Cyclic Definition and 309 Threads

Homework Statement If G is a finite abelian group that has one subgroup of order d for every divisor d of the order of G. Prove that G is cyclic. Homework Equations The Attempt at a Solution

Homework Statement Let H be a finite abelian group that has one subgroup of order d for every positive divisor d of the order of H. Prove that H is cyclicHomework Equations We want to show H={a^n|n is an integer}

Why is Z mod 2 x Z mod 3 isomorphic to Z mod 6 but Z mod 2 x Z mod 2 not isomorphic to Z mod 4?

Homework Statement Name each of the following cyclic alkanes, and indicate the formula of the compound. http://img230.imageshack.us/img230/1042/organicnaming.jpg Homework Equations The Attempt at a Solution a. The rectangle has four corners so four carbon for butane. The branch...

Homework Statement If G is an Abelian group and contains cyclic subgroups of orders 4 and 6, what other sizes of cyclic subgroups must G contain? Homework Equations A cyclic group of order n has cyclic subgroups with orders corresponding to all of n's divisors. The Attempt at a...

Homework Statement Prove any cyclic group with more than two elements has at least two different generators. Homework Equations A group G is cyclic if there exists a g in G s.t. <g> = G. i.e all elements of G can be written in the form g^n for some n in Z. The Attempt at a...

Homework Statement Show ZXZ/<1,1> is an infinite cyclic group. Homework Equations The Attempt at a Solution <1,1> = {...(-1,-1), (0,0), (1,1),...} implies ZXZ/<1,1> = {<1,0>+<1,1>, <0,1>+<1,1>} which is isomorphic to ZXZ. But ZXZ is not cyclic, is my description of the...

Homework Statement Show ZXZ is an infinite cyclic group. Under addition of course. Homework Equations The Attempt at a Solution So this obviously is an infinite cyclic group. Z is generated by <1> or <-1>. The problem I run into here is I think <(1,1)> will only generate elements of the form...

Let H be normal in G, H cyclic. Show any subgroup K of H is normal in G. I was thinking about using the fact that subgroups of cyclic groups are cyclic, and that subgroups of cyclic groups are (fully)Characteristic (is that true?). Then we would have K char in H and H normal in G. Hence K...

Describe the field F=\frac{\mathbb{F}_3[x]}{(p(x))} [p(x) is an irreducible polynomial in \mathbb{F}_3[x]]. Find an element of F that generates the cyclic group F^* and show that your element works. [p(x)=x^2+1 is irreducible in \mathbb{F}_3[x] if that helps]

Ok, here is something i thought i understood, but it turns out i am having difficulties fully grasping/proving it. Let \theta:G->G' be an isomorphism between G and G', where o(G)=m=o(G'), and both G and G' are cyclic, i.e. G=[a] and G'=[b] So my question is, when we want to find the...

Homework Statement Let G be a finite group of rotation of the plane about the origin. Prove that G is cyclic. The Attempt at a Solution What it means to be cyclic is that every element of the group can be written as a^n for some integer n. I can see this is true if i take some...

[b]1. What is/are the condition for a group with no proper subgroup to be cyclic? Homework Equations [b]3. this is just a general qustion I am asking in oder to prove something?

Today we learned about subgroups of cyclic groups G = <a>. During the discussion we reached this point: |<a^k>| = minimum L, L > 0, such that a^(kL) = 1. |G| = n. Then a^kL = a^bn, thus kL = bn, and thus L = n/gcd(k, n). However, I don't understand the bolded. My number theory is...

Homework Statement Let K be a field, and let K' be an algebraic closure of K. Let sigma be an automorphism of K' over K, and let F be the fix field of sigma. Let L/F be any finite extension of F. Homework Equations Show that L/F is a finite Galois extension whose Galois group...

Hello, I know trace is usually cyclic, but is partial trace cyclic too? Why? Thanks! Jenga

I'm currently going through Hungerford's book "Algebra", and the first proof I found a bit confusing is the proof of the theorem which states that every infinite cyclic group is isomorphic to the group of integers (the other part of the theorem states that every finite cyclic group of order m is...

I'm "aware" of current theories about the cyclic model but what's wrong with this hypothesis? The universe starts crunching forming a big black hole All the energy is sucked back into this big black hole The big bang starts again with all the matter and energy as the previous big bang...

I had the pleasure of attending a lecture given by Paul Steinhardt, a Princeton professor, regarding the big bang and cyclic universe models at Fermilab this evening. Steinhardt, having written a book called The Endless Universe, is obviously a fan of the cyclic universe camp and the main focus...

I also wonder about an other interesting residue relation Let P be a prime, let a^{2^n} be called a cyclic quadratic residue if there is integer m dependent on a such that a^{2^{n + mp}} = a^{2^n} for all integers p \mod P It seems that the sum of all such cylic residues is either 0 or...

Homework Statement A sample of an ideal gas goes through the process shown below. From A to B, the process is adiabatic; from B to C, it is isobaric with 98 kJ of energy entering the system by heat. From C to D, the process is isothermal; from D to A, it is isobaric with 158 kJ of energy...

Homework Statement A Cyclic proces of three parts for a monoatomic idealgas: 1-2: Isochor, where: p2 = 2*p1 v2 = v1 2-3: Adiabatic, from v2 = v3, where: v3 > v2 p3 = p1 3-1: unknow proces, where v4 = v1 p4 = p1 My problem is to determine Q1, Q2, Q3 and W1, W2, W3 and...

I'm trying to prove that GL(2,p^n) has a cyclic subgroup of order p^{2n} - 1. This should be generated by \left( \begin{array}{cc} 0 & 1 \\ -\lambda & -\mu \end{array} \right) where X^2 + \mu X + \lambda is a polynomial over F_{p^n} such that one of its roots has multiplicative order...

[SOLVED] Cyclic Sequence of Angles Fix an angle \theta. Let n be a positive integer and define \theta_n = n\theta \bmod 2\pi. The sequence \theta_1, \theta_2, \ldots is cyclic if if it starts repeating itself at some point, i.e. the sequence has the form \theta_1, \ldots, \theta_k, \theta_1...

Homework Statement Let G be a group and let #G=77. Prove the following: a) G is cyclic, if there is such an element a in G that a21≠1 and a22≠1 b) If there are such elements a and b, so that ord(a)=7 and ord(b)=11, then G=<a,b> 2. Homework Equations , 3. The Attempt at a Solution I...

I have a voltammegram (graph of potential vs current) and I want to find the Ipc of it. I'm not really sure how..is there a way to find it accurately or do i just have to estimate?

I'm trying to find out the roles of the following for cyclic voltammetry: working electrode auxiliary electrode reference electrode and potentiostat I kind of found out what they are.. but i am not sure of its exact role, like for the working electrode, it is the electrode at which the...

Homework Statement List the cyclic subgroups of U(30) Homework Equations The Attempt at a Solution In order to list the cyclic subgroups for U(30) , you need to lists the generators of U(30) U(30)={1,7,11,13,17,19,23,29} . all the elements of U(30) are not generaters. in...

The fractal sequence http://www.research.att.com/~njas/sequences/A054065 is of interest because it provides permutations of the numbers 1-n such that the decimal part of k*tau (k = {1,2,3,...n} is ordered from the lowest possible value to the highest. For instance if n = 3 the permutation...

Homework Statement List the elements of the subgroups <3> abd <15> in Z(18) Homework Equations The Attempt at a Solution <3>={0,3,6,9,12,15} . <15> ={0,15} Together , I can conclude that the number of elements amongst <3> and <15> add up to 7 elements.

Homework Statement Find all generators of Z(6), Z(8) , and Z(20) Homework Equations The Attempt at a Solution I should probably list the elements of Z(6), Z(8) and Z(20) first. Z(6)={0,1,2,3,4,5} Z{8}={0,1,2,3,4,5,6,7}...

Homework Statement Show that \left( \mathbb{Z}/32\mathbb{Z}\right)^{*} is not a cyclic group. Homework Equations The Attempt at a Solution A little calculator magic has showed that all elements in the group have order 8, but that doesn't seem like a very educational solution :). If...

Homework Statement Prove that no group can have its automorphism group cyclic of odd order. Homework Equations The Attempt at a Solution Aut(Z2) has order 1, which is odd...trivial, yes, but I thought I was DONE. However, my professor has said "well prove it EXCEPT for Z2"...

Homework Statement If an R module M is cyclic so M=Rm with annihilator(m)=(p), p prime then can we infer that M is isomorphic to R/(p) without any more infomation?

I came across a theorem the proof of which I don't quite understand. The theorem states that every infinite cyclic group is isomorphic to the additive group Z. So, the mapping f : Z --> G given with k |--> a^k, where G = <a> is a cyclic group, is an epimorphism, which is quite obvious...

Describe al group homomorphisms \phi : C_4 --> C_6 The book I study from seems to pass over Group Homomorphisms very fast. So I decided to look at Artin's to help and it uses the same definition. So I think I am just not digesting something I should be. I know it's defined as \phi (a*b)...

I came across a section of my notes that claimed the automorphism group of the cyclec groups C_2p where p=5,7,11 is cyclic, that is Aut(C_2p) is cyclic for p = 5,7,11. I wasn't able to see why this is so. Is it just a fact or is there some sort of proof of the above...? Thanks

An ideal gas undergoes a reversible, cycli process. First it expands isothermally from state A to state B. It is then compressed adiabatically to state C. Finally, it is cooled at constant volume to its original state, A. I have to calculate the change in entropy of the gas in each one of the...

prove that Z(v,T)=Z(u,T) iff g(T)(u)=v, where g(t) is prime compared to a nullify -T of u. (which means f(t) is the minimal polynomial of u, i.e f(T)(u)=0). (i think that when they mean 'is prime compared to' that f(t)=ag(t) for some 'a' scalar). i tried proving this way: suppose, g(T)(u)=v...

the cyclic universe... is the theory proposed by paul steinhardt can be explained even without the use of superstring theory? i mean can you apply other theories (such as lqg) to explain the cyclic universe idea, or it's entirely depended on strings and extra dimensions?

"Suppose V is an n-dimensional vector space over an algebraically closed field F. Let T be a linear operator on V. Prove that there exists a cyclic vector for T <=> the minimal polynomial is equal to the characteristic polynomial of T." (A cyclic vector is one such that (v,Tv,...,T^n-1 v) is a...

http://img20.imageshack.us/img20/2964/physics16ri.th.png The working substance of a cyclic heat engine is 0.75kg of an ideal gas. The cycle consists of two isobaric processes and two isometric processes as shown in Fig. 12.21 (image above). What would be the efficiency of a Carnot engine...

is this true for all cases? i know something can be abelian and not cyclic. thanks

When an unknown weight W was suspended from a spring with an unknown force constant k, it reached its equilibrium position and the spring was stretched 14.2cm because of the weight W. Then the weight W was pulled further down to a position 16cm (1.8cm below its equilibrium position) and...

Right I have been given the following problem and cannot resolve it. I have had an attempt but without much success. Could anyone help me with this exercise, please? Hints or a little more welcome :-) A cyclic hexagon is a hexagon whose vertices all lie on the circumference of a circle...

I'm really stuck on these two questions, please help! 1. Let G={invertible upper-triangular 2x2 matrices} H={invertbile diagonal matrices} K={upper-triangular matrices with diagonal entries 1} We are supposed to determine if G is isomorphic to the product of H and K. I have concluded...

Why is the vanishing of a cyclic integral the property of a state function?

Show that the set X = \{x : 0 < x < p^m, x \equiv 1 (\mathop{\rm mod}p)\} where p is an odd prime, together with multiplication mod p^m forms a cyclic group. It might help to write the x in X in the form: x = 1 + a_1p^1 + \dots + a_{m-1}p^{m-1} for (a_1,\, a_2,\, \dots ,\, a_{m-1}) \in...

This time I need a yes/no answer (but a definitive one!): Suppose we have a group of finite order G, and two cyclic subgroups of G named H1 and H2. I know the intersection of H1 and H2 is also a subground of G, question is - is it also cyclic? And can I tell who is the creator of it, suppose I...

At school we extracted limonene from orange peels and we had to make an IR spectroscopy for it but I don't see anywhere how we can know the product has a ring constitution... I see a lot of information about aromatic rings but nothing for an alkene ring... Can anybody help me?