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. M

    How cyclic coordinates affect the dimension of the cotangent manifold

    Our professor's notes say that "In general, in Hamiltonian dynamics a constant of motion will reduce the dimension of the phase space by two dimensions, not just one as it does in Lagrangian dynamics." To demonstrate this, he uses the central force Hamiltonian...
  2. T

    MHB Find T cyclic operator that has exactly N distinct T-invariant subspaces

    Let T be a cyclic operator on $R^3$, and let N be the number of distinct T-invariant subspaces. Prove that either N = 4 or N = 6 or N = 8. For each possible value of N, give (with proof) an example of a cyclic operator T which has exactly N distinct T-invariant subspaces. Am I supposed to...
  3. L

    Proving Any Group of Order 15 is Cyclic

    Homework Statement Prove that any group of order ##15## is cyclic. 2. The attempt at a solution I am looking at a link here: (http://www.math.rice.edu/~hassett/teaching/356spring04/solution.pdf) and I am confused why "there must be one orbit with five elements and three orbits with three...
  4. C

    Heat transferred in a cyclic process

    Homework Statement Assume that a gas obeys the VDW Virial expansion Pv = RT + (b-a/RT)P to first order in P and u = 3RT - aP/RT to first order in P, where v and u are molar quantities. In the following cycle (see attachment), the heat transferred to the gas is transferred by direct thermal...
  5. J

    What about 2nd law of thermodynamics in Cyclic Universe Model?

    Not everyone likes the idea of Universe ​​created from a point singularity, so recently grows in popularity the cyclic model - that our Universe will finally collapse and use obtained momentum to bounce (so-called Big Bounce) and become the new Big Bang. One might criticize that we "know" that...
  6. caffeinemachine

    MHB Tricky Linear Algebra Question. To show that an operator is 'cyclic'.

    Hello MHB, I am stuck at this problem for quite a long time now. Problem. Let $F_p$ denote the field of $p$ elements, where $p$ is prime. Let $n$ be a positive integer. Let $V$ be the vector space $(F_p)^n$ over the field $F_p$. Let $GL_n(F_p)$ denote the set of all the invertible linear...
  7. M

    Determining whether the unit circle group is a cyclic group

    1. Homework Statement Let S be the set of complex numbers z such that |z|=1. Is S a cyclic group? 3. The Attempt at a Solution I think this group isn't cyclic but I don't know how to prove it. My only idea is: If G is a cyclic group, then there is an element x in G such that...
  8. T

    Is Conformal Cyclic Cosmology a Viable Explanation for the Big Bang?

    Hey everyone, I wanted to share this new interview I found with Roger Penrose wherein he discusses his theory of conformal cyclic cosmology: http://www.ideasroadshow.com/issues/roger-penrose-2013-07-12 I was wondering, are you convinced by his theory? I find it intriguing but I'm...
  9. marcus

    Higgs-driven cyclic conformal cosmology (Steinhardt Turok Bars)

    I approach this cautiously but with interest. (How often do ideas like this work out?) http://arxiv.org/pdf/1307.1848v1.pdf and suggest you jump immediately to page 24 where there is a suggestive graph, Figure 1. ==quote page 24 of Bars Steinhardt Turok 1307.1848== Fig.(1) is an...
  10. Saitama

    What is the Efficiency of a Cyclic Process?

    Homework Statement (see attachment, ignore the arrows made with the pen) Homework Equations The Attempt at a Solution Efficiency of a cycle is defined as ##\eta=\frac{W}{Q}## where W is work done and Q is heat input. W can be easily calculated by finding the area enclosed...
  11. T

    Composition Factors cyclic IFF finite group soluble

    Hey, just trying to get my head around the logic of this. I can see that if composition factors are cyclic then clearly the group is soluble, since there exists a subnormal series with abelian factors, but I am struggling to see how the converse holds. If a group is soluble, then it has a...
  12. C

    How to do cyclic permutation on interatomic matrix elements?

    For example, how to obtain E_(yz,xz)(l,m,n) from E_(xy,xz)(l,m,n)?
  13. R

    Is every Subgroup of a Cyclic Group itself Cyclic?

    Homework Statement Are all subgroups of a cyclic group cyclic themselves? Homework Equations G being cyclic means there exists an element g in G such that <g>=G, meaning we can obtain the whole group G by raising g to powers. The Attempt at a Solution Let's look at an arbitrary...
  14. M

    Easy test if unitary group is cyclic

    Is there an easy way to see if a unitary group is cyclic? The unitary group U(n) is defined as follows U(n)=\{i\in\mathbb{N}:gcd(i,n)=1\}. Cyclic means that there exits a element of the group that generates the entire group.
  15. D

    Finding which direct sum of cyclic groups Z*n is isomorphic to

    I always see problems like "how many structurally distinct abelian groups of order (some large number) are there? I understand how we apply the theorem which tells us that every finite abelian group of order n is isomorphic to the direct sum of cyclic groups. We find this by looking at the...
  16. Q

    Bong Angles of Cyclic Hydrocarbons

    We're going over the reactivity of aromatic compounds so I was reviewing material over Sn1 and Sn2 reactions. In the book that we have, it says that the smaller the bong length between two carbons, or other atom, in a cyclic compound, the smaller the bond angle. That's where I get confused...
  17. X

    Cannot stand the eternity in cyclic models

    I have had this question for a long time and thought this forum might be the best place to answer.. If there is an infinite number of bangs happened before the big bang, our universe's bang should never have happened because it would require an endless number of bangs that -by definition- would...
  18. S

    Homomorphisms of Cyclic Groups

    So this is a pretty dumb question, but I'm just trying to understand homomorphisms of infinite cyclic groups. I understand intuitively why if we define the homomorphism p(a)=b, then this defines a unique homorphism. My question is why is it necessarily well-defined? I think I'm confused...
  19. H

    What is a reversible cyclic process?

    I know carnot's cycle is an example. but what is it exactly? a cycle in which ever part of process has a 'counter-process' please elaborate.
  20. R

    Calculate work and heat in a cyclic process ?

    Homework Statement Consider n moles of ideal gas kept in a cylinder with a piston. Two heat reservoirs 1 and 2 with the temperatures T1 < T2 are available, and at any given moment of time the heat exchange is established with only one of the reservoirs. In the initial equilibrium state the...
  21. D

    Exploring the Cyclic Nature of U(p^k) and its Relationship to Modular Arithmetic

    Prove that U(p^k) is cyclic p^k is an odd prime power. I've been working on this problem for a while and can't figure it out. The professor's hint is "to think about the solutions to x2 =1." (pk - 1)2 mod pk = 1 but I'm unsure how that is helpful. I know that that 2 generates every set by...
  22. D

    Question about generator of cyclic group

    Say we have a cyclic group G, and a generator a in G. This means [a] = G. We know the order of an element a, is the order of the group it generates, [a], and also this is the smallest integer s such that as=e, where e is the identity element. In this case, [a]=G, so s is just the order of G...
  23. A

    Which coordinate is cyclic in this case

    Consider a simple two particle system with two point masses of mass m at x1 and x2 with a potential energy relative to each other which depends on the difference in their coordinates V = V(x1-x2) The lagrangian is: L = ½m(x1')2 + ½m(x2')2 + V(x1-x2) Obviously their total momentum is conserved...
  24. M

    Cyclic permutation and operators

    Hi there I am working through the problems in R.I.G. Hughes book the structure and interpretation of quantum mechanics and have hit a wall in the last part of the following question: Show that Sx and Sy do not commute, and evaluate SxSy-SySx. Express this difference in terms of Sz, and...
  25. L

    Why does current change direction in cyclic voltammetry?

    You start at a certain voltage. Then you decrease this voltage to be more negative which reduces the analyte. Then you switch at a set voltage and increase the potential so that it is becoming more positive. Why does this switch the direction of the current so that on the reverse sweep the...
  26. B

    Cyclic thermodynamics processes

    Homework Statement Consider a cyclic process involving a gas. If the pressure of the gas varies during the process but returns to the original value at the end, is it correct to write ΔH=q Homework Equations The Attempt at a Solution I'm actually not sure. q is a path function and H...
  27. S

    Cyclic tests - tension compression torsion

    Hi, I have got experimental data for a steel material for a cyclic test perfoemd in the following scenario: 0-1% Torsion followed by 0-1% strain Tension followed by 1-2% Torsion followed by 1% to 0 % compression followed by 2 to 3% torsion followed by 0 to -1% compression...
  28. C

    Work done by cyclic process (thermodynamics)

    Homework Statement http://s9.postimage.org/5iw5rixyl/image.jpg (sorry doesn't let me embed) Homework Equations P1V1=P2V2 and PV=nRT I know that Work done by gas from a --> b = (nRT)*Integral(V2/V1) My question is do i use PV=nRT to find N? and if T is constant (isothermal) what do I plug in...
  29. M

    Riemann tensor cyclic identity (first Bianchi) and noncoordinate basis

    I got trouble to understand the cyclic sum identity (the first Bianchi identity) of the Riemann curvature tensor: {R^\alpha}_{[ \beta \gamma \delta ]}=0 or equivalently, {R^\alpha}_{\beta \gamma \delta}+{R^\alpha}_{\gamma \delta \beta}+{R^\alpha}_{\delta \beta \gamma}=0. I can understand the...
  30. J

    Proof of Order of b is a Factor of the Order of a in Cyclic Groups

    Proposition: If G= <a> and b ϵ G, then the order of b is a factor of the order of a. Proof: Let G be a group generated by a. That is, G=<a>. Let b ϵ G. Since G is cyclic, the element b can be written as some power of a. That is, b=ak for some integer k. Suppose the order of a is n. Hence...
  31. J

    Mapping generator to generator in cyclic groups.

    Attached is my attempt at a proof. Please critque! :shy: Thank you!
  32. T

    G is cyclic and |G| = p^n, p is prime <=> H,K Subgroups, H⊆K or K⊆H

    Homework Statement Show that the following conditions are equivalent for a finite group G: 1.G is cyclic and |G| = p^n where p is prime and n\geq 0 2.If H and K are subgroups of G, either H⊆K or K⊆H. The Attempt at a Solution 1 => 2. Let H,K be subgroups of G = <g> where o(g)...
  33. J

    Another problem involving cyclic groups.

    Show that in a finite cyclic group G of order n, writtten multiplicatively, the equation xm = e has exactly m solutions x in G for each positive integers m that divides n. Attempt... Proof: Let G be a finite cyclic group of order n, and suppose m is a positive integer that divides n. Let x be...
  34. N

    Abelian group with order product of primes = cyclic?

    It seems rather straight forward that if you have an abelian group G with \# G = p_1 p_2 \cdots p_n (these being different primes), that it is cyclic. The reason being that you have elements g_1, g_2, \cdots g_n with the respective prime order (Cauchy's theorem) and their product will have to...
  35. J

    Problem concerning cyclic groups.

    The question states: "Let G be a group and let Gn={gn|g ε G}. Under what hypothesis about G can we show that Gn is a subgroup of G?The set Gn is taking each element of G and raising it to a fixed number. I started my investigation by examining what happens if I take n=3 and considering the...
  36. N

    Roots of unity form a cyclic group

    In a lot of places, I can read that the roots of unity form a cyclic group, however I can find no proofs. Is the reasoning as follows: Let's work in a field of characteristic zero (I think that's necessary). Let's look at the nth roots of unity, i.e. the solutions of x^n - 1. There are n...
  37. H

    Do cyclic models of the universe

    I have a question as to the actual nature of cyclic models of the universe (e.g. Roger Penrose's Conformal Cyclic Cosmology or the Ekpyrotic universe) - essentially where the universe has no beginning or end it simply goes through cycles eternally in both time directions. So in these situations...
  38. P

    The Cyclic Model: Multidimenstional Membranes? Where do these come from?

    The Cyclic Model: Multidimenstional Membranes? Where do these come from? Hello all, I am intensely interested in Astrophysics and Cosmology as a 12 year old. Wandering across this topic and reading some of the replies, it occurred to me, that NO ONE has yet mentioned the Cyclic Model of...
  39. L

    Homomorphism of a cyclic subgroup is a cyclic subgroup ?

    Homework Statement Let \alpha:G \rightarrow H be a homomorphism and let x\inG Prove \alpha(<x>) =<\alpha(x)> Homework Equations α(<x>) = α({x^{r}: r ∈ Z}) = {α(x^{r}) : r ∈ Z} = {α(x)^{r}: r ∈ Z} = <α(x)>. I do not understand how can we take out the 'r' out of a(x^{r}) to...
  40. C

    Prove G is Cyclic: Prime p Order Group

    1. Let p be a prime and G a group whose order is p. Prove that G is cyclic. 2. I know that if p is prime, then the only possible subgroups of G are {e} and G itself. But, how to use this fact to show that G is cyclic?
  41. J

    Naming Cyclic Organic Compound

    Hey, How would I name (IUPAC) the isomer of C2H4O that is a triangular ring with C at two vertices and O at the third? Also, I'm trying to name the 11 isomers of C3H6O. So far I have 10 and I can't seem to get the last one. The ones I have are: propanal propan-2-one prop-2-en-1-ol...
  42. I

    How to tell if U(n) is cyclic?

    In my book U(n) is defined as all numbers less than n that are relatively prime to n. U(n) is cyclic for some n but not for all. I was wondering if there is theory behind how to tell if U(n) will be cyclic or, even better, what elements of U(n) generate U(n). Also, the formal name of this group...
  43. 2

    prove that the group U(n^2 -1) is not cyclic

    Sorry if I formatted this thread incorrectly as its my first post ^^ Homework Statement For every integer n greater than 2, prove that the group U(n^2 - 1) is not cyclic. Homework Equations The Attempt at a Solution I've done a problem proving that U(2^n) is not cyclic when...
  44. L

    Is the direct sum of cyclic p-groups a cyclic group?

    For arbitrary natural numbers a and b, I don't think the direct sum of Z_a and Z_b (considered as additive groups) is isomorphic to Z_ab. But I think if p and q are distinct primes, the direct sum of Z_p^m and Z_q^n is always isomorphic to Z_(p^m * q^n). Am I right? I've been freely using...
  45. B

    Proving Cyclic Decompositions: Let T be a Linear Operator on V

    Let T be a linear operator on the the finite dimensional space V, and let R be the range of T. (a) Prove that R has a complementary T-invariant subspace iff R is independent of the null space N of T. (b) If R and N are independent, prove that, N is the unique T-invariant subspace...
  46. T

    Cyclic Permutation in C: Create Alphabet Pattern with Nested For Loops

    Use nested for loops to produce the following pattern of cyclic permutations of the English alphabet: abcde...yz bcdef...za cdef...zab ... zabcde...xy HINT: you may find the modulo (remainder) operator % useful. I have an idea of how to do this but it would not use the modulo operator...
  47. B

    Cyclic groups and isomorphisms

    I have a question where it says prove that G \cong C_3 \times C_5 when G has order 15. And I assumed that as 3 and 5 are co-prime then C_{15} \cong C_3 \times C_5 , which would mean that G \cong C_{15} ? So every group of order 15 is isomorohic to a cyclic group of order 15...
  48. B

    Ptolemy's Theorem and Cyclic Quadrilateral

    Homework Statement In cyclic quadrilateral ABCD with diagonals intersecting at E, we have AB=5, BC=10, BE=7, and CD=6. Find CE. Homework Equations Ptolemy's Theorem: The product of the measures of its diagonals is equal to the sum of the products of the measures of the pairs of opposite...
  49. S

    Cyclic set: Difference between generator and unit

    Hello everyone, I've just begun a lesson on cyclic sets, but I am having problems determining a few concepts. One question will ask me to find the generators and the units of a cyclic set Z8. I have become confused and realized that I did not understand the difference between a generator and...
  50. B

    Isomorphic direct product cyclic groups

    Help! For p prime I need to show that C_{p^2} \ncong C_p \times C_p where C_p is the cyclic group of order p. But I've realized I don't actually understand how a group with single elements can be isomorphic to a group with ordered pairs! Any hints to get me started?
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