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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6
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191 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 ...
4
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0answers
42 views

Proving an eliptic curve is cyclic, and determining it's order

I need a solution with an explanation for the following. Thanks! Let $E/F_q$ be an elliptic curve and let $P ∈ E(F_q)$ be a point a. if $n=ord(P)>1/2(q^{0.5}+1)^2$ prove that $E(F_q)$ is cyclic ...
4
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0answers
48 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 ...
4
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0answers
129 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 ...
3
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0answers
32 views

Proving that a Galois group is cyclic

Let $K$ be a field containing a primitive $n$th root of unity and let $F = K(t)$ be the field of rational functions over $K$. I'm having trouble proving that for each $n > 1$ the field $F$ is Galois ...
3
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0answers
222 views

Using Lagrange's theorem, prove that a non-abelian group of order $10$ must have a subgroup of order $5$.

Using Lagrange's theorem, prove that a non-abelian group of order $10$ must have a subgroup of order $5$. Attempt: Let $G$ be a group of order $10$. By Lagrange's theorem, if there exist a ...
3
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0answers
38 views

Which abelian groups have only a single composition series?

Cyclic groups of composite powers don't: for example, $1=C_1\triangleleft C_3\triangleleft C_6 $ and $1=C_1\triangleleft C_2\triangleleft C_6 $ are both composition series for $C_6$. But cyclic ...
3
votes
0answers
78 views

prove that a is of order m if and only if $a^m = e$ and $a^k $is not $e$ for all $0 < k < m$.

$a$ is of order $m$ if and only if $a^m = e$ and $a^k \neq e$ for all $0 < k < m$. It is about order in Algebra. I sketch the proof. Is it correct? I need your help. First ($\Rightarrow$) Let ...
2
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0answers
39 views

Is there a short proof of the existence of $a$ so that $a$ is a primitive root for infinitely many primes $p$?

After looking for a general answer I found Artins conjecture, and I was happy to see so much is known. However I don't know nearly enough to follow the proof, yet it bothers me I can't prove the ...
2
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0answers
37 views

Progressed A problem on semidirect products where one component is cyclic : a specific problem I managed partially but stuck on the rest

I was recently presented this in my abstract algebra class and I have managed some of it on my own the rest is still a mystery: Let $ H $ be a group and $ K = \langle x\rangle $ be a cyclic group ...
2
votes
0answers
109 views

Artin's Algebra, Exercise 2.4.11. (1st edition)

I have been working through Artin's algebra book. Here's a simple exercise, but I want to make sure I am not missing anything important. Let $(G,\cdot)$ be a group and let $x,y \in G$ with orders ...
2
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0answers
41 views

How do I interpret “K2 mod K1”?

I've been doing a bit of work these past couple of nights on computing cyclic subspace decompositions, finding cyclic bases, and then computing the Jordan canonical form of matrices. My question is: ...
2
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0answers
168 views

Does group $G$ of order 42 have a normal cyclic subgroup of order 21?

Show that a group $G$ of order 42 has a normal cyclic subgroup of order 21. What I did so far is using Sylow's theorem to show that $G$ has a unique 7-sylow subgroup $S(7)$ (which is normal) and {1 ...
2
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0answers
42 views

Some subgroup of $GL_2(\mathbb{Q})$

Let's consider $GL_2(\mathbb{Q})$ and $C_2\times C_2 \times C_2$, $C_2$ - cyclic group of order 2. I can't show, that group $C_2\times C_2 \times C_2$ is not a subgroup of $GL_2(\mathbb{Q})$.I don't ...
2
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0answers
25 views

How do I know the cyclic subgroups of G generated by a are equal?

I had a question of understanding that I was hoping you guys could help me with. Suppose there exists a group in the integers. For the sake of my understanding, let's say the group is, G = $Z_{60}$. ...
2
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0answers
89 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 ...
2
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0answers
38 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 ...
2
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0answers
151 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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24 views

Show that ${C_\infty }/\left\langle {{c^n}} \right\rangle \simeq {C_n}$

Let ${C_\infty } = \left\langle c \right\rangle $ be an infinite cyclic group. Show that if $n > 0$, then $${C_\infty }/\left\langle {{c^n}} \right\rangle \simeq {C_n}$$where ${C_n}$ is a ...
1
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0answers
11 views

Conway's theorem on the number of orbits on the set of all ordered cycles in a $d$-valent graph

I am trying to understand Conway's theorem on the number of orbits on the set of all ordered cycles in a $d$-valent graph. I quote it from Cycles in graphs and groups by Kantor. Theorem $1$ ...
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60 views

If $G/Z(G)$ is cyclic, why is $G$ only abelian and not also cyclic?

If the factor group with respect to the center of $G$ is cyclic, then $(aZ(G))^n=gZ(G)$ for some $n$ and any $g$, where both $a$ and $g$ are from $G$ (and $a^n$ is, too). Because of the definition of ...
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22 views

Consider a set consisting of matrices and show it is a group.

I am giving $$A=\{1,i,-1,-i\}$$ and $$B=\{\begin{pmatrix} 1 & 0 \\ 0 & 1\end{pmatrix}, \begin{pmatrix} 0 & -1 \\ 1 & \phantom{-}0\end{pmatrix}, \begin{pmatrix} -1 & \phantom{-}0 \\ ...
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0answers
20 views

Number of elements in factorgroup

Consider the following Group Theory question: Find the number of elements in each of the factorgroups: $$ \frac{\mathbb{Z}/6\mathbb{Z}}{\left \langle \bar{2} \right \rangle}, ...
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0answers
34 views

combinatorics on cyclic sequences

Given $m\geq 1$, let $I=(a_1,\ldots,a_{3m})$ be a sequence such that $I$ contains exactly $m$ zeros, $m$ ones, and $m$ twos. Given $i=1,2$ and $j\leq 3m,k\leq m$ we can define ...
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0answers
32 views

If $\alpha\approx\sqrt p$ and $\beta\approx\log p$, is $\alpha\beta^{-1}\bmod p\approx p$ with probability $1-o(1)$?

Given $p$ a prime and a random $\alpha\in\Bbb Z_p$ with $\alpha\approx\sqrt{p}$ suppose we pick a random $\beta\approx\log p$ then what is the probability that remainder $\alpha\beta^{-1}\bmod p$ is ...
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57 views

Book Proof Problem - nth roots of unity are of the form $z=\text{cis}(\frac{2k\pi}{n})$

So I found this theorem and proof in my abstract algebra book in the section about cyclic groups. Here it goes: Theorem: If $z^n=1$, then the nth roots of unity are ...
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25 views

Count pairwise integer modulo

Assume $N$ and $M=(N-1)/2$ are both prime. For any $L(L\leq N)$ different integers $1\leq i_1<i_2<\ldots<i_L \leq N$, denote $A_m(i_1,\ldots,i_L)$, $1\leq m \leq M$ to be the number of ...
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0answers
35 views

Right cosets of the cyclic subgroup in $A_5$

Here's a homework question in Abstract Algebra that I have some troubles solving. Given that permutation $\sigma=(1 2 4)(1 5 2)$, and $\sigma\subset A_5$. Find the number of right cosets of the ...
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36 views

Colored beads on a loop

Suppose we have $p$ beads of $n$ different colors on a loop. $p$ is a prime number and we consider the loop to be the same if one is a rotation of the other. Then how many distinct beads are there? By ...
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34 views

Abelian Groups and orders

I came across this question, and was wondering what the notation G[n : a] means. Is x the generator? Any comments/explanations will be of great help. The question I found was: Let $G$ be an abelian ...
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0answers
37 views

Proving property of cyclic groups

A user asked the following question. It was closed as off-topic, or rather as missing context, but it seems the context close reason doesn't exist, so off-topic was chosen. Here it is: I am having ...
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0answers
19 views

Eccentricity of vertex in a permutation cycle graph

The permutation cycle graph is defined as follows. Permutation Cycle Graph: A permutation cycle graph for a given permutation $\pi$ of a finite set $V$, is its cycle graph $\Gamma$ such that ...
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123 views

Prove that a group $G$ has a faithful irreducible representation over a field $F$ if and only if the centre of $G$ is cyclic

An answer to this question would be greatly appreciated. Let $G$ be a finite group and $F$ a field such that the order $\mid G\mid$ is coprime to the characteristic of $F$. Prove that $G$ has a ...
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38 views

Clarification for question: Homomorphisms from $\Bbb Z / n \to \Bbb C_{\ne 0}$

Please don't solve the problem - it is for an assignment - this is just a question for clarification purposes. Let $G$ be a group and $\hat G$ be the set of homomorphisms from $G$ to the group of ...
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41 views

Question about exponents of groups

Okay, I'm trying to understand exponents of groups. I will start with the Set $Z_3$, where $Z_3$ is the integers mod3 under addition. Now, I want to set out to find the exponent of this group, ...
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65 views

question regarding group theory proof

Can someone please explain the sentence in red?, how does it follow?
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31 views

$\{m\in\mathbb Z_{2^n}:m\equiv 1 \mod 4\}$ is cyclic

How can I prove that the multiplicative group $G:=\{m\in\mathbb Z_{2^n}:m\equiv 1 \mod 4\}$ is cyclic? I tried claiming that $G=\langle5\rangle$ but failed.
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0answers
32 views

Getting the least possible order of a group from existing cyclic subgroups or elements.

Let's say I have finite group $G$. Let it have have following subgroups: $<a> = \{e, a\}$, $<b> = \{e, b\}$, $<c> = \{e, c, c^2\}$, $<d> = \{e, d, d^2, d^3\}$ I can for sure ...
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118 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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59 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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37 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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41 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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32 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
70 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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0answers
43 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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148 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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0answers
34 views

Need an very extensive explanation on what this problem is talking about

Group theory and my lecture notes says nothing about this but yet expects me to know it. I'm unfortunately not Galois or anyone around that and have no means to work what this even means on my own ...
0
votes
0answers
22 views

Element of infinite order and all generator of subgroups

Suppose that a has infinite order Find all generators of subgroup $\left \langle a^{3} \right \rangle$ Now, since a has infinite order then so does $\left ( a^{3} \right )^{n} $for if a has ...
0
votes
0answers
17 views

showing the inverse of an element is a generator of the non-inverse

Question: Let $G$ be a group and let $a \in G$. Prove that $\left \langle a^{-1} \right \rangle=\left \langle a \right \rangle$ Suppose $\left \langle a^{-1} \right \rangle$ so $\left ...
0
votes
0answers
32 views

What is the generator of the group of the units in the field $\mathbb{F}_3[x]/(x^3+2x+2)$?

I have shown that $\mathbb{F}_3[x]/(x^3+2x+2)$ is a field. Then the question asks what is the generator of the group of the units? no idea how to proceed from there....