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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Finding the generator of cyclic group $U(Z_{27})$

$U(\mathbb{Z}_{27})$ is a group of order $18$. $U(\mathbb{Z}_{27})=\{1,2,4,5,7,8,10,11,13,14,16,17,19,20,22,23,25,26\}$ How do I find the generators to prove that this group is cyclic? The final ...
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2answers
76 views

Prove that the subgroup of the quotient group is cycling and infinitely generated

$$M = \left\{\,\dfrac{m}{13^n}\biggm| m\in \mathbb{Z}, n\in\mathbb{N} \,\right\}, \quad G = M/\mathbb{Z}$$ Prove that any subgroup $H < G$, $H\neq G$ is cyclic and infinitely generated and that ...
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1answer
171 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 ...
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1answer
34 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 ...
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1answer
50 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
85 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
83 views

What is a generator of a finite cyclic group? (General)

I have asked a few questions about this but I am still confused. So, in general, what is a generator of a finite cyclic group and how is it found? I have seen in books and my notes a lot of ...
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1answer
94 views

In S4, find all the even permutation and show that the set of odd permutations isn't stable for binary operations in S4.

I want to find the even permutations of $S_4$ so i am supposed to find the transpositions right? but of what permutation exactly do i find the transpositions? And how do i know which ones are even? ...
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1answer
36 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
108 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
27 views

Dihedral group of order 2n

I would appreciate if someone could prove this for me: Let G be a dihedral group of order 2n and suppose H is a cyclic quotient group of G. Show that |H|is less than or equal 2.
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1answer
26 views

Symmetries of a $9$ puzzle (Rubik's Slide)

Consider this Rubik's slide. With these moves (and their inverses): $$\text{Vertical shift}\: v=(147)(258)(369)$$$$\text{Rotation}\: c=(12369874)$$$$\text{Horizontal shift}\: h=(123)(456)(789)$$ Also ...
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1answer
45 views

If G is abelian and simple ,then G is cyclic

True /False .IF G is abelian and simple ,then G is cyclic Solution True If G is an abelian simple group then G is isomorphic to Zp for some prime p
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1answer
72 views

Abstract Algebra: Cyclic Groups (Lattice Diagram)

Example 4.2: Lets find all the subgroups of the given group and draw the lattice diagram for the subgroup. Z12 Z36 Z8 In the book finding the subgroups is explained well but it does not explain ...
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1answer
62 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 ...
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1answer
50 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
24 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
42 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
117 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
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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
42 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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0answers
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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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0answers
139 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 ...
2
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0answers
36 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
24 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
63 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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0answers
34 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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0answers
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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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0answers
52 views

question regarding group theory proof

Can someone please explain the sentence in red?, how does it follow?
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29 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
23 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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0answers
21 views

groups and symmetry

Which of the following may be false? A) Any non-trivial element of $C_n$ generates $C_n$. B) Any subgroup of $C_n$ is cyclic. C) if $m|n$ then $C_n$ has at least one subgroup of order $m$. D) if ...
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0answers
55 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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49 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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0answers
27 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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0answers
34 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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0answers
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
61 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
39 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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0answers
137 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
57 views

A $p^n$th root of unity is in $N_{F/k}(F^\times)$ if and only if there is an extension of $F$ which is a cyclic extension of $k$ of degree $p^{n+r}$

Hello, I have to solve this homework but I have completely no idea. please help me, it is very difficult for me. any attempt will be welcomed and appreciated. Conditions : $n, r$ are fixed number ...
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0answers
25 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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0answers
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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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0answers
40 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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0answers
224 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
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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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0answers
372 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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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 ...