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.

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154 views

Let G be a group of order $n$, where $n$ is a positive integer relatively prime to $\varphi(n)$. Show that G is cyclic.

Let G be a group of order $n$, where $n$ is a positive integer relatively prime to $\varphi(n)$. Show that G is cyclic. You may only assume the Feit-Thompson theorem here and prove in the following ...
6
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1answer
80 views

Necessary and Sufficient Condition for $\phi(i) = g^i$ as a homomorphism - Fraleigh p. 135 13.55

Let $g \in \text{ group } G $ and $n \in N$. Let $\phi : \mathbb{Z_n} \rightarrow G$ be defined by $\phi(i) = g^i$ for $0 \le i \le n$. Give a necessary and sufficient condition (in terms of g and n) ...
2
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1answer
28 views

Proper Subgroup is Cyclic Proof

Let G be a group with |G| = p^2, where p is prime. Show that every proper subgroup of G is cyclic. Proof: Let H be a proper subgroup of G where |G| = p^2 and p is prime. Then |H| divides |G| by ...
2
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1answer
49 views

if $\operatorname{ord}(\alpha)=n_1$ and $\operatorname{ord}(\beta)=n_2$, then what is $\operatorname{ord}(\alpha\beta)$?

$\newcommand{\ord}{\operatorname{ord}}$ If $\alpha,\beta \in \mathbb{GF}(q)$ and $\ord(\alpha)=n_1$ and $\ord(\beta)=n_2$, then what is $\ord(\alpha\beta)$ ? Edit: if $k=\ord(\alpha\beta)$ then ...
2
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1answer
82 views

Cyclic (sub)groups

I have two questions that are confusing me. Why is the cyclic group generated by $g$ the smallest subgroup containing $g$? If $g$ generates an infinite subgroup, why is it called cyclic? I mean, ...
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1answer
35 views

Number of elements in a group

The group $G$ consists of the binary strings of length $5$ under addition $\mod 2$ in each component. (It is isomorphic to $(\mathbb Z_2)^5$, the direct product of $5$ copies of $\mathbb Z_2$.) Let ...
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1answer
75 views

Counting Elements of Order $2$ in a Direct Product of Cyclic Groups

I am not sure if I am oversimplifying this question or not, but here it is: Suppose we have the following direct product of groups: $$G=\mathbb Z/60\mathbb Z \times \mathbb Z/45\mathbb Z ...
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1answer
34 views

Discrete math Group - Isomorphism and Automorphism

Let G be a Cyclic group Prove or disprove: A.let $ a,b \in G \quad $ so the function $ f:G \to G,f(a^k) = b^k$ is Automorphism of G(which means G is Isomorphism to herself) B.let a,b generators ...
0
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1answer
36 views

Direct product of groups is cyclic or not?

Let $\Bbb Z$ be the additive group of integers and $S = \{-1,1\}$ be a group under multiplication. Is the product $\Bbb Z \times S$ cyclic? Why or why not? I am really confused on this question ...
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1answer
22 views

Cycles “converging” to an infinite cycle?

I recently had as an assignment, to find cycles $\sigma,\tau\in S_{\mathbb{N}}$ (i.e. permutations over the naturals) such that $ord(\sigma)=ord(\tau)=2$ and $\tau\circ\sigma$ has order infinity. This ...
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1answer
201 views

Finding the number of Bracelet five beads different colors

Consider a bracelet of 5 colored beads. How many different bracelets are there if all the 5 colors are different? Take into account rotations and flipping. Verify the ...
0
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1answer
40 views

Prove that $\langle a\rangle = G$, where $G$ a cyclic group of order 24, $a \in G$, $a^8 \ne e$, $a^{12} \ne e$.

Let $G$ a cyclic group, $|G| = 24$. Let $a\in G$, such that $a^8 \ne e$ and $a^{12}\ne e$. Prove that $\langle a \rangle = G$. So far i have found that $a^2,a^3,a^4,a^6 \ne e$. So to solve the ...
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1answer
110 views

Why isn't $\langle a ; a^2 \rangle$ (or $\langle a;a^3, a^7\rangle$) a presentation of $C_4$?

I've just read the first few pages of Combinatorial Group Theory by Magnus, Karrass, and Solitar, and based on their definitions there, and more specifically, the reasoning given in the hint to ...
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0answers
60 views

Find all subgroups of $\mathbb{Z_2} \times \mathbb{Z_2} \times \mathbb{Z_4}$ isomorphic to the Klein $4$-group - Fraleigh p. 110 Exercise 11.12

I tried to fill in the steps but I'm still confounded by this solution. $|\mathbb{Z_2} \times \mathbb{Z_2} \times \mathbb{Z_4}| = 2\cdot2\cdot4$. $order(\mathbb{Z_2} \times \mathbb{Z_2} \times ...
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0answers
32 views

Subsets of cyclic group with distinct pairwise differences

Given a number $m\in\mathbb N$, let $\mathbb Z_m=\{0,1,\dots,m-1\}$ denote the ring of integers modulo $m$ (although we won't need multiplication, so any cyclic group of order $m$ will do). Given a ...
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49 views

Nontrivial homomorphism for $Z_a \times Z_b $to $Z_c \times Z_d$ - Fraleigh p. 134 13.35

This isn't a duplicate of this. Let $(A, B) \in \mathbb{Z_a \times Z_b}$. Hinging on p. 2, I guess homomorphism is $h(A,B) = (A \text{ mod } c, B \text{ mod } d)$. I'm unsettled. p. 2 sprang it up ...
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118 views

$(\mathbb{Z}/p^r\mathbb{Z})^{\ast}$ is a cyclic group

I would like to prove that the group $(\mathbb{Z}/p^r\mathbb{Z})^{\ast}$ of the invertible elements of $\mathbb{Z}/p^r\mathbb{Z}$ with $p>2$ prime and $r>0$ is cyclic. My text suggests to ...
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32 views

What primes can ramify and decompose in $k(\mu_{p^m}) \mid k$?

Let $k$ be a number field and $p$ be a rational prime. Then consider the extension $k(\mu_{p^m}) \mid k$ of adjoining all roots of unity of degree $p^m$ to $k$. Assuming that $\mu_{p^m} \cap k$ is ...
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82 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.
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42 views

Prove that a group is non-cyclic

I have some problems with an exercise of group theory. It sounds like this: Let $G$ be a finite group of order $p^2q^2$ with $p$, $q$ prime numbers, $p$ is odd, $p<q$. I know that $G$ doesn't ...
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42 views

Describe up to isomorphism the semidirect product

I have the following problem: I have to describe up to isomorphism the semidirect product $C_6 \rtimes C_2$, where $C_6$ denotes cyclic group of order six. I think I have to use external semidirect ...
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45 views

Divisors of the totient function and congruences

Based on the question Question about the totient function and congruence classes, I would like to ask two new questions: Question 1 Does it hold that if $k\mid\varphi(n)$ then the elements of the ...
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24 views

Cyclic $\mathbb{Z}_{2p^k}^*$ group

I'm new in group theory and have some problems. I have got difficulties with understanding my notes from lecture. Maybe you can help me? I have this proof that $\mathbb{Z}_{2p^k}^*$ ($p>2$ prime) ...
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32 views

Cyclic Code Galois

I have a generator polynomial over GF(3) for a cyclic code. What is the fastest way to find the minimum distance of this code?
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26 views

Terminology for multilinear functionals the sum of whose cyclic shifts is zero?

Let $\varphi$ be an $n$-linear functional on a vector space $X$. Suppose that $\varphi$ has the property that, for all $(x_1,\ldots,x_n) \in X^n$, we have $$ \varphi(x_1,\ldots,x_n) + ...
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0answers
58 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 ...
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35 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 ...
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130 views

Finding generators for commutative encryption

The paper Information Sharing Across Private Databases presents a protocol for finding an intersection of two private sets. They use a commutative encryption that is quite similar to the ...
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22 views

Prove identity involving alternating groups

Prove the following identity: where $I_{A_n}(x_1,...,x_n)$ is a cyclic index the natural action of the alternating group $A_n$ on the set ${1,...,N}$ (assuming that $I_{A_0} = 1$).
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59 views

Free groups vs. free abelian groups

I'm trying to solve this question in page 74 of Hungerford's book: A free abelian group is a free group (Section I.9) if and only if it is cyclic. I have no idea how to proceed, a solution or a ...
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31 views

Is the subgroup of orthogonal matrices in $GL_3(\mathbb R)$ cyclic? What's generating it?

Is the subgroup of $GL_3(\mathbb R)$ denoted by $Q=\{A \in GL_3(\mathbb R)\mid AA^t=I_3\}$ cyclic? What's generating it? I showed $Q$ is a group, and realized it's the group of all orthogonal ...
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153 views

Let G be the dihedral group of order 14

Let G be the dihedral group of order 14 i) Let A=$C_2$ be a cyclic group of order 2. Find all homomorphisms $G\to A$ ii) Let B=$C_7$ be a cyclic group of order 7. Find all homomorphisms $G\to B$ I ...
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0answers
43 views

Let $G = \langle x\rangle$ be cyclic of order $n$. Prove that $\langle x^r\rangle\leq \langle x^s\rangle$ iff $r$ is a multiple of $s$ modulo $n$.

Let $G = \langle x\rangle$ be cyclic of order $n$. Prove that $\langle x^r\rangle \leq \langle x^s\rangle$ iff $r$ is a multiple of s modulo $n$. Give the complete subgroup lattice of $G = \langle ...
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278 views

Is the multiplicative group of non-zero elements of a finite field cyclic? What are the generators of $(Z_p \setminus \{0\}$ for a prime $p$?

How to decide if the multiplicative group of non-zero elements of a finite field is cyclic or not? Based on our experience with $(Z_7 \setminus \{0\}, \cdot)$, which is generated by $3$ or $5$, and ...
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0answers
22 views

A question about the order of $2$ in the reduced residue system of a $\frac{p\#}{2}$ where $p\#$ is the primorial of $p$

I've noticed a pattern in the order of $2$ in the reduced residue system for $\frac{p\#}{2}$ where $p\#$ is the primorial for a prime $p$. For $\frac{3\#}{2}$, $\operatorname{ord}_{3}(2)=2$ and ...
0
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0answers
161 views

Homomorphisms and Automorphisms between cyclic groups of prime order

Let $A = B \times C$ where $B$ and $C$ are cyclic of order $p$ and $p^2$ respectively, where $p$ a prime. How many endomorphisms are there? How many of these endomorphisms are automorphisms?