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. N

    Static Equivalent force for cyclic load (fatigue related)

    Homework Statement I have a suspension mount (square tube with unknown thickness) with a bolt going through it that undergoes an 800 lb load laterally into the bracket in an on/off fashion (NOT fully reversed) at a rate of 50 Hz. Ultimately we are trying to calculate the dimensions of the bolt...
  2. J

    A What is the Current Status of Cyclic Cosmology and its Relation to M-Theory?

    Its been formalized for ~15 yrs now by Steinhardt & Turok; Wiki sez it has problems, but will not elaborate. My concern is that despite their denial, their version of CC is built on branes, which are of course a Very speculative basis, since it originates in M-theory. Worse, S&T do not seem tb...
  3. PsychonautQQ

    Isomorphism to certain Galois group and cyclic groups

    Homework Statement Let c be a pth root of unit where p is prime. Then the Galois group G(Q(c):Q) is isomorphic to Z_p*. Show that if there is some m that divides p-1, then there is an extension K of Q such that G(K:Q) is isomorphic to Z_q* Homework EquationsThe Attempt at a Solution I suspect...
  4. P

    I Can Hyperbolic Space be affecting our view of the universe?

    Usually when gravitational lensing is discussed, the examples are those of matter bending spacetime into a positive curvature. https://commons.wikimedia.org/wiki/File:Gravitational_lens-full.jpg In these cases, distortion of light is clearly evident as images of galaxies from behind these...
  5. M

    MHB Find the angle, cyclic quadrilaterals

    Here is a circle with center $O$ :cool: Its is given that $\angle ABD=50$ & to find the magnitudes of $\angle ACD$ & $\angle ACB$ Now what I know is (Nerd) $\angle ACD=50$ due to the inscribed angle theorem, Can you help me to find the other angle which I don't know how to find ,stating the...
  6. M

    MHB How to Solve Angle Equivalence in Cyclic Quadrilaterals?

    Here's the problem (Wait) (Sweating) So, Any Ideas on how to begin ? (Happy)
  7. G

    B Cyclic quadrilateral and alternate segment

    As the secant AE is moved downwards, the exterior angle remains equal to the same interior angle, with the result that as the secant becomes a tangent, the cyclic quadrilateral disappears and the exterior angle becomes equal to the angle in the alternate segment. pdf is attached.It is...
  8. C

    Inverse tangents in cyclic order

    Homework Statement The problem- if $$\theta= tan^{-1}(\frac{a(a+b+c)}{bc})+tan^{-1}(\frac{b(a+b+c)}{ac})+tan^{-1}(\frac{c(a+b+c)}{ab})$$ , then find $$tan\theta$$ Homework EquationsThe Attempt at a Solution I tried to use these as sides of a triangle and use their properties, but other than...
  9. M

    MHB How to Find the Product of Cyclic Groups in an Abelian Group?

    Hey! :o Let $M$ be the abelian group, i.e., a $\mathbb{Z}$-module, $M=\mathbb{Z}_{24}\times\mathbb{Z}_{15}\times\mathbb{Z}_{50}$. I want to find for the ideal $I=2\mathbb{Z}$ of $\mathbb{Z}$ the $\{m\in M\mid am=0, \forall a\in I\}$ as a product of cyclic groups. We have the following...
  10. M

    MHB Proving $M$ is Cyclic: Simple $R$-Module & Isomorphism with Maximal Ideal $J$

    Hey! :o Let $R$ be a commutative ring with unit and $M$ a $R$-module. If $M$ is a simple $R$-module, i.e., the only $R$-submodule are $O$ and $M$, then $M$ is cyclic and isomorphic to $R/J$ where $J$ is a maximal ideal of $R$. Could you give me some hints how we could show that $M$ is cyclic...
  11. M

    MHB Proving That $G$ is Abelian When $G/Z$ is Cyclic

    Hey! :o Let $Z\subseteq Z(G)$ such that $G/Z$ is cyclic. I want to show that $G$ is abelian. We have the following: $$Z(G)=\{g\in G\mid ga=ag \ \forall a \in G\} \\ G/Z=\{gz\mid g\in G\}, z\in Z$$ Since $G/Z$ is cyclic we have that $(gz)^n=1$. To show that $G$ is abelian, we want to...
  12. M

    How to solve crossover issue during cyclic Voltammetry?

    How to solve crossover issue during cyclic Voltammetry? We are using Electrochemical Test instrument to get CV, RDE results. When we start CV or RDE test, we are having crossover issue instead of nice results. We test nearly all possibilities including pure Pt plate, thin film samples, nanorods...
  13. 1

    Evidence for a Cyclic Universe

    http://physics.princeton.edu/~steinh/lambda16.pdf In this research article the authors suggest a cyclic universe, specifically one involving collisions of higher dimensional branes (an idea taken out of string theory), could indirectly explain why the observed cosmological constant is so small...
  14. G

    Cyclic Quotient Group: Is My Reasoning Sound?

    Hi everyone. So it's apparent that G/N cyclic --> G cyclic. But the converse does not seem to hold; in fact, from what I can discern, given N cyclic, all we need for G/N cyclic is that G is finitely generated. That is, if G=<g1,...,gn>, we can construct: G/N=<(g1 * ... *gn)*k> Where k is the...
  15. mykamakiri

    Total mass and/or future of dark energy = cyclic universe?

    In an article called "From big bang to big bounce" published in New Scientist in 2008, author Anil Ananthaswamy outlines two different theories that lead to our universe being cyclic. 1: "Cosmologists are still very much in the dark about dark energy. Some theoretical models speculate that the...
  16. DeldotB

    Determine if a group is cyclic

    Hello all! If I have a group of order 20 that has three elements of order 4, can this group be cyclic? What if it has two elements? I am new to abstract algebra, so please keep that in mind! Thanks!
  17. P

    Higgs cyclic model from Steinhardt, Turok, Bars

    Its been suggested that the metastibility of the Higgs may lead to a new cyclic cosmology to replace inflation. http://arxiv.org/abs/1307.8106 Can anyone give a layman's guide to how this works and they propose to solve the problems of the big bang that inflation is supposed to solve: flatness...
  18. S

    Entropy change for reversible and cyclic process

    Homework Statement An ideal diatomic gas is initially at temperature ##T## and volume ##V##. The gas is taken through three reversible processes in the following cycle: adiabatic expansion to the volume ##2V##, constant volume process to the temperature ##T##, isothermal compression to the...
  19. K

    Cyclic Permutations: εijk, Even or Odd?

    εijk is the permutation symbol and cyclic permutations, for example 123→231→312, are always even, thus ε123=ε231=ε312=+1, but: ε132=ε213=ε321=-1 I understand the first 2, but ε321 is even, no? and also all this series is cyclic, it's not all even and...
  20. M

    Calculating Cyclic Energy from Acceleration (Relative)

    Hello, I am looking at relating two situations under which cyclic energy is applied to a material. Condition 1: A material has been subjected to a force of 1G at 0.1Hz for 47 days. Condition 2: The same material has been subjected to a force of 4.5G at 60Hz for 3600Seconds. Is it possible to...
  21. H

    Penrose's cyclic universe - question

    I've just watched the lecture of Penrose on his cyclic universe theory here: I fact I understood that he claims that any kind of matter dissapears in a couple of Googol years due to Hawking radition; so there is no matter left at the end, which leads to a reduced degree of freedom in terms of...
  22. HaLAA

    Show the group of units in Z_10 is a cyclic group of order 4

    Homework Statement Show that the group of units in Z_10 is a cyclic group of order 4 Homework EquationsThe Attempt at a Solution group of units in Z_10 = {1,3,7,9} 1 generates Z_4 3^0=1, 3^1=3, 3^2=9, 3^3= 7, 3^4= 1, this shows <3> isomorphic with Z_4 7^0=1 7^1= 7, 7^2= 9 7^3=3 7^4=1, this...
  23. PsychonautQQ

    Onto Homomorphism to cyclic group

    Homework Statement If P: G-->C_6 is an onto group homomorphism and |ker(p)| = 3, show that |G| = 18 and G has normal subgroups of orders 3, 6, and 9. C_6 is a cyclic group of order 6. Homework Equations none The Attempt at a Solution I determined that |G| = 18 by taking the factor group...
  24. PWiz

    Reduction of NADP in cyclic photophosphorylation

    I understand that an electron jumps to an excited state after absorbing a photon with the right energy (frequency) in photosystem 1 and exits the structure of the primary pigment, moves through different electron acceptors and returns to photosystem 1 (now at a lower energy state). What I don't...
  25. Charles Stark

    Proof of Cyclic Graph Edges = Vertices Formula

    I noticed that for cyclic graphs the number of edges is equal to the number of verticies. Is there a proof out there for this statement? Just curious... I was able to find the proof of the formula for finding the number of edges for complete graphs, I couldn't find anything related to the above.
  26. D

    Irreducible linear operator is cyclic

    I´m having a hard time proving the next result: Let T:V→V be a linear operator on a finite dimensional vector space V . If T is irreducible then T cyclic. My definitions are: T is an irreducible linear operator iff V and { {\vec 0} } are the only complementary invariant subspaces. T...
  27. L

    Does glucose in its cyclic structure react with HI ?

    does glucose in its cyclic structure react with HI to form CH3-CH2-CH2-CH2-CH2-CH3? (open chain structure of glucose reacts with HI to form CH3-CH2-CH2-CH2-CH2-CH3)
  28. J

    Solving Cyclic Group Questions: How Many Elements of Order What?

    I was hoping someone could check the following solutions to these 3 basic questions on cyclic groups and provide theorems to back them up. 1. How many elements of order 8 are there in C_{45}? Solution: \varphi(8)=4 2. How many elements of order 2 are there in C_{20}\times C_{30}? Solution...
  29. 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.
  30. 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...
  31. 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...
  32. M

    MHB A cyclic group with only one generator can have at most two elements

    Hey! :o Show that a cyclic group with only one generator can have at most two elements. I thought the following: When $a \neq e$ is in the group, then $a^{-1}$ is also in the group. So, when $a$ is a generator, then $a^{-1}$ is also a generator. Is this correct?? (Wondering) But I how can I...
  33. 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...
  34. R

    MHB Finding subgroups and their generators of cyclic group

    List every generator of each subgroup of order 8 in \mathbb{Z}_{32}. I was told to use the following theorem: Let G be a cyclic group of order n and suppose that a\in G is a generator of the group. If b=a^k, then the order of b is n/d, where d=\text{gcd}(k,n). However, I am unsure how this...
  35. 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?
  36. J

    Cyclic Group - Isomorphism of Non Identity Mapping

    Homework Statement Prove that if G is a cyclic group with more than two elements, then there always exists an isomorphism: ψ: G--> G that is not the identity mapping. Homework Equations The Attempt at a Solution So if G is a cyclic group of prime order with n>2, then by Euler's...
  37. V

    How to prepare sample for cyclic voltammetry measurement

    I am doing research on synthesis of copper nano particles. I would like to have a cyclic voltammetry (CV) of this material but I don't know how to prepare sample. Please tell me the ways to carry out this measurement. Thank you so much!
  38. D

    Cyclic Group Generators <z10, +> Mod 10 group of additive integers

    So I take <z10, +> this to be the group Z10 = {0,1,2,3,4,5,6,7,8,9} Mod 10 group of additive integers and I worked out the group generators, I won't do all of them but here's an example : <3> gives {3,6,9,2,5,8,1,4,7,0} on the other hand <2> gives {2,4,6,8,0} and that's it! but...
  39. B

    Cyclic symmetry - harmonic load components

    I have a homework problem where I have to solve for the displacements of the attached system using cyclic symmetry. To do this, I know that I have to find the harmonic load components of the system. One thing that my professor did not make clear (or if he did, I missed it) is how to determine...
  40. S

    Cyclic voltammetry of GOX

    Hey, this is not a homework question really but more a research issue my fellow students and I have run into. So basically, we have a project where we have cross-linked glucose oxidase to a polypyrrole surface on a gold electrode. The solution additionally contain PBS as well as ferricyanide...
  41. Saladsamurai

    Cyclic Symmetry Analysis: Capturing Features

    I am wondering what kind of approaches people take to a cyclic symmetry analysis in FEM when you have multiple repeating features that don't divide to the same integer. Take the example below for example. I have a N_HOLES = 136 and I have N_SLOTS = 52. I am not sure what to do here. The...
  42. J

    C/C++ C++ function to tell whether a group is cyclic

    Is there anything wrong with my logic and is there any way to further optimize this potentially long-running function? I've put a lot of comments to explain what's going on. template <typename ObType, typename BinaryFunction> bool isCyclic(const std::set<ObType> & G, BinaryFunction & op...
  43. I

    What are cyclic devices exactly?

    What are cyclic devices exactly? Are heat pumps and refrigerators both examples of cyclic devices? Also is a heat engine the same as a heat pump?
  44. B

    For every positive integer n there is a unique cyclic group of order n

    Hi, I can't understand why the statement in the title is true. This is what I know so far that is relevant: - A subgroup of a cyclic group G = <g> is cyclic and is <g^k> for some nonnegative integer k. If G is finite (say |G|=n) then k can be chosen so that k divides n, and so order of g^k...
  45. twistor

    Does BICEP2 kill Conformal Cyclic Cosmology?

    Is CCC killed by BICEP2 or there is a way in which it survives?
  46. O

    MHB Is $\Bbb Z_2 \times \Bbb Z_2$ a cyclic group?

    in Z_3 x Z_4 find all elements of cyclic subgroups(<1,2) generated by (1,2) this is just confusing me :(
  47. O

    MHB What are Cyclic Subgroup Generators and How Do We Determine Them?

    i am having a difficulity understanding the concept of cyclic subgroup generators. may I be given an explanation with examples if possible of how you determine whether a function is a subgroup and when they say list all cyclic subgroups eg <Z_10,+>. show that Z_10 is generated by 2 and 5
  48. C

    Character table for cyclic group of order 7

    Homework Statement a)Write down all irreducible representations of ##\mathbb{Z}_7##. b)How many of the irreducible representations are faithful? Homework Equations Group structure of ##\mathbb{Z}_7 = \left\{e^{2\pi in/7}, \cdot \right\}## for ##n \in \left\{0,...,6\right\}## The Attempt at a...
  49. N

    Deriving the Cyclic Rule in Thermodynamics

    Homework Statement Derive the cyclic rule in thermodynamics. ##\frac{\partial p}{\partial T} \cdot \frac{\partial T}{\partial V} \cdot \frac{\partial v}{\partial p}=-1##Homework Equations The Attempt at a Solution OK, so I write out the total differential of ##p##: ##dp=\frac{\partial...
  50. Saladsamurai

    Can cyclic symmetry be accurately modeled in ANSYS software?

    I am trying to understand a little further how software such as ANSYS implements cyclic symmetry in an analysis. A colleague of mine spoke to a support engineer and I think that he may have misinterpreted what was said. He is now under the impression that when we invoke a cyclic symmetry...
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