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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I am currently working on a FE simulation project where a tensile test of DP800 steel is subjected to cyclic loading. It utilizes Yoshida Uemori model (YUM) to formulate the modulus of elasticity (E) to simulate the hysteresis caused by cyclic loading. We are using a USDFLD subroutine in...
2. Efficiency and Temperature in Heat Engine Cycles: Approaching Parts B, C, and D

I didn't have much trouble with part a but I'm struggling with b,c, and d. I considered the efficiency formula for a heat engine e = work done by engine/ qh but i am unsure of how to approach it. for part c) not sure how i can get to Tc without knowing Th for d) my gut is telling me 5/2 but i...
3. POTW Product of Two Finite Cyclic Groups

For each positive integer ##m##, let ##C_m## denote a cyclic group of order ##m##. Show that for all positive integers ##m## and ##n##, there is an isomorphism ##C_m \times C_n \simeq C_d \times C_l## where ##d = \operatorname{gcd}(m,n)## and ##l = \operatorname{lcm}[m,n]##.
4. A Steady state confined flow field: Is it cyclic?

For a fluid that is confined to a finite region with no sources and sinks, are the only options for the flow field a) static, and b) cyclic? The example I have in mind is Rayleigh convection in a shallow dish heated from below, where convection cells are formed beyond a certain temperature...
5. I Is Penrose's Cyclic Cosmology Model Only Applicable on a Local Scale?

A recent thread asked about Penrose's proposal on cyclic cosmology. It was closed due to lack of any remotely acceptable sourcing, even after prompting. Much of the original professional publication on this is not available on arxiv. However the following includes a summary of conformal cyclic...
6. A What is the generator of the cyclic group (Z,+)?

I do not understand why ##(Z.+)## is the cyclic group? What is a generator of ##(Z,+)##? If I take ##<1>## I will get all positive integers. If I take ##<-1>## I will get all negative integers. I should have one element which generates the whole group. What element is this?
7. MHB Aa.17 Prove that G is cyclic

Let G be a group of order 25. a, Prove that G is cyclic or $g5=e$ for all $g 2 G$. Generalize to any group of order $p2$ where p is prime. Let $g\in G$. If $g=e$, then clearly $g^5=e$. So $g^6=e$. Then $|g|$ divides $25$, i.e., $|g| = 1,5,\textit{ or } 25$. But $|g|\ne1$ since we assumed...
8. Proving a sum of three squared terms, cyclic in #a,b,c#, is equal to 1

Problem statement : I copy and paste the statement of the problem from the text. (Given ##\boldsymbol{a+b+c=0}##) Attempt : I am afraid I couldn't make any meaningful progress. With ##a = -(b+c)##, I substituted for ##a## in the whole of the L.H.S, both numerators and denominators. I multiplied...
9. Proving one equation, cyclic in variables ##a,b,c##, to another

Problem statement : Let me copy and paste the problem statement from the text : Attempt at solution : I could not solve the problem reducing the L.H.S into the R.H.S. However, I could solve the problem by expanding the R.H.S. into the L.H.S., though it is less than satisfactory. Below is my...
10. Reducing an algebraic fraction, cyclic in three variables, to another

Problem : Let me copy and paste the problem statement as it appears in the text, as shown above. Attempt : I can sense there is an "elegant" way of doing this, but I don't know how. I show below my attempt using ##\text{Autodesk Sketchbook}##. I hope am not violating anything. Ok so I have...
11. Chemistry Oxydation of cyclic enol with KMnO4

Hi, please help me write the mechanism of this reaction. Usually id have an alpha hydrogen for me to do an internal proton transfer, but in this case i don't know what to do. I know the last step is wrong but i don't know how to obtain a carbonyl orcarboxylic acid form this. Thank you so much...
12. I Cyclic rotation of the cross product involving derivation

Dear PF, so we know that cross product of two vectors can be permutated like this: ## \vec{ \alpha } \times \vec{ \beta }=-\vec{ \alpha} \times \vec{ \beta} ## But in a specific case, like ## \vec{p} \times \vec{A} = \frac{ \hbar }{ i } \vec{ \nabla } \times \vec{A} ## the cyclic permutation of...
13. A Cyclic coordinates in a two body central force problem

(Goldstein 3rd edition pg 72) After reducing two body problem to one body problem >We now restrict ourselves to conservative central forces, where the potential is ##V(r)## function of ##r## only, so that the force is always along ##\mathbf{r}##. By the results of the preceding section, I've...
14. B Is an infinite universe compatible with cyclic models?

Cyclic models for reference. I will take simple Big Bounce as an example of what I have in mind. In Big Bounce there periods of expansion and periods of contraction which result in a never-ending series of Big Bangs. However if Universe is infinite in extent this would require infinite amount...
15. Cyclic Process for Ideal Gases

Hello there, is my solution for part d logically correct? Here is my attempt at the solution : Part a : where : P1 = 3P2 Part b : Since P1=3P2, therefore, T1=3T, where T=300K. Thus, T1=900K Part c : Because the final pressure at the end of the cycle is exactly the same as the pressure at...
16. Solve the problem in the given cyclic quadrilateral

now for part ##19.1##, My approach is as follows, using cosine rule; ##DF= r^2 + r^2- 2r^2 cos E## We know that angle ##E## + angle ## ∅##= ##180^0## ## ∅## is acute, therefore angle ##E## would be negative. (If ## ∅=60^0## for e.g then it follows that ##E=120^0##) Thus we shall have, ##DF^2=...
17. A BGV-theorem and Penrose's Conformal Cyclic Cosmology

Dear all, Some time ago I stumbled upon the famous BGV-theorem, - https://en.wikipedia.org/wiki/Borde–Guth–Vilenkin_theorem - https://arxiv.org/abs/gr-qc/0110012 which states that on spacetimes which have, on average, a positive Hubble constant, one can find timelike geodesics which cannot be...
18. MHB Find All Possible Values of $AD$ in Cyclic Quadrilateral

$ABCD$ is a cyclic quadrilateral such that $AB=BC=CA$. Diagonals $AC$ and $BD$ intersect at $E$. Given that $BE=19$ and $ED=6$, find all the possible values of $AD$.
19. B Proof of Cyclic Quadrilateral AEDT in Circle ABCD

##TA## and ##TD## are tangent line of circle ##ABCD## and ##TB \parallel DC##. Show ##A,E,D,T## are cyclic quadrilateral. I know ##x=\angle TAD= \angle TDA = \angle ACD= \angle TEA## And ##\angle ATD=180-2x## But I don't know how to prove ##\angle AED=x##. Or there's another easily method? Thanks.
20. Work done BY the gas in a cyclic thermodynamic process

Since the assignment asks the work done by the gas, that should be equal to P1*(V2-V1) aka the area under the P1 line. Do I have to subtract the work done to the system or is this the solution already? If so, why do I need P2?
21. Relations on the Kinetic velocities in a cyclic reaction

Hi to all, I whant to ask a question about theoretical chemistry. Let us consider a cyclic reaction ##\alpha A\rightarrow \beta B\rightarrow \gamma C\rightarrow \alpha A## where ##\alpha,\beta;\gamma## are the stochiometric coefficients and ##A,B,C## chemical molecules ... there are relations...
22. MHB Prove that the 12-th roots of unity in C form a cyclic group

Hey guys, Sorry that it's been a decent amount of time since my last posting on here. Just want to say upfront that I am extremely appreciative of all the support that you all have given me over my last three or four posts. Words cannot express it and I am more than grateful for it all. But, in...
23. T is cyclic iff there are finitely many T-invariant subspace

Homework Statement "Let ##T## be a linear operator on a finite-dimensional vector space ##V## over an infinite field ##F##. Prove that ##T## is cyclic iff there are finitely many ##T##-invariant subspaces. Homework Equations T is a cyclic operator on V if: there exists a ##v\in V## such that...
24. Proving that Aut(K), where K is cyclic, is abelian

Homework Statement Let ##G## be a group and ##K## a finite cyclic normal subgroup of ##G##. a) Prove that ##\operatorname{Aut}(K)## is an abelian group b) Prove that ##G' \subseteq C_G (K)##, where ##G'## is the commutator subgroup of ##G##. Homework EquationsThe Attempt at a Solution I'm...
25. I Is any attention being given to Conformal Cyclic Cosmology?

Conformal Cyclic Cosmology, or CCC, is a hypothesis put forward by Roger Penrose in the early 2000s. My understanding of physics is lacking so my explanation will not be that clear, but I will summarize it here. Essentially, the existence of a previous spacetime, or "aeon," is postulated. This...
26. Prove that the roots of unity is a cyclic group

Homework Statement Let ##\mu=\{z\in \mathbb{C} \setminus \{0\} \mid z^n = 1 \text{ for some integer }n \geq 1\}##. Show that ##\mu = \langle z \rangle## for some ##z \in \mu##. Homework EquationsThe Attempt at a Solution My thought would be just to write out all of the elements of ##\mu## in...
27. Characterizing subgroups of a cyclic group

Homework Statement Show that for every subgroup ##H## of cyclic group ##G##, ##H = \langle g^{\frac{|G|}{|H|}}\rangle##. Homework EquationsThe Attempt at a Solution At the moment the most I can see is that ##|H| = |\langle g^{\frac{|G|}{|H|}}\rangle|##. This is because if...
28. Order of element and order of cyclic group coincide

Homework Statement Let ##G## be a group and ##x \in G## any element. Prove that if ##|x| = n##, then ##|x| = |\{x^k : k \in \mathbb{Z} \}|##. Homework EquationsThe Attempt at a Solution Let ##H = \{x^k : k \in \mathbb{Z} \}##. I claim that ##H = \{1,x,x^2, \dots , x^{n-1} \}##. First, we show...
29. I Cyclic variables for Hamiltonian

A single particle Hamitonian ##H=\frac{m\dot{x}^{2}}{2}+\frac{m\dot{y}^{2}}{2}+\frac{x^{2}+y^{2}}{2}## can be expressed as: ##H=\frac{p_{x}^{2}}{2m}+\frac{p_{y}^{2}}{2m}+\frac{x^{2}+y^{2}}{2}## or even: ##H=\frac{p_{x}^{2}}{2m}+\frac{p_{y}^{2}}{2m}+\frac{\dot{p_{x}}^{2}+\dot{p_{x}}^{2}}{4}##...
30. Proof of Subgroup Property for Cyclic Group G: Homework Help

Homework Statement Let G be a group. Assume a to be an element of the group. Then the set <a> = {ak I k∈ℤ} is a subgroup of G. I am confused as to why the proof makes the assumption that <a> is a subset of the set G. Homework EquationsThe Attempt at a Solution The proof I think is like the...
31. Show that Q x Q is not cyclic

Homework Statement Prove that ##\mathbb{Q} \times \mathbb{Q}## is not cyclic. Homework EquationsThe Attempt at a Solution For contradiction suppose that ##\mathbb{Q} \times \mathbb{Q}## is cylic. Hence it is generated by some element ##(r,q)## where ##r \ne 0## and ##q \ne 0##. Then for some...
32. Infinite cyclic group only has two generators

Homework Statement Let ##H = \langle x \rangle##. Assume ##|x| = \infty##. Show that if ##H = \langle x^a \rangle## then ##a = \pm 1## Homework EquationsThe Attempt at a Solution Here is my attempt: Suppose that ##H = \langle x^a \rangle##. Then, for arbitrary ##b \in \mathbb{Z}##, ##x^b =...
33. Showing that cyclic groups of the same order are isomorphic

Homework Statement Prove that any two cyclic groups of the same finite order are isomorphic Homework EquationsThe Attempt at a Solution So I began by looking at the map ##\phi : \langle x \rangle \to \langle y \rangle##, where ##\phi (x^k) = y^k##. So, I went through and showed that this is...
34. I Facts of a finite cyclic group

Problem: If ##H = \langle x \rangle## and ##|H| = n##, then ##x^n=1## and ##1,x,x^2,\dots, x^{n-1}## are all the distinct elements of ##H##. This is just a proposition in my book with a proof following it. What I don't get is the very beginning of the proof: "Let ##|x| = n##. The elements...
35. Help with calulating a cyclic rate

Hello, sorry if i do not ask the question correctly but any help would be great. i need to find the speed of an item and I'm not sure what i need to find this out? well a cyclic rate of this item? Thanks
36. SN1 reactions of cyclic ethers

The Attempt at a Solution Firstly, since -Br is the better leaving group among the three, it's either (a) or (b). Further, since -Br is present at α position in (b), which also has an acidic α hydrogen to the same carbon, it would react better with a weak nucleophile (favouring SN1) than in (a)...
37. MHB Proving Finite subgroups of the multiplicative group of a field are cyclic

I am looking at this proof and I am stuck on the logic that $a^{p}$ = 1. For example, consider the group under multiplication without zero, ${Z}_{5}$, wouldn't 2^4 = 1 imply that the order is 4 not 5? We know that if G is a finite abelian group, G is isomorphic to a direct product...
38. L

I Paul Steinhardt's cyclic model

Hello. I think I don't understand very well the Paul Steinhardt's cyclic model of Universe(s). According to Paul Steinhardt, 2 universes get closer. Then, there's the big bounce, which products effects like a big bang. If 2 universes get closer, they have a (relative) speed ( speed is the...
39. L

I How many generators can a cyclic group have by definition?

Hi, so I have just a small question about cyclic groups. Say I am trying to show that a group is cyclic. If I find that there is more than one element in that group that generates the whole group, is that fine? Essentially what I am asking is that can a cyclic group have more than one generator...
40. A

I Why are cyclic coordinates named such?

In Lagrangian mechanics, if the Lagrangian is not a function of one of the generalised coordinate, then it is called a cyclic coordinate. Why is it called such? What is the significance of the term 'cyclic'?
41. I Dark Energy and the Cyclic Universe?

Now two points here. (1) Most Cyclic Universe theories I've heard require the universe to reverse course and fall back into a Big Crunch to recycle again. Now that Dark Energy has been discovered, the chances of a Big Crunch have gone away. (2) Some theories suggest that the universe started...
42. Cyclic Thermodynamic processes

I'm a high school student with basic knowledge about thermodynamics. I have always come across systems under going reversible cyclic processes. Are there any cases for irreversible cyclic processes? Thanks in advance.
43. Every infinite cyclic group has non-trivial proper subgroups

Homework Statement Every infinite cyclic group has non-trivial proper subgroups Homework EquationsThe Attempt at a Solution I know that if we have a finite cyclic group, it only has non-trivial proper subgroups if the order of the group is not prime. But I'm not sure how to make this argument...
44. Abelian group as a direct product of cyclic groups

Homework Statement Consider G = {1, 8, 12, 14, 18, 21, 27, 31, 34, 38, 44, 47, 51, 53, 57, 64} with the operation being multiplication mod 65. By the classification of finite abelian groups, this is isomorphic to a direct product of cyclic groups. Which direct product? Homework EquationsThe...
45. I The proof of the above theorem is similar to the proof of the above statement.

We define a cyclic group to be one all of whose elements can be written as "powers" of a single element, so G is cyclic if ##G= \{a^n ~|~ n \in \mathbb{Z} \}## for some ##a \in G##. Is it true that in this case, ##G = \{ a^0, a^1, a^2, ... , a^{n-1} \}##? If so, why? And why do we write a cyclic...
46. If G is cyclic, and G is isomorphic to G', then G' is cycli

Homework Statement Title Homework EquationsThe Attempt at a Solution This would seem to be very easy problem, since it's intuitively obvious that if two groups are isomorphic, and one is cyclic, then the other is cyclic too. However, I can't seem to formalize it with math. Here is an idea. We...
47. Cyclic group has 3 subgroups, what is the order of G

Homework Statement Suppose a cyclic group, G, has only three distinct subgroups: e, G itself, and a subgroup of order 5. What is |G|? What if you replace 5 by p where p is prime? Homework EquationsThe Attempt at a Solution So, G has three distinct subgroups. By Lagrange's theorem, the order of...
48. Is the group of positive rational numbers under * cyclic?

Homework Statement Is the group of positive rational numbers under multiplication a cyclic group. Homework EquationsThe Attempt at a Solution So a group is cyclic if and only if there exists a element in G that generates all of the elements in G. So the set of positive rational numbers would...
49. Verifying a Proof about Maximal Subgroups of Cyclic Groups

Homework Statement Show that if ##G = \langle x \rangle## is a cyclic group of order ##n \ge 1##, then a subgroup ##H## is maximal; if and only if ##H = \langle x^p \rangle## for some prime ##p## dividing ##n## Homework Equations A subgroup ##H## is called maximal if ##H \neq G## and the only...
50. Futurama: How plausible is the cyclic model, is it mainstream?

https://en.wikipedia.org/wiki/Cyclic_model it brings to mind as well the many worlds theory the cyclic model is shown in a futuruma episode where the professor builds a only forward time machine(consistent with relativity) but he goes so far into the future that he ends in the past of the...