathbb{R} - \{ 0\}## has one solution while ##\mathbb{C} - \{0 \}## has three. My question, how can I use ##x^2 = -1## to show that they are not...
  11. K

    I Group Theory: Elements as Ops, Representations as Math?

    Given a group, can we regard its elements as statements of what the operation does, while its representations are the mathematical translation? For instance, given a square, I'd say that ##a## is an operation that rotates the square by 90°, and the representation of ##a## would be the matrix...
  12. BillTre

    Global Carbon Content of Groups of Organisms

    This article describes how total carbon in organisms is distributed among different groups. Science news article here. Original PNAS article here. Plants win, not surprisingly (primary producers). Bacteria are next (Archaea which are similar to bacteria in many ways have much less carbon)...
  13. MermaidWonders

    MHB Thioester Isomer Count: ${C}_{4}{H}_{8}OS$ - 4 Possibilities

    How many isomers are there with the following description? - Thioesters with the formula ${C}_{4}{H}_{8}OS$? I was able to draw 2 of them, but apparently, the answer key showed and stated that there are 4. I am confused about why the following two are possibliities: I thought that thioesters...
  14. diegzumillo

    I Breaking down SU(N) representation into smaller groups

    Hi all I have a shallow understanding of group theory but until now it was sufficient. I'm trying to generalize a problem, it's a Lagrangian with SU(N) symmetry but I changed some basic quantity that makes calculations hard by using a general SU(N) representation basis. Hopefully the details of...
  15. Math Amateur

    MHB Correspondence Theorem for Groups .... Another Question ....

    I am reading the book: "Advanced Modern Algebra" (Second Edition) by Joseph J. Rotman ... I am currently focused on Chapter 1: Groups I ... I need help with an aspect of the proof of Proposition 1.82 (Correspondence Theorem) ... Proposition 1.82 reads as follows...
  16. Math Amateur

    I Correspondence Theorem for Groups .... Another Question ....

    I am reading the book: "Advanced Modern Algebra" (Second Edition) by Joseph J. Rotman ... I am currently focused on Chapter 1: Groups I ... I need help with an aspect of the proof of Proposition 1.82 (Correspondence Theorem) ... Proposition 1.82 reads as follows: In the above proof by...
  17. Math Amateur

    MHB The Correspondence Theorem for Groups .... Rotman, Proposition 1.82 ....

    I am reading the book: "Advanced Modern Algebra" (Second Edition) by Joseph J. Rotman ... I am currently focused on Chapter 1: Groups I ... I need help with an aspect of the proof of Proposition 1.82 (Correspondence Theorem) ... Proposition 1.82 reads as follows: In the above proof by...
  18. Math Amateur

    I Correspondence Theorem for Groups .... Rotman, Propn 1.82 ....

    I am reading the book: "Advanced Modern Algebra" (Second Edition) by Joseph J. Rotman ... I am currently focused on Chapter 1: Groups I ... I need help with an aspect of the proof of Proposition 1.82 (Correspondence Theorem) ... Proposition 1.82 reads as follows: In the above proof by...
  19. L

    A Tensor symmetries and the symmetric groups

    In one General Relativity paper, the author states the following (we can assume tensor in question are tensors in a vector space ##V##, i.e., they are elements of some tensor power of ##V##) To discuss general properties of tensor symmetries, we shall use the representation theory of the...
  20. hideelo

    A Reality conditions on representations of classical groups

    I'm reading "Division Algebras and Quantum Theory" by John Baez https://arxiv.org/abs/1101.5690 In the last paragraph of section 5 (Applications) he says the following "SU(2) is not the only compact Lie group with the property that all its irreducible continuous unitary representations on...
  21. pellis

    A Who wrote "Ch 6 Groups & Representations in QM"?

    Who really wrote the best introductory account of representation theory in QM that I've seen so far ? [Likely mis-attribution discussed here below; prefixed "Advanced" to reach lecturers who are more likely to know the answer to this question.] It's available via...
  22. The Bill

    I What are the groups for NxNxN puzzle cubes called?

    The group of moves for the 3x3x3 puzzle cube is the Rubik’s Cube group: https://en.wikipedia.org/wiki/Rubik%27s_Cube_group. What are the groups of moves for NxNxN puzzle cubes called in general? Is there even a standardized term? I've been trying to find literature on the groups for the...
  23. J

    Programs European Cosmology groups for PhD

    Hi, I am planning to apply to some PhD programs in Cosmology in Europe. I've already identified some potential groups, but I'm sure that I'm still missing a few. I am interested in Cosmology in general (not a particular aspect/research area of it). Could you kindly inform me of any European...
  24. U

    Determining a group, by checking the group axioms

    Homework Statement For the following sets, with the given binary operation, determine whether or not it forms a group, by checking the group axioms. Homework Equations (R,◦), where x◦y=2xy+1 (R*,◦), where x◦y=πxy and R* = R - {0} The Attempt at a Solution For question 1, I found a G2...
  25. H

    A On the equivalent definitions for solvable groups

    We have the 3 equivalent definition for solvable groups: There exists a chain of subgroups 1 < G1 ...< Gi + < G i+1 < Gr = G such that Gi is normal in Gi+1 and Gi+1/Gi is abelian. Another definition is there exists 1 < H1 ...< Hi + < H i+1 < Hs = H such that Hi is normal in Hi+1, and...
  26. T

    I Are my thoughts about groups correct?

    I would kindly appreciate any corrections to my conclusions, because I need to get this subject straight for learning QFT in a satisfactory way. From what I have been reading about Lie groups so far, I have concluded the following: 1 - A group is independent of a representation, but we usually...
  27. B

    Linearly Independent Sets in Abelian Groups

    Homework Statement [/B] ##X## is linearly independent if and only if every nonzero element of the subgroup ##\langle X \rangle## may be written uniquely in the form ##n_1 x_1 + ... n_k x_k## (##n_i \in \Bbb{Z} \setminus \{0\}##, and ##x_1,...,x_k \in X## are distinct). Homework Equations [/B]...
  28. A

    I How can there only be two possible four-element groups?

    How can you prove that there can only be 2 possible four-element group?
  29. davenn

    Stargazing 2 significant spot groups currently visible

    hi gang there are currently 2 significant spot groups visible traversing the face of the solar disk The centre-left string is active region 2671 and the region near the right edge ( eastern limb) is active region 2672. AR2672 will continue to rotate across the disk across the next 2 weeks...
  30. S

    MHB How do I merge two age groups per 10k from an abortion rate table?

    I would like to find the 12 to 19 abortion rate per 10,000 women from the following table: This would be the merging of age groups "12-17" with "18-19" to some how get the 12-19 abortion rate. EDIT: "Distribution of abortion age" was translated from "Répartition des ges l'avortement" in...
  31. I

    Quantum QFT: groups, effective action, fiber bundles, anomalies, EFT

    Hi, I am looking for textbooks in QFT. I studied QFT using Peskin And Schroeder + two year master's degree QFT programme. I want to know about the next items: 1) Lorentz group and Lie group (precise adjectives, group representation and connection between fields and spins from the standpoint of...
  32. Luck0

    I Question about Haar measures on lie groups

    I'm not sure if this question belongs to here, but here it goes Suppose you have to integrate over a lie group in the fundamental representation. If you pass to the adjoint representation of that group, does the Haar measure have to change? I think that it should not change because it is...
  33. Math Amateur

    MHB Galois Groups .... A&F Example 47.7 .... ....

    I am reading Anderson and Feil - A First Course in Abstract Algebra. I am currently focused on Ch. 47: Galois Groups... ... I need some help with an aspect of the Example 47.7 ... Example 47.7 and its proof read as follows: In the above example, Anderson and Feil write the following...
  34. Math Amateur

    B Galois Groups .... A&F Example 47.7 .... ....

    I am reading Anderson and Feil - A First Course in Abstract Algebra. I am currently focused on Ch. 47: Galois Groups... ... I need some help with an aspect of the Example 47.7 ... Example 47.7 and its proof read as follows: In the above example, Anderson and Feil write the following: "...
  35. Math Amateur

    MHB Galois Groups .... A&W Theorem 47.1 .... ....

    I am reading Anderson and Feil - A First Course in Abstract Algebra. I am currently focused on Ch. 47: Galois Groups... ... I need some help with an aspect of the proof of Theorem 47.1 ... Theorem 47.1 and its proof read as follows: At the end of the above proof by Anderson and Feil, we...
  36. Math Amateur

    I Galois Groups .... A&W Theorem 47.1 .... ....

    I am reading Anderson and Feil - A First Course in Abstract Algebra. I am currently focused on Ch. 47: Galois Groups... ... I need some help with an aspect of the proof of Theorem 47.1 ... Theorem 47.1 and its proof read as follows: At the end of the above proof by Anderson and Feil, we...
  37. Math Amateur

    MHB Groups of Automporphisms - Aut(C) .... Anderson and Feil Ch. 24 ....

    I am reading Anderson and Feil - A First Course in Abstract Algebra. I am currently focused on Ch. 24: Abstract Groups ... ... I need some help in understanding some claims in Chapter 24 by Anderson and Feil ... ...Anderson and Feil claim that \text{Aut} ( \mathbb{C} ) is a group with only...
  38. Math Amateur

    I Groups of Automorphisms - Aut(C) ....

    I am reading Anderson and Feil - A First Course in Abstract Algebra. I am currently focused on Ch. 24: Abstract Groups ... ... I need some help in understanding some claims in Chapter 24 by Anderson and Feil ... ...Anderson and Feil claim that ##\text{Aut} ( \mathbb{C} )## is a group with only...
  39. F

    Proving Subgroups in Abelian Groups

    Homework Statement Let G be a group. if H = ##{x \epsilon G : x = x^{-1}}##, that is H consists of all elements of G which are their own inverses, prove that H is a subgroup of G. Homework Equations to show H is a subgroup of G, show that H is closed under the operation of G and every element...
  40. Math Amateur

    MHB Groups of Automorphisms - remarks by Anderson and Feil ....

    I am reading Anderson and Feil - A First Course in Abstract Algebra. I am currently focused on Ch. 24: Abstract Groups ... ... I need some help in understanding some claims in Chapter 24 by Anderson and Feil ... ...Anderson and Feil claim that \text{Aut}( \mathbb{R} ) is the trivial group...
  41. Math Amateur

    I Groups of Automporphisms - remarks by Anderson and Feil ....

    I am reading Anderson and Feil - A First Course in Abstract Algebra. I am currently focused on Ch. 24: Abstract Groups ... ... I need some help in understanding some claims in Chapter 24 by Anderson and Feil ... ...Anderson and Feil claim that ##\text{Aut}( \mathbb{R} )## is the trivial group...
  42. FallenApple

    A Testing if probability is the same for two groups

    Ok, so in a logistic regression context, I need to test if the probability of ##Y_{i}=1 ## is the same for two different groups at different ages where age is a continuous variable. This is actually complicated because of nonlinearity. Can I default to testing if the odds of ##Y_{i}=1 ## is...
  43. B

    Groups that cannot be the direct product of subgroups

    Homework Statement I am trying to show that neither ##Z_{p^n}## nor ##\mathbb{Z}## can be written as any family of its proper subgroups. Homework EquationsThe Attempt at a Solution First, I believe this solution (http://www.auburn.edu/~huanghu/math7310/7310-hw2-answer.pdf see problem 6) is...
  44. Mr Davis 97

    Theorem of Finitely Generated Abelian Groups

    Homework Statement Are the groups ##\mathbb{Z}_8 \times \mathbb{Z}_{10} \times \mathbb{Z}_{24}## and ##\mathbb{Z}_4 \times \mathbb{Z}_{12} \times \mathbb{Z}_{40}## isomorphic? Why or why not? Homework EquationsThe Attempt at a Solution I think I am misunderstanding the Theorem of Finitely...
  45. FallenApple

    A Interpreting Poisson Regression Estimates across groups

    Say for example I want to see the rate of injury for firefighter vs police vs soldier. ##InjuryCount_{i}## The number of injuries recorded for the ith person over time ##T_{i} ## Time the person was followed. Varies from person to person. ##I(f)_{i}## indicator for ith person of being a...
  46. Mr Davis 97

    The order of an element of a direct product of groups

    Homework Statement Let ##A## and ##B## be finite groups, and ##A \times B## be their direct product. Given that ##(a,1)## and ##(1,b)## commute, and that ##(a,1)^n = (a^n,1)## and ##(1,b)^n = (1,b^n)## for all a and b, show that the order of ##(a,b)## is the least common multiple of the orders...
  47. dkotschessaa

    A Homology groups with points identified

    I am looking for some general guidance on questions of the form: "Using a ## \Delta ## complex, compute the homology groups of the quotient space obtained fromt the 2-sphere ##S^2## by identifying three of its distinct points." Similarly I have a question about a torus with two points...
  48. Greg Bernhardt

    I Peter Woit's free QT, Groups and Representations ebook

    Enjoy! http://www.math.columbia.edu/~woit/QM/qmbook.pdf
  49. R

    MHB Isomorphic Groups: Z2 X Z3 & G = (1,2,4,8,10)

    hi Show that Z2 X Z3 IS ISOMORPHIC TO THE GROUP G = (1,2,4,8,10)
  50. M

    I Lie groups left invariant vector fields

    hello every one . can someone please find the left invariant vector fields or the generator of SO(2) using Dr. Frederic P. Schuller method ( push-forward,composition of maps and other stuff) Dr Frederic found the left invariant vector fields of SL(2,C) and then translated them to the identity...
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