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. R

    Abelian groups of order 70 are cyclic

    Homework Statement Show that every abelian group of order 70 is cyclic.Homework Equations Cannot use the Fundamental Theorem of Finite Abelian Groups.The Attempt at a Solution I've tried to prove the contrapositive and suppose that it is not cyclic then it cannot be abelian. But that has lead...
  2. T

    Hypothetical cyclic process- does it violate the thermodynamic laws?

    Consider the following cyclic process: Each cycle 800J of Energy is transferred from a reservoir at 800K and 600J of energy from a reservoir at 600K. 400J of heat is rejected to a reservoir at 400K and 1000J of work is done. I think that the process doesn't violate the first or second laws...
  3. H

    Finding the number of elements in a cyclic group

    How do we go about finding the number of elements of a cyclic subgroup that's generated by an element in the main group. For example: The subgroup Z30 generated by 25. I would think this subgroup would be {0,1,5,25} but there's supposed to be 6 elements and not four. Whats going on?
  4. H

    Are Relatively Prime Elements Generators of Cyclic Groups?

    Homework Statement Zn={0,1,...,n-1}. show that an element k is a generator of Zn if and only if k and n are relatively prime. Homework Equations The Attempt at a Solution it makes sense but I am having a hard time proving this.
  5. H

    On Arrhenius plots in cyclic voltammetry

    I am doing a catalytic study on my Pt nanoparticles. My experiment set-up is a three-electrode cell with sulfuric acid as electrolyte for methanol electrooxidation reaction. Now, i want to calculate the apparent activation energy and for that I need to get the voltammograms at various...
  6. P

    Need help checking a proof on a cyclic group

    1. Prove that (Q,+) is not cyclic Here is what I have, and I need help knowing if this proof makes sense, is thorough enough, or is completely wrong. Note that (Q,+) is rationals Suppose, by contradiction, that (Q,+) is cyclic, p/q E (Q,+) and q=/=0 => (Q,+) can be generated by <p/q>...
  7. G

    Understanding why cyclic rule works

    I don't quite understand why cyclic rule works (from Pchem) (del x/ del y)_z = part of x with respect to y, hold z constant I don't know why is it negative 1? del x/ del y)_z * del y/ del z)_x * del z/ del x)_y = -1
  8. G

    Does Order Matter in Cyclic Subgroups/Groups?

    Since certain operations are not commutative, when a group G = <a, b>, does the order matter (so that <a, b> is not necessarily equal to <b, a>)?
  9. A

    Calculating Heat Transfer and Work in Cyclic Processes | Refrigerator Example

    Homework Statement Over several cycles, a refrigerator does 1.51 x 10^4 J of work on the refrigerant. The refrigerant in turn removes 7.55 x 10^4 J as heat from the air inside the refrigerator. a. how much energy is transferred as heat to the outside air? b. what is the net change in the...
  10. G

    Cyclic abelian group of order pq

    I'm looking at the exercises of Hungerfod's Algebra. Some looks easy but it seems the proofs are not so obvious. Here's one I'm particularly having a hard time solving: Let G be an abelian group of order pq with (p,q)=1. Assume that there exists elements a and b in G such that |a|= p and |b|...
  11. N

    Is the Cyclic Model of the Universe a Viable Alternative to the Big Bang Theory?

    Paul Steinhardt and Neil Turok, Princeton and Cambridge, respectively, explain their new cyclic model of the universe in THE ENDLESS UNIVERSE, 2007. Like most on this forum, I took the big bang and subsequent inflation as the best explanation for how this universe got started. I now see these...
  12. F

    How to Prove a Group is Cyclic?

    Homework Statement How do i go about proving that a group is cyclic? Homework Equations The Attempt at a Solution
  13. C

    Proving converse of fundamental theorem of cyclic groups

    Homework Statement If G is a finite abelian group that has one subgroup of order d for every divisor d of the order of G. Prove that G is cyclic. Homework Equations The Attempt at a Solution
  14. C

    Proving H is Cyclic: Finite Abelian Group

    Homework Statement Let H be a finite abelian group that has one subgroup of order d for every positive divisor d of the order of H. Prove that H is cyclicHomework Equations We want to show H={a^n|n is an integer}
  15. T

    Isomorphisms between cyclic groups? (stupid question)

    Why is Z mod 2 x Z mod 3 isomorphic to Z mod 6 but Z mod 2 x Z mod 2 not isomorphic to Z mod 4?
  16. 3

    Identifying Cyclic Alkanes: C6H12 - C9H18 - C10H20

    Homework Statement Name each of the following cyclic alkanes, and indicate the formula of the compound. http://img230.imageshack.us/img230/1042/organicnaming.jpg Homework Equations The Attempt at a Solution a. The rectangle has four corners so four carbon for butane. The branch...
  17. D

    Cyclic subgroups of an Abelian group

    Homework Statement If G is an Abelian group and contains cyclic subgroups of orders 4 and 6, what other sizes of cyclic subgroups must G contain? Homework Equations A cyclic group of order n has cyclic subgroups with orders corresponding to all of n's divisors. The Attempt at a...
  18. D

    Proving Cyclic Group Generators: An Exploration

    Homework Statement Prove any cyclic group with more than two elements has at least two different generators. Homework Equations A group G is cyclic if there exists a g in G s.t. <g> = G. i.e all elements of G can be written in the form g^n for some n in Z. The Attempt at a...
  19. S

    Action of a cyclic group on modules

    Homework Statement Let G = <x> be a cyclic group of prime order p and let M be a vector space over \mathbb{Q} with basis S = \{m_0,m_1,\dots,m_{p-1}\}. G acts on the S in a natural way by cyclic permutations and this action is linearly extended to an action of G on M. Now, the resulting...
  20. D

    Show ZXZ/<1,1> is an infinite cyclic group.

    Homework Statement Show ZXZ/<1,1> is an infinite cyclic group. Homework Equations The Attempt at a Solution <1,1> = {...(-1,-1), (0,0), (1,1),...} implies ZXZ/<1,1> = {<1,0>+<1,1>, <0,1>+<1,1>} which is isomorphic to ZXZ. But ZXZ is not cyclic, is my description of the...
  21. D

    Show ZXZ is an infinite cyclic group.

    Homework Statement Show ZXZ is an infinite cyclic group. Under addition of course. Homework Equations The Attempt at a Solution So this obviously is an infinite cyclic group. Z is generated by <1> or <-1>. The problem I run into here is I think <(1,1)> will only generate elements of the form...
  22. M

    Cyclic Normal Groups: Proving Normality of Subgroups in Cyclic Groups"

    Let H be normal in G, H cyclic. Show any subgroup K of H is normal in G. I was thinking about using the fact that subgroups of cyclic groups are cyclic, and that subgroups of cyclic groups are (fully)Characteristic (is that true?). Then we would have K char in H and H normal in G. Hence K...
  23. M

    Irreducible polynomial, cyclic group

    Describe the field F=\frac{\mathbb{F}_3[x]}{(p(x))} [p(x) is an irreducible polynomial in \mathbb{F}_3[x]]. Find an element of F that generates the cyclic group F^* and show that your element works. [p(x)=x^2+1 is irreducible in \mathbb{F}_3[x] if that helps]
  24. E

    Subgroups of a Cyclic Normal Subgroup Are Normal

    Homework Statement If H ≤ G is cyclic and normal in G, prove that every subgroup of H is also normal in G. The attempt at a solution Let H = <h>. We know that for g in G, hi = ghjg-1 by the normality of H. A simple induction shows that hin = ghjng-1, so that <hi> = g<hj>g-1. Now all I need...
  25. S

    Isomorphisms between cyclic groups

    Ok, here is something i thought i understood, but it turns out i am having difficulties fully grasping/proving it. Let \theta:G->G' be an isomorphism between G and G', where o(G)=m=o(G'), and both G and G' are cyclic, i.e. G=[a] and G'=[b] So my question is, when we want to find the...
  26. S

    Proving a group of rotaions is cyclic

    Homework Statement Let G be a finite group of rotation of the plane about the origin. Prove that G is cyclic. The Attempt at a Solution What it means to be cyclic is that every element of the group can be written as a^n for some integer n. I can see this is true if i take some...
  27. B

    Cyclic and non proper subgroups

    [b]1. What is/are the condition for a group with no proper subgroup to be cyclic? Homework Equations [b]3. this is just a general qustion I am asking in oder to prove something?
  28. S

    What is the connection between L and gcd(k,n) in cyclic subgroups?

    Today we learned about subgroups of cyclic groups G = <a>. During the discussion we reached this point: |<a^k>| = minimum L, L > 0, such that a^(kL) = 1. |G| = n. Then a^kL = a^bn, thus kL = bn, and thus L = n/gcd(k, n). However, I don't understand the bolded. My number theory is...
  29. P

    Proving Cyclic Extension of Finite Galois Group L/F

    Homework Statement Let K be a field, and let K' be an algebraic closure of K. Let sigma be an automorphism of K' over K, and let F be the fix field of sigma. Let L/F be any finite extension of F. Homework Equations Show that L/F is a finite Galois extension whose Galois group...
  30. J

    Is partial trace still cyclic?

    Hello, I know trace is usually cyclic, but is partial trace cyclic too? Why? Thanks! Jenga
  31. radou

    Infinite cyclic groups isomorphic to Z

    I'm currently going through Hungerford's book "Algebra", and the first proof I found a bit confusing is the proof of the theorem which states that every infinite cyclic group is isomorphic to the group of integers (the other part of the theorem states that every finite cyclic group of order m is...
  32. M

    Universe cyclic model and energy loss

    I'm "aware" of current theories about the cyclic model but what's wrong with this hypothesis? The universe starts crunching forming a big black hole All the energy is sucked back into this big black hole The big bang starts again with all the matter and energy as the previous big bang...
  33. Nabeshin

    Big Bang v. Cyclic Universe models

    I had the pleasure of attending a lecture given by Paul Steinhardt, a Princeton professor, regarding the big bang and cyclic universe models at Fermilab this evening. Steinhardt, having written a book called The Endless Universe, is obviously a fan of the cyclic universe camp and the main focus...
  34. R

    What Are Cyclic Quadratic Residues and Their Sums Modulo Prime Numbers?

    I also wonder about an other interesting residue relation Let P be a prime, let a^{2^n} be called a cyclic quadratic residue if there is integer m dependent on a such that a^{2^{n + mp}} = a^{2^n} for all integers p \mod P It seems that the sum of all such cylic residues is either 0 or...
  35. T

    Thermodynamics - cyclic pressure/volume process

    Homework Statement A sample of an ideal gas goes through the process shown below. From A to B, the process is adiabatic; from B to C, it is isobaric with 98 kJ of energy entering the system by heat. From C to D, the process is isothermal; from D to A, it is isobaric with 158 kJ of energy...
  36. L

    What are the Thermodynamic Processes for a Monoatomic Ideal Gas?

    Homework Statement A Cyclic proces of three parts for a monoatomic idealgas: 1-2: Isochor, where: p2 = 2*p1 v2 = v1 2-3: Adiabatic, from v2 = v3, where: v3 > v2 p3 = p1 3-1: unknow proces, where v4 = v1 p4 = p1 My problem is to determine Q1, Q2, Q3 and W1, W2, W3 and...
  37. H

    What is the cyclic subgroup order of GL(2,p^n) generated by the given matrix?

    I'm trying to prove that GL(2,p^n) has a cyclic subgroup of order p^{2n} - 1. This should be generated by \left( \begin{array}{cc} 0 & 1 \\ -\lambda & -\mu \end{array} \right) where X^2 + \mu X + \lambda is a polynomial over F_{p^n} such that one of its roots has multiplicative order...
  38. E

    Is there a pattern to determine which angles will create a cyclic sequence?

    [SOLVED] Cyclic Sequence of Angles Fix an angle \theta. Let n be a positive integer and define \theta_n = n\theta \bmod 2\pi. The sequence \theta_1, \theta_2, \ldots is cyclic if if it starts repeating itself at some point, i.e. the sequence has the form \theta_1, \ldots, \theta_k, \theta_1...
  39. N

    Proving G is Cyclic & G=<a,b> with #G=77

    Homework Statement Let G be a group and let #G=77. Prove the following: a) G is cyclic, if there is such an element a in G that a21≠1 and a22≠1 b) If there are such elements a and b, so that ord(a)=7 and ord(b)=11, then G=<a,b> 2. Homework Equations , 3. The Attempt at a Solution I...
  40. H

    Finding Ipc of Voltammgram - Cyclic Voltammetry

    I have a voltammegram (graph of potential vs current) and I want to find the Ipc of it. I'm not really sure how..is there a way to find it accurately or do i just have to estimate?
  41. H

    Cyclic Voltammetry: Roles of Working, Auxiliary, Ref. Electrodes & Potentiostat

    I'm trying to find out the roles of the following for cyclic voltammetry: working electrode auxiliary electrode reference electrode and potentiostat I kind of found out what they are.. but i am not sure of its exact role, like for the working electrode, it is the electrode at which the...
  42. B

    What are the cyclic subgroups of U(30)?

    Homework Statement List the cyclic subgroups of U(30) Homework Equations The Attempt at a Solution In order to list the cyclic subgroups for U(30) , you need to lists the generators of U(30) U(30)={1,7,11,13,17,19,23,29} . all the elements of U(30) are not generaters. in...
  43. R

    Construction of a cyclic sequence re the Golden Ratio

    The fractal sequence http://www.research.att.com/~njas/sequences/A054065 is of interest because it provides permutations of the numbers 1-n such that the decimal part of k*tau (k = {1,2,3,...n} is ordered from the lowest possible value to the highest. For instance if n = 3 the permutation...
  44. B

    What are the elements of the subgroups <3> and <15> in Z(18)?

    Homework Statement List the elements of the subgroups <3> abd <15> in Z(18) Homework Equations The Attempt at a Solution <3>={0,3,6,9,12,15} . <15> ={0,15} Together , I can conclude that the number of elements amongst <3> and <15> add up to 7 elements.
  45. B

    Finding Generators for Cyclic Groups Z(6), Z(8), and Z(20)

    Homework Statement Find all generators of Z(6), Z(8) , and Z(20) Homework Equations The Attempt at a Solution I should probably list the elements of Z(6), Z(8) and Z(20) first. Z(6)={0,1,2,3,4,5} Z{8}={0,1,2,3,4,5,6,7}...
  46. D

    Showing that a group isn't cyclic.

    Homework Statement Show that \left( \mathbb{Z}/32\mathbb{Z}\right)^{*} is not a cyclic group. Homework Equations The Attempt at a Solution A little calculator magic has showed that all elements in the group have order 8, but that doesn't seem like a very educational solution :). If...
  47. happyg1

    Can a Group Have a Cyclic Automorphism Group of Odd Order?

    Homework Statement Prove that no group can have its automorphism group cyclic of odd order. Homework Equations The Attempt at a Solution Aut(Z2) has order 1, which is odd...trivial, yes, but I thought I was DONE. However, my professor has said "well prove it EXCEPT for Z2"...
  48. P

    R Module M is Cyclic: Isomorphic to R/(p)?

    Homework Statement If an R module M is cyclic so M=Rm with annihilator(m)=(p), p prime then can we infer that M is isomorphic to R/(p) without any more infomation?
  49. radou

    Proof of Infinite Cyclic Group Isomorphism to Z

    I came across a theorem the proof of which I don't quite understand. The theorem states that every infinite cyclic group is isomorphic to the additive group Z. So, the mapping f : Z --> G given with k |--> a^k, where G = <a> is a cyclic group, is an epimorphism, which is quite obvious...
  50. G

    Group homomorphisms between cyclic groups

    Describe al group homomorphisms \phi : C_4 --> C_6 The book I study from seems to pass over Group Homomorphisms very fast. So I decided to look at Artin's to help and it uses the same definition. So I think I am just not digesting something I should be. I know it's defined as \phi (a*b)...
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