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. E

    Isomorphism and Cyclic Groups: Proving Generator Mapping

    Homework Statement I need to prove that any isomorphism between two cyclic groups maps every generator to a generator. 2. The attempt at a solution Here what I have so far: Let G be a cyclic group with x as a generator and let G' be isomorphic to G. There is some isomorphism phi: G...
  2. B

    Cyclic Quadrilaterals: Understanding Angle Equality and Ptolemy's Theorem

    Well this isn't a homework question (I'm just trying to refresh my memory from the plane geometry I did in high school) and so, I was reading through cyclic quadrilaterals on wikipedia and I don't see how certain angles are equal. Here are two images taken from wikipedia...
  3. E

    Cyclic Subgroups of P15: Homework Solutions

    Homework Statement Consider the set P15 of all integer numbers less than 15 that are mutually prime with 15: P15 = {1, 2, 4, 7, 8, 11, 13, 14}. It is a group under multiplication modulo 15. (a) P15 has six cyclic groups. Find them. my answer: <3>=<6>=<9>=<12>= {0, 3, 6 , 9, 12}...
  4. S

    Proving the Existence of Subgroups in Cyclic Groups

    Homework Statement Let G be a finite cyclic group of order n. If d is a positive divisor of n, prove that the equation x^d=e has d distinct solutions Homework Equations n=dk for some k order(G)=nThe Attempt at a Solution solved it: <g^k>={g^k, g^2k,...,g^dk=e} and for all x in <g^k> x^d=e...
  5. D

    Proving G Abelian iff G/Z(G) is Cyclic

    This isn't some homework question its more theory based that I'm struggling with from class and we will probably have homework on it. If G is some arbitrary group, why is G Abelian <---> the factor group of G/Z(G) is cyclic? My professor mentioned something about one direction being trivial...
  6. A

    Thermodynamics: Cyclic Processes

    Thermodynamics: Cyclic Processes (solved) Sorry for the false alarm guys. It looked like I did use the wrong equation while finding the internal energy for case 1. referred to this. Thanks. https://www.physicsforums.com/showthread.php?t=412577 Given that 1 mol of ideal monoatomic gas at p= 1...
  7. bcrowell

    More on Penrose's Conformal Cyclic Cosmology

    We had a discussion of Penrose's conformal cyclic cosmology last month: https://www.physicsforums.com/showthread.php?t=427567 His popular-level book Cycles of Time was published first in the UK, but is now available in the US. I got a copy and have read it, so I can report on a few of the...
  8. M

    Proving Order of Cyclic Group with Elements a & b is Finite

    Homework Statement G is a group. Let a,b be elements of G. If order(ab) is a finite number n, show order(ba) = n as well. Homework Equations order(a) = order(<a>) where <a> is the cyclic group generated by a. The Attempt at a Solution I do not know. I thought it may be related to...
  9. E

    Elementary cyclic normal group theory

    Homework Statement If G is a finite group and let H be a normal subgroup of G with finite index m=[G:H]. Show that a^m\in H for all a\in G. Homework Equations order of a group equal the order of element. The Attempt at a Solution no idea.
  10. N

    Theoretical question on cyclic function

    a(x) is continues on R with cycle T ,a(x+T)=a(x) u(x) is non trivial soluion of y'=a(x)y \lambda=\int_{0}^{T}a(x)dx which of the following claims is correct: A. if \lambda>0 then \lim_{x\rightarrow\infty}u(x)=\infty B. if \lambda=0 then u(x) is a cyclic function i don't have...
  11. J

    Cyclic subgroup of a particular form proof with the group infinitely large.

    Homework Statement G is additive group. If the order of G is infinity, then G is cyclic iff each subgroup H of G is of the form nG for some interger n. Homework Equations cyclic property. The Attempt at a Solution i kind of know that nG is the answer but why G has to be infinite...
  12. F

    What Are the Subgroups of Z3 x Z3?

    Homework Statement Find all of the subgroups of Z3 x Z3 Homework Equations Z3 x Z3 is isomorphic to Z9 The Attempt at a Solution x = (0,1,2,3,4,5,6,7,8) <x0> or just <0> = {0} <1> = {identity} <2> = {0,2,4,6} also wasn't sure if I did this one correctly x o x for x2 <3> =...
  13. K

    How to prove that a group of order prime number is cyclic without using isomorphism?

    How to prove that a group of order prime number is cyclic without using isomorphism/coset? Can i prove it using basic knowledge about group/subgroup/cyclic(basic)? I just learned basic and have not yet learned morphism/coset/index. Can you guys kindly give me some hints or just answer...
  14. P

    Group Theory Question involving nonabelian simple groups and cyclic groups

    Homework Statement Let A be a normal subgroup of a group G, with A cyclic and G/A nonabelian simple. Prove that Z(G)= AHomework Equations Z(G) = A <=> CG(G) = A = {a in G: ag = ga for all g in G} My professor's hint was "what is G/CG(A)?" The Attempt at a Solution A is cyclic => A is...
  15. T

    Groups of prime order are cyclic. (without Lagrange?)

    I know full well the proof using Lagrange's thm. But is there a direct way to do this without using the fact that the order of an element divides the order of the group? I was thinking there might be a way to set up an isomorphism directly between G and Z/pZ. Clearly all non-zero elements...
  16. R

    # of automorphisms of a cyclic group

    Homework Statement Find all the automorphisms of a cyclic group of order 10. Homework Equations The Attempt at a Solution I think it might be useful if I could first figure out how many automorphisms are there. There are 4 elements of order 5, 4 elements of order 10, 1 element...
  17. marcus

    Cycles of time-Penrose says his cyclic cosmology obeys thermodynamics.

    Cycles of time--Penrose says his cyclic cosmology obeys thermodynamics. Roger Penrose has devised a cyclic cosmology which he sees as not violating the second law of thermodynamics. There are several online videos of him lecturing about it. I'll get some links. I'm curious to know if others...
  18. T

    Cyclic Subgroup H=<9> of Z30: List and Find Elements

    1. (a) List all elements in H=<9>, viewed as a cyclic subgroup of Z30 (b) Find all z in H such that H=<z> I'm thinking that H=<9> = {1,7,9} (viewed as a cyclic subgroup of Z30) is this correct? And could someone explain what (b) is asking in other terms?
  19. M

    Proving the Existence of One Cyclic Subgroup in a Cyclic Group of Order n

    Homework Statement Let G be a cyclic group of order n, and let r be an integer dividing n. Prove that G contains exactly one subgroup of order r. Homework Equations cyclic group, subgroup The Attempt at a Solution Say the group G is {x^0, x^1, ..., x^(n-1)} If there is a subgroup...
  20. M

    Cyclic Forces in the Solar System & Zodiac

    Hy. From what it is known today, are there any forces that go up and down in cycles ? This regarding planet earth. I talk about whatever forces in the solar system and zodiac. Something like a few days or weeks up then down, then up then down etc. Thanks!
  21. E

    Abstract Algebra- homomorphisms and Isomorphisms, proving not cyclic

    1. Suppose that H and K are distinct subgroups of G of index 2. Prove that H intersect K is a normal subgroup of G of index 4 and that G/(H intersect K) is not cyclic. 2. Homework Equations - the back of my book says to use the Second Isomorphism Theorem for the first part which is... If K...
  22. R

    Thermodynamics : Work in a Cyclic Process

    Homework Statement Three moles of an ideal gas are taken around the cycle abc. For this gas, Cp= 29.1 J/mol K. Process ac is at constant pressure, process ba is at constant volume, and process cb is adiabatic. The temperatures of the gas in states a, c, and b are Ta= 300K, Tb= 490K, Tc=...
  23. R

    Group Theory, cyclic group proof

    Homework Statement Prove that Z sub n is cyclic. (I can't find the subscript, but it should be the set of all integers, subscript n.)Homework EquationsLet (G,*) be a group. A group G is cyclic if there exists an element x in G such that G = {(x^n); n exists in Z.} (Z is the set of all...
  24. J

    Largest Cyclic Subgroup of S_n

    I became interested in this question a few weeks ago, I couldn't find much on it basically I've realized it's equivalent to finding for each n a partition of n say x_1,x_2,...,x_k such that x_1+x_2+...+x_k=n and lcm(x_1,...,x_k) is maximum (because you can then take the subgroup...
  25. E

    Organic chemistry : cyclic alkane from esters

    Hi, I am trying to understanding how esters can form a cyclic alkane. Could someone explain the mechanism for this? I would guess that we'd have to hydrolyze the ester in acidic condition to carboxylic acid. The hydroxide then can attack the carbonyl to form a ring, but that would give us an...
  26. L

    Linear Algebra - Cyclic Decomposition

    Linear Algebra - Cyclic Decomposition, Rational Canonical Form Homework Statement I am given a 5x5 real matrix A, and I am looking for an invertible matrix P so that P^{-1}AP is in rational form. Homework Equations The Attempt at a Solution I calculated the characteristic...
  27. A

    Temprature-Entropy cyclic process

    Homework Statement There is a temperature-Entropy graph (T-S) (attachment),which illustrates a hypothetical cyclic process. a) Calculate the heat input or output along each of the paths. b) Find an expression for the efficiency η of the complete cycle in terms of T1 and T2 only...
  28. Y

    Groups of permutations and cyclic groups

    1: Is a group of permutations basically the same as a group of functions? As far as I know, they have the same properties: associativity, identity function, and inverses. 2: I don't understand how you convert cyclic groups into product of disjoint cycles. A cyclic group (a b c d ... z) := a->b...
  29. K

    How many distinct H cosets are there?

    Homework Statement Consider the cyclic group Cn = <g> of order n and let H=<gm> where m|n. How many distinct H cosets are there? Describe these cosets explicitly. Homework Equations Lagrange's Theorem: |G| = |H| x number of distinct H cosets The Attempt at a Solution |G| = n...
  30. K

    Analyzing Cyclic Subgroups in S6

    Homework Statement I have to determine whether some groups are cyclic. The first is the subgroup of S6 generated by (1 2 3)(4 5 6) and (1 2)(2 5)(3 6) Homework Equations Lagrange's Theorem? The Attempt at a Solution I don't really know how to tackle this problem. I have only...
  31. T

    Abstract algebra cyclic subgroups

    Homework Statement Suppose that G is a group with exactly eight elements of order 10. How many cyclic subgroups of order 10 does G have? Homework Equations The Attempt at a Solution I really don't have a clue how to solve this, any help would be greatly appreciated.
  32. graybass

    Cyclic Universe: Exploring the Theory of Multiple Big Crunches and the Big Rip

    Any you guys read the proposal coming out of Chapel Hill about a fresh model on the Cyclic Universe? Supposedly a good theory on how the Universe will have a lot of separate "Big Crunches" just before the "Big Rip" I just glanced over it. I'll read it in depth later today. GB
  33. M

    Submodule of a cyclic R-module

    hi, i want to show that If R is a PID then a submodule of a cyclic R-module is also cyclic. do i need to use fundamental theorem for finitely generated R-module over R PID ? thanks in advance
  34. K

    Exploring Group Properties and Examples: Cyclic, Abelian, and Non-abelian Groups

    Homework Statement Let G1 and G2 be groups, let G = G1 x G2 and define the binary operation on G by (a1,a2)(b1,b2):=(a1b1,a2b2) Prove that this makes G into a group. Prove G is abelian iff G1 and G2 are abelian. Hence or otherwise give examples of a non-cyclic abelian group of order 8...
  35. M

    Period of superposed cyclic integer rows

    Take two rows of respective length m and n: a1, a2, a3,..., am and b1, b2, b3, ..., bn. Then produce as follows the generated array Gai to contain these elements: a1, a1+a2, a1+a2+a3, ..., a1+..+am, a1+..+am+a1, a1+..+am+a1+a2, ... Alike produce the generated array Gbj to contain...
  36. E

    Shell thickness of a pressure vessel required subjected to a cyclic pressure

    I was wondering if anyone could help me with the following question, please: A thin walled pressure vessel is to be used as a pressure accumulator in a number of situations all involving a number of different operation conditions some of which create cyclic stresses. The dimentions of the...
  37. B

    Have a proof re. cyclic groups, need a little explaining

    Homework Statement Let a,b be elements of a group G. show that if ab has finite order, then ba has finite order. Homework Equations The Attempt at a Solution provided proof: Let n be the order of ab so that (ab)n = e. Multiplying this equation on the left by b and on the right by...
  38. T

    Can I calculate the (multiplicative) inverse of any element in a cyclic group?

    Homework Statement The original problem has to do with telling messages encrypted with a version of the ElGamal public key crypto system apart. It relies on exponentiation in an arbitrary cyclic group G of prime order p with generator g. The public key is y = g^x where x is the private key...
  39. B

    Help clarifying a question regarding (i think) cyclic groups

    Homework Statement Let G be a group with a finite number of elements. Show that for any a in G, there exists an n in Z+ such that an=e.Homework Equations a hint is given: consider e, a, a2,...am, where m is the number of elements in G, and use the cancellation laws.The Attempt at a Solution so...
  40. P

    Normal Cyclic Subgroup in A_4: Proving Normality and Identifying Elements

    Homework Statement Is the Cyclic Subgroup { (1), (123), (132)} normal in A_{4} (alternating group of 4) Homework Equations The Attempt at a Solution So I believe if I just check if gH=Hg for all g in A_4 that would be suffice to show that it is a normal subgroup, but that seems...
  41. A

    The Universe's Possible Cyclic Existence

    Is it possible that the universe we live in has a ciclyc way of existing? I mean,could our universe born with te big bang,then expand,stop expanding,to slowly attract matter in one point than implode;and after all this to explode again in another big bang?
  42. V

    Question on cyclic groups (addition mod n)

    I am trying to show show that there is no homomorphism from Zp1 to Zp2. if p1 and p2 are different prime numbers. (Zp1 and Zp2 represent cyclic groups with addition mod p1 and p2 respectively). I am not sure how to do this but here are some thoughts; For there to be a homomorphism we...
  43. Q

    Is the Cyclic Model of the Universe a Credible Theory?

    I was reading in some "Discover" magazine on the universe in which they were talking on the pre-Big Bang universe with this thing called the "cyclic model" developed by two astrophysicists by the names of Neil Turok and Paul Steinhardt. The theory basically states that the Big Bang is caused...
  44. L

    Is there cyclic motion of galaxies?

    Im aware that the planets in our solar system all orbit the sun on the same plane. and if we look at the milky way all the stars are aligned on the same plane. if we go out one further (ie to look at galaxies) , I am wondering if the motion of galaxies and how they are aligned has any structure...
  45. S

    Finding the Diameter of a Cyclic Quadrilateral

    Homework Statement Given a cyclic quadrilateral with side lengths 1, 2, 3 and d (in that order) where d is the diameter of the circle, find d. The Attempt at a Solution I tried using Ptolemy's theorem and Brahmagupta's formula, but to no avail. Can I get pointed in the correct...
  46. F

    Proving a Group is Cyclic - A Guide for Beginners

    Hi everyone. How could I prove if something is a cyclic group? I was wondering because I can prove is something is a group, a subgroup, and a normal subgroup, but I have no Idea as to how to prove something is a cyclic group. Ex: Suppose K is a group with order 143. Prove K is cyclic...
  47. D

    Prove that a group of order 34 with no more than 33 automorphisms is cyclic

    Homework Statement I need to show that for a group G of order 34 that if the order of the automorphism group is less than or equal to to 33, then G is cyclic. Homework Equations none The Attempt at a Solution I'm mainly trying to do a proof by contradiction. First I assumed that G...
  48. D

    Is the Steinhardt-Turok Cyclic Model Gaining Support?

    I typically post in the QM section, but I was reading an article about the cyclic model and wanted input on if this model of Steinhardt–Turok is widely accepted, is gaining support, has been upgraded, or replaced by something more current?
  49. T

    Cyclic variation of engine torque

    ! Cyclic variation of engine torque If the cylinders fire sequentially according to the fire order 1-2-4-3 What is the pattern of the cyclic variation of each cyclinder engine torque and the resultant engine torque?
  50. C

    Linear algebra: cyclic vector space

    Homework Statement Prove that V is cyclic relative to a linear transformation T, T:V->V if and only if the minimal polynomial of T is the same as the characteristic polynomial of T. Homework Equations The Attempt at a Solution i have finished the => direction (proved that if...
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