Use with the (group-theory) tag. A group is cyclic if it can be generated by a single element, $a$. Every element has the form $a^i$ for some $i\in\mathbb{Z}$, and so these groups are abelian.
4
votes
1answer
46 views
Group homomorphisms into a field
Let $G$ be a finite group, and let $k$ be a field, which should be algebraically closed, I think. How to describe all homomorphisms $G\rightarrow k^*$ (i.e. one-dimensional representations: ...
0
votes
2answers
54 views
How many elements are in the kernel of the homomorphism $f:(\mathbb{Z}/154\mathbb{Z})^\star \to (\mathbb{Z}/154\mathbb{Z})^\star $ where $f(x)=x^5$?
How many elements are in the kernel of the homomorphism $f:(\mathbb{Z}/154\mathbb{Z})^* \to (\mathbb{Z}/154\mathbb{Z})^* $ where $f(x)=x^5$? The group operation in this case is multiplication with ...
4
votes
3answers
117 views
Quadratic subfield of cyclotomic field
Let $p$ be prime and let $\zeta_p$ be a primitive $p$th root of unity. Consider the quadratic subfield of $\mathbb{Q}(\zeta_p)$. For instance, for $p=5$ we get the quadratic subfield to be ...
2
votes
3answers
92 views
Group with an order?
Let $G$ be a group and $g\in G$ with $\operatorname{ord}(g)=40$. Find the order of $a = g^8$, $b= g^5$, and what is $ab$?
I know how to solve the proff $\operatorname{ord}(a)=m$, $\operatorname{ord} ...
2
votes
0answers
33 views
Find the cycle index of cyclic group $C_p$ where $p$ is prime
Find the cycle index of cyclic group $C_p$ where $p$ is prime.
No idea where to start...any help pointing me in the right direction would be appreciated.
1
vote
3answers
74 views
Why is $| \rm{Aut}(\mathbb{Z}_n) | = \phi(n)$?
If $G$ is a finite cyclic group of order $n$, prove that $| \rm{Aut}(G) | = \phi(n)$, where $\phi(n)$ is the Euler's totient function.
Can someone please help me with this?
0
votes
3answers
58 views
Groups - Prove that if $G/Z(G)$ is cyclic then $G$ is abelian
Prove that if $G/Z(G)$ is cyclic then $G$ is abelian. Using this fact and $G$ is a nontrivial group of prime power order, deduce that a group of order $p^2$ , $p$ prime, is abelian.
7
votes
2answers
194 views
Abelian group admitting a surjective homomorphism onto an infinite cyclic group
I am working on the following problem:
Let $G$ an Abelian group and $f: G \to \Bbb Z$ a surjective
homomorphism. Prove that $G \cong \ker(f) \times \Bbb Z$
By means of the First Isomorphism ...
1
vote
3answers
70 views
Must $G$ be cyclic?
If $G$ is a group and $H_1,H_2$ are cyclic subgroups of $G$ with $G=H_1H_2=H_2H_1$ then must $G$ be cyclic?
Please help me !
1
vote
1answer
65 views
Euler's formula and subgroups of $\mathbb Z_n$
Prove that in $\mathbb Z/n\mathbb Z$ for every divisor $d$ of $n$ there is a unique subgroup of order $d$ using the following results: $\sum_{d\mid n}\varphi(d)=n$ and the number of generators of ...
4
votes
2answers
108 views
Determining whether two groups are isomorphic
I am reading "a first course in algebra" and there, i am trying to solve the exercises, but there is something i don't understand. How do we understand whether two groups are isomorphic or not? For ...
1
vote
2answers
60 views
Orders of elements in cyclic groups
I think I'm a bit confused about the order of elements in cyclic groups.
If we suppose $G$ is a group of order 35, and let $x∈G$ such that x≠e, from Lagrange's Theorem $x$ will be of order 5, 7, ...
2
votes
1answer
41 views
Show the extension is not cyclic
Let $D\in\mathbb{Z}$ be a squarefree integer and let $a\in\mathbb{Q}$ be a nonzero rational number. Please show that $\mathbb{Q}(\sqrt{a\sqrt{D}})$ cannot be a cyclic extension of degree 4 over ...
1
vote
2answers
88 views
Sylow Theorems Question
Let $p$ be the smallest prime that divides the order of a finite group $G$. If $H$ is a Sylow $p$-subgroup of $G$ and is cyclic. Prove that $N(H)=C(H)$.
0
votes
1answer
61 views
Adding an element to a subgroup
By adding an element $g_j$ to a subgroup $H_{j-1}=\{g_1,g_2,\ldots g_{j-1}\}$, do we get the subgroup $H_j=\langle g \rangle H_{j-1}$? I dont understand how the new group is simply a cyclic extension ...
5
votes
5answers
126 views
What is the number of distinct homomorphism from $\Bbb Z/5 \Bbb Z$ to $\Bbb Z/7 \Bbb Z$
What is the number of distinct homomorphism from $\Bbb Z/5 \Bbb Z$ to $\Bbb Z/7 \Bbb Z$ and how to find it?
I came across the above problem and do not know how to get it? Can someone point me ...
3
votes
2answers
68 views
Subgroups of the roots of unity.
Let $G=\mathbb{C}^*$ and let $\mu$ be the subgroup of roots of unity in $\mathbb{C}^*$. Show that any finitely generated subgroup of $\mu$ is cyclic. Show that $\mu$ is not finitely generated and find ...
3
votes
3answers
169 views
Help me to prove that group is cyclic
Prove that a group of order 5 must be cyclic, and every Abelian group
of order 6 will also be cyclic.
Let G be the group of order 5.
To prove group of order 5 is cyclic do we have prove it by every ...
3
votes
2answers
109 views
When are the product of cyclic groups also cyclic? [duplicate]
In general, products of cyclic groups are not cyclic. If $C_n$ is the cyclic group of order $n$, $C_{22} \times C_{33}$ is not cyclic. But is there an easy way to tell when such a product will be ...
1
vote
3answers
56 views
Understanding a Theorem regarding Order of elements in a cyclic group
This is part of practice midterm that I have been given (our prof doesn't post any solutions to it) I'd like to know whats right before I write the midterm on Monday this was actually a 4 part ...
1
vote
2answers
94 views
What does it mean to have no proper non-trivial subgroup
I am reading a first course in abstract algebra and there is a claim that says a group $G$ with no proper non trivial subgroups is cyclic. But I don't understand what does it mean to have no proper ...
3
votes
2answers
81 views
Cyclic subgroups and generators
Sorry for posting a second question on this topic, abstract algebra is taking a bit longer to get my head around.
I'm trying to work out if this is cyclic or not, and find all generators or show no ...
14
votes
1answer
176 views
Fibonacci Sequence in $\mathbb Z_n$.
Consider a Fibonacci sequence, except in $\mathbb Z_n$ instead of $\mathbb Z$:
$$F(1) = F(2) = 1$$ $$F(n+2) = F(n+1) + F(n)$$
It is easy to see that each of these sequences must cycle through some ...
1
vote
3answers
91 views
How can I conclude that an automorphism is the identity map?
Let $\phi$ be an automorphism on a group $G$. If $\phi$ maps any one non-identity element in $G$ to itself, is $\phi$ necessarily the identity map? What if $G$ is cyclic?
1
vote
3answers
171 views
Show that all abelian groups of order 21 and 35 are cyclic.
Show that all abelian groups of order 21 and 35 are cyclic. I have no idea on how to start. Can anyone give some hints?
8
votes
5answers
363 views
Is $\mathbb{Z}^2$ cyclic?
Is $\mathbb{Z}^2$ cyclic? What does it means for a group to be cyclic? Is it just that it has one generator?
Thanks
2
votes
1answer
84 views
Cyclic groups and generators
For each of the groups $\mathbb Z_4$,$\mathbb Z_4^*$ indicate which are cyclic. For those that are cyclic list all the generators.
Solution
$\mathbb Z_4=${0,1,2,3}
$\mathbb Z_4$ is cyclic and all ...
1
vote
3answers
153 views
The Direct Product of a finite number of cyclic groups, is again a cyclic group?
Can anyone please brief me on this question, do you prove or disprove? If it is to be disproved can you give a counterexample? if there is no generator, then what is an example of such a Direct ...
2
votes
2answers
92 views
Prove that a group G is not cyclic if, and only if, G is a union of proper subgroups.
Can anyone please help me prove this question.
=> So first we can assume that G is not cyclic. Then we need to show that G is a union of proper subgroups. How can I do this?
3
votes
2answers
61 views
Product of Elements in a Finite Cyclic Group with Odd or Even Order
Assuming that G is a finite cyclic group, let "a" be the product of all the elements in the group.
i. If G has odd order, then a=e. Is this because there are an even number of non-trivial elements ...
3
votes
2answers
151 views
An infinite cyclic group has a unique subgroup of index $m$ for any integer $m \geq 1$.
Let $G$ be an infinite cyclic group. Show that for any integer $m \geq 1$, there exists a unique subgroup of index $m$. My attempt is consider $G=(\mathbb{Z},+)$. Consider a subgroup $m\mathbb{Z}$ of ...
1
vote
2answers
73 views
Looking for a simple proof that groups of order $2p$ are up to isomorphism $\mathbb{Z}_{2p}$ and $D_{p}$ .
I'm looking for a simple proof that up to isomorphism every group of order 2p (p prime) is either $\mathbb{Z}_{2p}$ or $D_{p}$ (The Dihedral group of order 2p).
I should note that by simple I mean ...
3
votes
5answers
178 views
If $|G|>1$ is not prime, there exists a subgroup of $G$ which is not $G$ or $e$.
My question is in the title: Prove this statement.
Immediately before this question, I proved that $\langle g \rangle$ is a subgroup of $G$. The question specifies $G$ is a finite group.
My attempt:
...
1
vote
2answers
93 views
What is $\mathbb{Z}/m\mathbb{Z}/n (\mathbb{Z}/m\mathbb{Z})$?
Is $\mathbb{Z}/m\mathbb{Z}/n (\mathbb{Z}/m\mathbb{Z})=\mathbb{Z}/(m\mathbb{Z} +n \mathbb{Z})$?
Thanks.
2
votes
2answers
97 views
How do I find the elements of $S=\langle (12),(1324)\rangle$?
Let $G=S_4$, I want to find the elements of
$$S=\langle (12),(1324)\rangle$$
How do I do this? I know that $$\langle(1324)\rangle≤S$$and I know I have to find out orders of $(12)$ and $(1324)$ ...
1
vote
4answers
136 views
group generator
Need to prove this :
Let $a$ element in $\mathbb Z_n$ , then $a$ is the generator of group $(\mathbb Z_n,+)$ iff $\gcd(a,n)=1$
Have no idea how to prove this, would appreciate some guidance.
1
vote
2answers
75 views
Cyclic subgroups of finite groups
Let's say I want to list all the cyclic subgroups of $G$. Let's say $G = \mathbb{Z}^*_{10}$. Then I know all the elements in $G$ are $1, 3, 7, 9$ so all I need know is to find the cyclic subgroups ...
1
vote
1answer
134 views
Congruence relationship used for primitive residue classes modulo n result
I'm trying to understand a proof for a theorem that states conditions under which the group of primitive residual classes modulo $n$ is cyclic. This proof uses the following result attributed to Gauß:
...
1
vote
1answer
112 views
Non-trivial 93rd roots of unity in finite fields [duplicate]
Possible Duplicate:
Finding the values of $n$ for which $\mathbb{F}_{5^{n}}$, the finite field with $5^{n}$ elements, contains a non-trivial $93$rd root of unity
For which of the following ...
5
votes
1answer
111 views
Find the number of homomorphisms
In each of the following examples determine the number of homomorphisms between the given groups:
$(a)$ from $\mathbb{Z}$ to $\mathbb{Z}_{10}$;
$(b)$ from $\mathbb{Z}_{10}$ to ...
1
vote
0answers
38 views
Matrix Representation of Integer Series
I would like some feedback regarding this process or the meaning of
this process.
Let say that I have a discrete time series:
S = [1 2 3 4 5]
And that I represent this serie by a stochastic matrix M ...
-7
votes
0answers
38 views
$\langle x^n\rangle \cap \langle x^m\rangle = \langle x^{ lcm(m,n)}\rangle$ [duplicate]
Possible Duplicate:
Intersection of cyclic subgroups: $(x^m) \cap (x^n) = (x^{lcm(m,n)})$
If $G=\langle x\rangle$ is a finite cyclic group that $\langle x^n\rangle \cap \langle x^m\rangle = ...
1
vote
0answers
26 views
About rotations of sets of vertices of a regular $p$-gon.
This is something I've been thinking about lately, and I don't seem to understand the problem well enough. There is a motivation to this problem, but I don't think giving it would be productive since ...
-1
votes
3answers
153 views
Intersection of cyclic subgroups: $(x^m) \cap (x^n) = (x^{lcm(m,n)})$
This group theory problem has stumped me. I want to prove that if $G=(x)$ is a finite cyclic group that $(x^n) \cap (x^m) = (x^{\operatorname{lcm}(m,n)})$ for all integers $m$ and $n$, where $(x)$ is ...
2
votes
1answer
49 views
The order of a subgroup of a cyclic subgroup
Let $G = Z^*_p$ under multiplication, with $p$ being a prime $> 3$ and $|G|=p-1$ and $G$ is cyclic. $H = \{a^2\mid a \in G\}$. Want to prove $H < G$ and $|H| = (p-1)/2$.
I can show that $H < ...
1
vote
2answers
59 views
Computing the order and cyclicity of quotients of direct products.
Determine the order of $(\mathbb{Z} \times \mathbb{Z} )/ \langle(2,2)\rangle$ and $(\mathbb{Z} \times \mathbb{Z} )/ \langle(4,2)\rangle$. Are the groups cyclic?
I've read many solutions on the ...
2
votes
2answers
50 views
Checking that $\rho$ is a representation
Let $z$ be a generator of the cyclic group $\mathbb{Z}_3 = \{ 1,z,z^2 \}$. Prove that a representation $\rho$ of $\mathbb{Z}_3$ in the $2$-dimensional complex vector space $\mathbb{C}^2$ can be ...
0
votes
1answer
82 views
Isomorphism of cyclic rings
Let $C(l)$ be a cyclic ring with elements $0,e,2e,...,(n-1)e$. Addition is defined by adding coefficients. Multiplication is defined such that $e^2=le$. So if we multiply two elements together, say ...
1
vote
2answers
73 views
Generators of a finite additive cyclic group
Let $C$ be an abelian additive group and write e for a generator of $C$. The elements of $C$ are then $0,e,2e,3e,\dots,(n-1)e$. If $C$ is finite, prove that the element $ke$ is another generator of ...
1
vote
1answer
87 views
Two equal cyclic subgroups of $S_n$ have conjugate generators
The statement is relatively simple but the proof is giving me some trouble. Any help will be very much appreciated:
Let $a,b \in S_n$. Assuming $<a> = <b>$ show that $b$ is a conjugate of ...
