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.
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...
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...
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?
How does one find the most probable central location of something/someone when four points of their earlier location have been recorded and drafted? For example, if one is given points A, B, D, and C, how would they find the central location? This method must use cyclic quadrilaterals, and four...
I'm trying to do this ferrocene lab, and we did cyclic voltammetry on our purified product and I calculated these results:
E_1/2= 664.6 mV
I_pc/I_pa=-.6221
Delta Ep=29.2 mV
Can anyone tell me what these numbers mean? I don't really understand what voltammetry tells you about your...
If I have an abelian group A that is the direct sum of cyclic groups, say
A=[tex]C_5 \oplus C_35[\tex], would I be right in saying the annihilator of A (viewed as a Z-module) is generated by (5,35)? If not, how do I find it?
Here's an interesting question which is related to proofs, one of the hardest chapters of math:
If a circle can be drawn to pass through the 4 vertices of a Quadrilateral, we call this a "cyclic quadrilateral". What special properties do you think a cyclic quadrilateral has that wouldn't be...
Here is a new paper by Niel Turok and Paul J Steinhardt.
http://uk.arxiv.org/PS_cache/hep-th/pdf/0403/0403020.pdf
The initial reading makes one ask a simple question, if I was to be at a far away location, say at the QSO of farthest detected Galaxy, and I looked back to the location of...
Hi everybody!
Question #1
What is the definition of a Klein group? The K_4 group has a table that looks like this:
\begin{array}{c|cccc}
*&e&a&b&c \\\hline
e&e&a&b&c\\
a&a&e&c&b\\
b&b&c&e&a\\
c&c&b&a&e
\end{array}
What is the strict definition of a Klein group? That every...
The cyclic model was introduced in this paper:
"A cyclic model of the universe"
http://arxiv.org/abs/hep-th/0111030
and is a idea of Steinhardt and Turok. (Some might some day recognize the great quantities of ideas that Steinhardt has introduced in the last 30 years).
The model proposes an...
A cyclic group of order 15 has an element x such that the set {x^3, x^5, x^9} has exactly two elements. The number of elements in the set {x^(13n) : n is a positive integer} is :
3.
WHy is the answer 3? Thanks!
1/7 = .142857... (repeated)
2/7 = .285714...
3/7 = .428571...
4/7 = .571428...
5/7 = .714285...
6/7 = .857142...
So, you get all n/7 from the same 'cycle' of 6 digits.
Let's call 7 a 'cyclic' integer.
The next cyclic integers are
17, 19, 23,...
They are all prime. But 11, 13, or 37 are...