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.
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...
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...
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]##.
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...
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...
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?
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...
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...
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...
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...
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...
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...
(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...
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...
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...
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=...
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...
$ABCD$ 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 $AD$.
##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.
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?
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...
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...
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...
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...
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...
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...
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...
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...
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}##...
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...
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...
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 =...
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...
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...
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
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)...
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...
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...
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...
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'?
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...
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.
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...
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...
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...
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...
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...
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...
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...
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...