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. Fallen Angel

    MHB Finite Group Inverses: Proving $N_{ABC}=N_{CBA}$

    Hi, I bring a new algebraic challenge ;) Let $G$ be a finite group and $U,V,W\subset G$ arbitrary subsets of $G$. We will denote $N_{UVW}$ the number of triples $(x,y,z)\in U\times V \times W$ such that $xyz$ is the unity of $G$, say $e$. Now suppose we have three pairwise disjoint sets...
  2. M

    MHB Direct sum of free abelian groups

    Show the direct sum of a family of free abelian groups is a free abelian group. My first thought was to just say that since each group is free abelian we know it has a non empty basis. Then we can take the direct sum of the basis to be the basis of the direct sum of a family of free abelian...
  3. TrickyDicky

    Automorphisms groups and spaces

    Is there any example of an automorphisms group of a space that coincides with the space, i.e. a space that is its own automorphisms group?
  4. T

    Quick question about subgroups of "odd" dihedral groups

    Will odd dihedral groups (e.g. d6, d10, d14) only have the identity, σ, τ and itself as subgroups as any power of σ with τ generates the entire group? If so would the subgroup lattice of d14 just be: d14 → σ & τ → e Thanks!
  5. H

    Exploring One-Sided vs. Two-Sided Identity Elements in Groups

    Hi I'm taking a math course at university that covers introductory group theory. The textbook's definition of the identity element of a group defines it as two sided; that is, they say that a group ##G## must have an element ##e## such that for all ##a \in G##, ##e \cdot a = a = a \cdot e## ...
  6. nomadreid

    The advantage of modular arithmetic, e.g. cyclic groups?

    In starting to look into the mathematical side of encryption , I note the heavy dependence upon modular arithmetic. What is the advantage is this? For example, why are finite cyclic groups and rings preferable? Note: I know zilch about programming; I am approaching it from the mathematical side.
  7. B

    Direct Product of Cyclic Groups

    Hello everyone, I was wondering if the following claim is true: Let ##G_1## and ##G_2## be finite cyclic groups with generators ##g_1## and ##g_2##, respectively. The group formed by the direct product ##G_1 \times G_2## is cyclic and its generator is ##(g_1,g_2)##. I am not certain that it...
  8. JonnyMaddox

    Lie Groups, Lie Algebra and Vectorfields

    Hi I'm learning about Lie Groups to understand gauge theory (in the principal bundle context) and I'm having trouble with some concepts. Now let a and g be elements of a Lie group G, the left translation L_{a}: G \rightarrow G of g by a are defined by : L_{a}g=ag which induces a map L_{a*}...
  9. PsychonautQQ

    Working with Factor Groups

    Homework Statement Let G be a finite group and let K be normal to G. If the factor group G/K has an element of order n, show that G has an element of order n. Homework Equations None The Attempt at a Solution Lets say Kg is the element in G/K with order n. That means: (Kg)^n = K and...
  10. PsychonautQQ

    Factor Groups: What am I thinking about wrongly here?

    say K is normal in G hence we have a factor group G/K. let g be an element of G where |g| = n. so Kg^n = K since g^n = 1. and using the properties of factor groups, we know Kg^n = (Kg)^n hence (Kg)^n = K So we know that the order of Kg divides n. Is this correct thinking? Factor groups are...
  11. Kushwoho44

    Fluid Mechanics: Dimensionless Groups Question, explain an outlier

    Homework Statement Hi guys, I need to explain the outlier point here, which has been shaded in the excel spread sheet when comparing the two dimensionless groups, The dimenionsless group, drag-coefficient is given by Drag/(density*V^2*D^2) and dimensionelss group, spin parameter, is given by...
  12. B

    Proving that the Dihedral Groups are non-cylic

    Hello everyone, I am suppose to show that all the Dihedral groups (##D_n##, for ##n >2##) are noncyclic. I know that every cyclic group must be abelian. So, what I intended on showing was that at least two elements in ##D_n## are not commutative. Here are my thoughts: Because we are dealing...
  13. N

    Explaining Electroweak Theory Decomposition to a Beginner

    I have come across physicists representing electroweak theory as some kind of decomposition (i.e. U(1)xSU(2)). I was wondering if someone could explain this 'crossing' to me a little further. Fair warning, my understanding of group/gauge theory is v rudimentary at this point.
  14. B

    What am I missing?What is the Proof for Cyclic Groups Being Subgroups?

    Hello everyone, I am trying to understand the proof given in this link: https://proofwiki.org/wiki/Subgroup_of_Cyclic_Group_is_Cyclic I understand everything up until the part where they conclude that ##r## must be ##0##. Their justification for this is, that ##m## is the smallest integer...
  15. B

    Hopf Algebras in Quantum Groups

    In the theory of quantum groups Hopf algebras arise via the Fourier transform: "A third point of view is that Hopf algebras are the next simplest category after Abelian groups admitting Fourier transform" At least for nice functions, a Fourier series is just a Laurent series on a circle (which...
  16. J

    Groups of homomorphisms of abelian groups

    Hello everybody! I've just started with studying group homorphisms and tensor products, so i am still not very sure if i undertstand the subject correct. I am stuck with a question and i would ask you for some help or hints how to proceed... What i have to do is to describe...
  17. W

    Homology Functor, Prod. Spaces, Chain Groups: Refs Needed

    Hi all, Went to a seminar today, arrived a few minutes late; hope someone can tell me something about this topic and/or give a ref so that I can read on it . I know this is a lot of material; if you can refer me to at least some if, I would appreciate it : 1)Basically, understanding how/why the...
  18. PsychonautQQ

    Modern Algebra Factor Groups question

    Homework Statement If K is normal in G and |g| = n for some g in G, show that the order of Kg in G/K divides n. Homework Equations None The Attempt at a Solution Okay so I feel like I have a solution but I don't use all the information given so I'm trying to find holes in it... g^n = 1...
  19. topsquark

    MHB Generalizing Rigid Motions Group w/ Metric

    Define: Euc(n) = \{ T \in End( \mathbb{R}^n )| ~ ||Tx - Ty|| = ||x - y||~\forall x,y \in \mathbb{R}^n \} This is defined as the Euclidean group of rigid motions. Can we generalize this group to be defined with any metric (well actually inner product, I suppose)? Obviously it won't be...
  20. Math Amateur

    MHB Help Peter Prove Simple Proposition on Simple Groups & Maximal Normal Subgroups

    I need help with the proof of an apparently simple Proposition in Aigli Papantonopoulou's book: Algebra: Pure and Applied. The proposition in question is Proposition 5.2.3 and reads as follows:Can someone please help me and provide an explicit and formal proof of this proposition.Since...
  21. M

    MHB Verifying Sentences about Groups: Answers and Hints

    Hey! :o I have to determine whether the following sentences are correct or not. Any two groups with three elements are isomorphic. At any cyclic group, each element is a generator. Each cyclic group has at least one non trivial proper subgroup. The group $G=\{ 1, i, -1, -i \}$ with respect...
  22. Math Amateur

    MHB Factor Groups and Simple Groups - Fraleigh: Theorem 15.18

    I need some help with the proof of Theorem 15.18 in Fraleigh: A First Course in Abstract Algebra. The text of Theorem 15.18 reads as follows: In the above text we read: " ... ... Now \gamma^{-1} of any non-trivial proper normal subgroup of G/M is a proper normal subgroup of G properly...
  23. metapuff

    Are All Indecomposable Groups Cyclic?

    A group is said to be indecomposable if it cannot be written as a product of smaller groups. An example of this is any group of prime order p, which is isomorphic to the group of integers modulo p (with addition as the group operation). Since the integers modulo p is a cyclic group (generated by...
  24. Math Amateur

    MHB Second Isomorphism Theorem for Groups

    I am revising the Isomorphism Theorems for Groups in order to better understand the Isomorphism Theorems for Modules. I need some help in understanding Dummit and Foote's proof of the Second Isomorphism Theorems for Groups (Diamond Isomorphism Theorem ? why Diamond ?). The relevant text from...
  25. O

    Find US HEP Research Groups | Best Experimental High Energy Physics

    Where to go to study experimental high energy physics in the US? What are the best groups working on experiment HEP to join with?
  26. A

    Complete List of 230 Space Groups in CSV Format for Easy Access

    Does anyone know of anywhere on the internet where I may find a spreadsheet containing the names of the 230 space groups? In a perfect world this would simply be a csv file containing the schoenflies and hermann-maguin notations; but I can't be too picky. Thanks
  27. D

    Changing undergraduate lab groups?

    I'm currently working in a very reputable lab and am wondering if it would be wise to switch to a smaller lab if the opportunity were to arise. The reason I am considering this is because in my current lab I do mostly programming and some 3d modelling. Whereas perhaps in a smaller lab I...
  28. PcumP_Ravenclaw

    What Are the Permutations and Principles in Group Theory?

    Hello everybody, https://www.physicsforums.com/showthread.php?t=768109 Please see above post for the questions after scrolling down fully. My answers to questions 1) 4 7 9 2 6 8 1 5 3 is the unique functtion and (1 4 2 7)(3 9)(5 6 8) are the permutations. 2) I don't understand the...
  29. K

    What are the algebra prerequisites for Lie groups?

    I don't know if this is the correct section for this thread. Anyway, I'm taking a graduate course in General Relavity using Straumann's textbook. I skimmed through the pages to see his derivation of the Schwarzschild metric and it assumes knowledge of Lie groups. I've never had an abstract...
  30. C

    Is there any groups working on FF's like CTEQ on PDF's?

    I know some groups on PDF's like CTEQ, MSTW, NNPDF... Is there any groups working on the parameterization of fragmentation functions, and publish any codes that can be called?
  31. C

    Experiments and groups about the parameterization of PDFs & FFs

    I want to do a survey about the parameterization of PDFs & FFs, and make a summary (say, in tables and figure) about the experiments and groups working on it, and their results. Where can I found these materials? Regards!
  32. PcumP_Ravenclaw

    Please verify a problem on Groups and permutations

    Alan F beardon, Algebra and Geometry chapter 1 6. For any two sets A and B the symmetric difference AΔB of A and B is the set of elements in exactly one of A and B; thus AΔB = {x ∈ A ∪ B : x / ∈ A ∩ B} = (A ∪ B)\(A ∩ B) . Let be a non-empty set and let G be the set of subsets of (note that G...
  33. M

    Beginning Group Theory, wondering if subset of nat numbers are groups?

    I know this post is in the topology thread of this forum, for group theory, this seemed like the reasonable choice to post it in. I realize group theory is of great importance in physics and I'm trying to eventually understand Emmy Noether's theorem. I'm learning group theory on my own, and...
  34. H

    Homomorphism between Lie groups

    Hello I've been reading some Group theory texts and would like to clarify something. Let's say we have two Lie groups A and B, with corresponding Lie algebras a and b. Does the fact that a and b share the same Lie Bracket structure, as in if we can find a map M:a->b which obeys...
  35. Math Amateur

    MHB Modules - Decomposibiity of abelian groups

    I am reading An Introduction to Rings and Modules With K-Theory in View by A.J. Berrick and M.E. Keating (B&K). I need help with understanding Example 2.1.3 (ii) (page 39) which concerns L as a submodule of the quotient module \mathbb{Z}/p^r \mathbb{Z} ... ... Example 2.1.3 (ii) (page 39)...
  36. PsychonautQQ

    Understanding the Cyclic Property of Groups

    Homework Statement My online notes stated that it |g| = |G| where g is an element of G then |G| is cyclic. Can somebody help me understand why this is true?
  37. C

    How Is the Killing Metric Normalized for Compact Simple Groups?

    The killing form can be defined as the two-form ##K(X,Y) = \text{Tr} ad(X) \circ ad(Y)## and it has matrix components ##K_{ab} = c^{c}_{\ \ ad} c^{d}_{\ \ bc}##. I have often seen it stated that for a compact simple group we can normalize this metric by ##K_{ab} = k \delta_{ab}## for some...
  38. Deveno

    MHB The Universal Property of the Direct Product in Groups

    With groups, one often seeks to create larger groups out of smaller groups, or the reverse: break down large groups into easier-to-understand pieces. One construction often employed in this regard is the direct product. The normal way this is done is like so: The direct product of two groups...
  39. J

    Proving o(An) = o(a) for Finite Abelian Groups | G, N, a | Group Theory Homework

    Homework Statement Let G be a finite group with N , normal subgroup of G, and a, an element in G. Prove that if (a) intersect N = (e), then o(An) = o(a). Homework Equations The Attempt at a Solution (aN)^(o(a)) = a^(o(a)) * N = eN = N, but is the least power such that (aN)^m = N...
  40. G

    Computing Čech cohomology groups

    I'm trying to understand Čech cohomology and for this I'm looking at the example of ##S^1## defined as ##[0,1]/\sim## with ##0 \sim 1##. To compute everything, I have the cover ##\mathcal U## consisting of the sets $$U_0= (0, 2/3) \, , \, U_1= (1/3, 1) \, , \, U_2= (2/3, 1] \cup [0, 1/3)$$...
  41. D

    Certain number of people arranged in several groups

    Homework Statement Find the number of ways of 9 people can be divided into two groups of 6 people and 3 people. Homework Equations The Attempt at a Solution my working is there are 6! arrangement for 6 people and 3! arrangment for 3 people. then the group of 3 people can be...
  42. L

    Isomorphic Groups: Proving Theorem True?

    Two groups are isomorphic if they has same number of elements and if they has same number of elements of same order? Is it true? Where can I find the prove of this theorem?
  43. Math Amateur

    MHB Finitely generated groups

    I am reading Chapter 2: Commutative Rings in Joseph Rotman's book, Advanced Modern Algebra (Second Edition). I am currently focussed on Theorem 2.68 [page 117] concerning finitely generated groups I need help to the proof of this theorem. Theorem 2.68 and its proof read as follows:In the...
  44. bigfooted

    Any recommendations for a good self-study book on Galois groups?

    I want to get a decent introduction into group theory and Galois groups. Can somebody recommend a good book that I can use for self-study? The book of Stewart - Galois Theory looks promising.
  45. Math Amateur

    MHB Groups as Groupoids with One Object

    I am reading Paolo Aluffi's book: "Algebra: Chapter 0" I am currently focussed on Chapter 2: "Groups: First Encounter". On page 41, Aluffi defines a group as follows: ------------------------------------------------------------------------------ Definition: A group is a groupoid with a...
  46. R

    What are the SU groups? I.e SU(2), SU(3)?

    I'm revising for my introductory particle physics exam and I've noticed SU(3) pop up a few times, it was never really explained and I don't really understand what it means. For example when talking about the conservation law of Colour, it says the symmetry is phase invariance under SU(3)...
  47. P

    Galois Theory, Differential Equations, and Lie Groups?

    I apologize for the informal and un-rigourous question. I have heard, in passing, that doing Galois Theory over Lie Groups instead of discrete groups is connected to solutions of differential equations instead of algebraic equations. First of all, is this correct? If so, what is this...
  48. W

    Tensors of Free Groups and Abelian groups

    Hi, let S be any set and let ##Z\{S\}## be the free group on ##S##, i.e., ##Z\{S\}## is the collection of all functions of finite support on ##S##. I am trying to show that for an Abelian group ##G## , we have that : ## \mathbb Z\{S\}\otimes G \sim |S|G = \bigoplus_{ s \in S} G ##, i.e., the...
  49. Mandelbroth

    Why Are Homology Groups Not MUCH Larger?

    So far, I think algebraic topology is turning out to be the best thing since sliced bread. However, I'm having a bit of difficulty with homology, for one particular reason. Consider, as an example, the first homology group of ##S^1##. The definition of the free abelian group (or, in general...
  50. B

    Why do several Lie groups have the same Lie algebra when acting on a space?

    Why is it that several lie groups can have the same Lie algebra? could it have to do with the space where they act transitively? Could two different Lie groups acting transitively on the same space have the same Lie algebra?
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