What is Cyclic: Definition and 323 Discussions

In group theory, a branch of abstract algebra, a cyclic group or monogenous group is a group that is generated by a single element. That is, it is a set of invertible elements with a single associative binary operation, and it contains an element g such that every other element of the group may be obtained by repeatedly applying the group operation to g or its inverse. Each element can be written as a power of g in multiplicative notation, or as a multiple of g in additive notation. This element g is called a generator of the group.Every infinite cyclic group is isomorphic to the additive group of Z, the integers. Every finite cyclic group of order n is isomorphic to the additive group of Z/nZ, the integers modulo n. Every cyclic group is an abelian group (meaning that its group operation is commutative), and every finitely generated abelian group is a direct product of cyclic groups.
Every cyclic group of prime order is a simple group, which cannot be broken down into smaller groups. In the classification of finite simple groups, one of the three infinite classes consists of the cyclic groups of prime order. The cyclic groups of prime order are thus among the building blocks from which all groups can be built.

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  1. B

    Cyclic groups generator problem

    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...
  2. E

    Find All Automorphisms of Cyclic Group of Order 10

    Homework Statement Find all the automorphisms of a cyclic group of order 10. Homework Equations ψ(a)ψ(b)=ψ(ab) For G= { 1, x, x^2,..., x^9}, and some function ψ(a) = x^(a/10) The Attempt at a Solution I know that a homomorphism takes the form Phi(a)*phi(b) = phi (ab)...
  3. 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...
  4. Math Amateur

    Representations of the cyclic group of order n

    I am reading James and Liebeck's book on Representations and Characters of Groups. Exercise 1 of Chapter 3 reads as follows: Let G be the cyclic group of order m, say G = < a : a^m = 1 >. Suppose that A \in GL(n \mathbb{C} ) , and define \rho : G \rightarrow GL(n \mathbb{C} ) by...
  5. maverick_starstrider

    Relativistic angular momentum and cyclic coordinates

    I'm getting myself confused here. If my relativistic Lagrangian for a particle in a central potentai is L = \frac{-m_0 c^2}{\gamma} - V(r) should \frac{d L}{d \dot{\theta}} not give me the angular momentum (which is conserved)? Instead I get \frac{d L}{d \dot{\theta}} = -4 m...
  6. M

    The Cyclic Universe: A Theory of Formation and Evolution

    To begin, and believe me, it will become apparent, I have no training or education in this field. I do however find it fascinating. Can someone with a greater understanding please tell me if my ideas are laughable or feasable? Has anybody else proposed these ideas? Okay, so...
  7. P

    Proving Cyclic Quadrilateral: Opposite Angles Sum 180°

    Homework Statement I have got a question that might be easy for you. But, not for me. I have attached a pic. Please tell me if the quadrilateral inside the circle is a cyclic quadrilateral or not. If it is, then please explain how to prove that it's opposite angles sum up to 180 degrees...
  8. K

    Axisymmetric vs cyclic symmetry boundary conditions

    Hi, Can anyone explain the difference between axisymmetric and cyclic symmetry boundary conditions? Isn't it the same i.e. bith cyclic symmetry and axisymmetric?
  9. S

    Vibration analysis of a structure with cyclic symmetry

    Lets us say we are doing a vibration analysis of a structure with cyclic symmetry In very brief (as pointed out by AlephZero in one of his excellent reply) the whole motion can be represented by complex numbers which describe the motion of one segment. Now, my question is: 1)Is it...
  10. H

    Is time linear or cyclic in a flat, zero energy universe?

    I'm writing a paper for a philosophy elective on the Cosmological argument. One of my counter arguments is that a causal loop (treated as a paradox in the Cosmological argument in favor of a creator) is not a paradox if time is cyclic in nature rather than linear. I treat the fact that the...
  11. S

    Exploring Cyclic Symmetry: A Jet Engine Primer

    Hi, Please can anyone explian what is cyclic symmetry? I'm new to this term and have encountered this in jet engine example Vishal
  12. 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...
  13. T

    Does k Divide R's Order When R's Additive Group is Cyclic?

    If R is a finite ring and its additive group is cyclic, then R = <r> = {nr : n an integer} for some r in R. r^2 is in R so r^2 = kr for some integer k. Does k have to divide the order of R?
  14. J

    Proving a quadrilateral is cyclic and finding the radius of the circle

    Homework Statement The Attempt at a Solution So my first thought is that the only way to solve this problem is to apply a characterization of a cyclic quadrilateral. We know that the perpendicular bisectors of a cyclic quadrilateral are concurrent. So here's my thoughts: Construct...
  15. Z

    Calculate Diffusion Coefficient K4Fe(CN)6 Cyclic Voltammetry

    Homework Statement Calculate the diffusion coefficient (cm2/s) of ferricyanide if cyclic voltammograms conducted on a solution of 1 mM KClO4 + 5 mM K4Fe(CN)6 at scan rates of 1, 2, 5, 10, 20, and 50 mV/s, resulted in peak currents of 76, 100, 175, 243, 348 and 552 mA. The electrode used for...
  16. L

    Why do so many people claim cyclic models brake down due to entropy?

    To my intuition, I find more easy to believe that the universe is infinitely old and the big bag was just one of an infinite number of similar events in a larger universe, rather then the universe had a beginning. However, after watching many documentaries, and reading a lot of popular...
  17. 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...
  18. I

    Proof of Cyclic Subgroup Equivalence for Finite Groups

    Homework Statement Suppose a \in <b> Then <a> = <b> iff a and b have the same order (let the order be n - the group is assumed to be finite for the problem). Proof: Suppose a and b have the same order (going this direction I'm trying to show that <a> is contained in <b> and <b> is...
  19. L

    Showing the unit group is cyclic

    Homework Statement Let p be a positive prime and let Up be the unit group of Z/Zp. Show that Up is cyclic and thus Up \cong Z/Z(p − 1). The Attempt at a Solution What do they mean by the unit group? Is that just the identity? Is it the group [p]? I'm lost without starting the question...
  20. 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...
  21. 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...
  22. B

    Cyclic Universe - Fixed in Size but Divided into 8 Sections

    What is wrong with assuming a fixed size universe, dividing it into 8 octants, and allowing the various octants to expand and contract against each other? In such a situation, could our octant be currently expanding and due to the pressure of our expansion cause one or more adjacent octants to...
  23. J

    Cyclic Fusion Reactor_Colliding Beams_Final Edition

    Cyclic Fusion Reactor_Colliding Beams_Final Edition PDF file
  24. S

    Proof that a certain group G contains a cyclic subgroup of order rs

    Homework Statement Let G be an abelian group and let H and K be finite cyclic subgroups with |H|=r and |K|=s. Show that if r and s are relatively prime, then G contains a cyclic subgroup of order rsHomework Equations Fundemental theorem of cyclic group which states that the order of any...
  25. N

    Cyclic Codes of Length 4 over Z3: 9 Codewords

    Homework Statement How many cyclic codes of length 4 are there over Z3? Write down two such codes that are different, but each have 9 codewords. Homework Equations number of codewords of length 4 over Z3 = 3^4 = 81 Not sure about how to refine this to just being the number of cyclic...
  26. K

    How Does the Cyclic Decomposition Theorem Simplify Matrix Analysis?

    I took an Intermediate Linear Algebra course all last year (two semesters worth) and we covered the CDT. My professor didn't teach it well, and I got my first B- in university because of it (didn't affect my GPA but still irritating). I didn't understand a lot of the canonical form stuff...
  27. P

    Is the group of permutations on the set {123} Cyclic? Justification required

    Homework Statement Consider the group of permutation on the set {123}. Is this group cyclic? Justify your answer Homework Equations The Attempt at a Solution I wrote out the cayley table for this group, and noticed that if we take (123)^3 = e . Seeing as we can get back to the...
  28. Z

    Proving a Group is Cyclic: What is the Generator of G?

    Homework Statement For each integer n, define f_{n} by f_{n}(x) = x + n. Let G = {f_{n} : n \in \mathbb{Z}}. Prove that G is cyclic, and indicate a generator of G. Homework Equations None as far as I can tell. The Attempt at a Solution Doesn't this require us to find one element of G such...
  29. 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...
  30. B

    Classes of polynomials whose roots form a cyclic group

    Hi, I'm currently doing a project and this topic has come up. Are there any known famous classes of polynomials (besides cyclotomic polynomials) that fit that description? In particular, I'm more interested in the case where the polynomials have odd degree. I know for example that the roots of...
  31. J

    Cyclic Fusion Reactor. Passing through each other colliding beams.

    Recently I have placed here the new - viable by my opinion Concept how to produce fusion. By some reasons I have decided not to file the patent application and so for discussing now I am placing here the description of Cyclic Reactor on base of that Concept. Ioseb (Joseph) Chikvashvili
  32. C

    New Cyclic Model: Exploring Possibilities & Laws

    I do not have any mathematical proof of this being possible, but am hoping to include equations soon. For now, it is purely conceptual. According to the Steinhardt-turok model, our universe is on a 3-brane located next to another 3-brane. I will assume this is the case. I am also presuming...
  33. S

    Finding U13 Cyclic Numbers: A Faster Way?

    Homework Statement Is U13 cyclic? The Attempt at a Solution I know the elements are {1,2,3,4,5,6,7,8,9,10,11,12}. I have eliminated 1,2,3,4,5 and I am working on 6. I am doing it this way: 60=1 61=6 62=10 63=8 64=9 65=2 ..and so on, but I did, for example, 62=36-13=23=10...
  34. P

    If every two maximal subgroups of G are conjugate in G, prove that G is cyclic.

    1. Homework Statement Let G be a group in which each proper subgroup is contained in a maximal subgroup of finite index in G. If every two maximal subgroups of G are conjugate in G, prove that G is cyclic. 2. Homework Equations This problem arises as the problem #6 in section 5.4 of...
  35. Y

    Is the Product of A and An in a Cyclic Group of Order n Outside the Group?

    Given a cyclic group of order n, with all its elements in the form : A, A2, A3, ..., An where A is an arbitrary element of the group. According to the definition of group, "The product of two arbitrary elements A and B of the group must be an element C of the group", That is...
  36. F

    Battery Standby and Cyclic Use

    I've been doing a project recently that requires a UB1250ZH Universal Battery rated 12V and 5Ah. Although I understand the concept of Ah, I'm having a difficult time in figuring out what the ratings for standby and cyclic use mean. For an example, it says - Standby Use: 12.6-13.8V, the...
  37. Z

    Why is Z Cyclic? Exploring the Definition of Cyclic Groups in Math

    This might sound like a silly question, but based on Definition: A group G is called cyclic if there is g\in G such that \langle g \rangle = G And if we take (\mathbb{Z},+) the set of integers with addition as the operation, then why is it considered cyclic? Because the problem I am having...
  38. D

    Cyclic Compounds: Which Shape is the Most Stable?

    Homework Statement C3H6 cyclopropane C4H8 cyclobutane C6H12 cyclohexane Looking at their bond angles, which do you think is most stable? The Attempt at a Solution I'm trying to examine their shapes - triangle, sqaure and hexagon, but I don't know which shape is more stable...
  39. marcus

    Latest Penrose video on Conformal Cyclic Cosmology

    http://pirsa.org/11040063/ Big Perimeter audience, appreciative interest, long question period after. He undercuts his critics (who said the concentric circles could have appeared by chance) by criticising their methods and makes at least one valid point. I would say that the critics still...
  40. C

    Cyclic Subgroups in Symmetric and Cyclic Groups

    Suppose K= < x > is a cyclic group with 2 elements and H= S3 is symmetric group with 6 elements. Find all different cyclic subgroups of G= H x K. Now since K is generated by x with 2 elements, I have K= {1,x} and H= {1, (12), (13), (23), (123), (132)} What I am confused about is finding...
  41. B

    Abstract Algebra and cyclic subgroups

    Homework Statement from Algebra by Michael Artin, chapter 2, question 5 under section 2(subgroups) An nth root of unity is a complex number z such that z^n =1. Prove that the nth roots of unity form a cyclic subgroup of C^(x) (the complex numbers under multiplication) of order n...
  42. D

    Cyclic Universe, Big Rip version

    I wonder if someone has ever considered the following scenario: 1. "Standard" Big Rip model (phantom energy-dominated) 2. Close to the Big Rip, Universe becomes crossed with Cosmological horizons 3. Based on semiclassical approach, these horizons emit Hawking radiation 4. As a (slightly...
  43. M

    |G|=4. Prove the group is either cyclic or g^2=e

    Let G be a group with |G|=4. Prove that either G is cyclic or for any x in G, x^2=e.
  44. radou

    Cyclic group generator problem

    Homework Statement Let <a> be a cyclic group, where ord(a) = n. Then a^r generates <a> iff r and n are relatively prime. The Attempt at a Solution OK, let n and r be relatively prime. Then ord(a^r) = n. We need to show that for some j, there exists an integer k such that (a^r)^k = a^j...
  45. radou

    Two more simple cyclic group problems

    Homework Statement 1) let <a> be a cyclic group of order n. If n and m are relatively prime, then the function f(x) = x^m is an automorphism of <a>. 2) Let G be a group and a, b be in G. Let a be in <b>. Then <a> = <b> iff a and b have the same order. The Attempt at a Solution 1) It...
  46. radou

    A very simple cyclic group problem

    Homework Statement As the title suggests - this is very simple, I only want to check. Let G be a cyclic group of order n. Then, for every integer k which divides n, there are elements in G of order k. The Attempt at a Solution Now, G = <a>, and a^n = e by definition. Let k be an...
  47. X

    Find the order of the cyclic subgroup of D2n generated by r

    Homework Statement Find the order of the cyclic subgroup of D2n generated by r. Homework Equations The order of an element r is the smallest positive integer n such that r^n = 1. Here is the representation of Dihedral group D2n = <r, s|r^n=s^2=1, rs=s^-1> The elements that are in D2n...
  48. R

    Organic Chemistry: Stabillity of Nitrogen Containing Cyclic Compounds

    Homework Statement Pyridine and pyrrolidine react rapidly with dilute aqueous HCl to form the corresponding hydrochloride salts which are easily purified, isolated and stored in a charge. However, pyrrole, which is another nitrogen-containing heterocycle, does not form a hydrochloride salt...
  49. S

    Reaction Mechanism for 1,3-Dibromopropane to Cyclopropane

    What is the reaction mechanism for 1,3-dibromopropane + 2Na ---> cyclopropane? Thanks :)
  50. B

    How do branes emerge in cyclic universe?

    how do branes emerge in cyclic universe?? ok so in the cyclic model of the universe...two branes are colliding and this is causing big bangs every few trillion years. this solves nicely the initial singularity and explains many things but i have two questions 1) how do these branes...
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