Cyclic Definition and 309 Threads

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.

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...

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> =...

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...

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...

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...

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...

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...

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?

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...

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!

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...

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=...

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...

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...

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...

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...

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...

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...

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...

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.

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

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

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...

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...

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...

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...

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...

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...

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...

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?

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...

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...

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...

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...

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...

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?

! 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?

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...

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...

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?

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.

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...

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>...

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

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>)?

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...

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|...

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...

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