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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67 views

$r$-cycle to a power $k$ is also an $r$-cycle if and only if $\gcd(k, r) = 1$

Let $\sigma$ be an $r$-cycle in $S_n$ and let $k\in\Bbb Z$. Show that $\sigma^k$ is also an $r$-cycle if and only if $\gcd(k,r)=1$.
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1answer
56 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
146 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
51 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 ...
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1answer
87 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
24 views

Find the Existence and Uniqueness of a Cyclic subgroup

Giving that $|H|$ is cyclic. If $|H|=n$ then for each $a>0$ such that $a|n$ there exist a unique subgroup of $H$ of order $a$. This subgroup is Cyclic subgroup $\langle x^d\rangle$ where $d= ...
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1answer
38 views

How many cycles $A$ and $B$ can form this cycle

How many cycles $A$ and $B$ can form this cycle: $AB=(axyguimjrcwk)(bvqphsleofzt)(d)(n)$ I can see that $A$ and $B$ must share the cycle $(dn)$, and I believe due to ordering, both $A$ and $B$ must ...
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1answer
232 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
35 views

Trying to prove that a group is Cyclic

Suppose that the order of $G$ is divisible by at least two distinct primes. Also, let $g\in G$ that order of $g$ is divisible by every prime divisor of $o(G)$ and $\forall x\in G$, $o(x)\mid o(g)$ ...
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1answer
41 views

Order of a cyclic group?

When finding the order of a cyclic group, do we determine so by counting the number of elements in that group generator by the cyclic group?
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45 views

Finding subgroups of order $4$ of $\mathbb{Z}_2\times\mathbb{Z}_4$

Now, the question asks me what the subgroups of order $4$ are of this relation and then to give them as sets and identify the group of order $4$ that each of the subgroups is isomorphic to. How do I ...
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1answer
38 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
27 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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$(\mathbb{Z}/p^r\mathbb{Z})^{\ast}$ 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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44 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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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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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 ...
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66 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 ...
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44 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 ...
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34 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: ...
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83 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 ...
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40 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 ...
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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 = $Z_{60}$. ...
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74 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
37 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
119 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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36 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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32 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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60 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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24 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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68 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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52 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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30 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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38 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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29 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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64 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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42 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
145 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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29 views

Let G be an abelian group. Let V be an irreducible faithful CG-module. Prove that dimV = 1 and G is cyclic.

I was wondering if I could get some help with the following problem. I know how to prove it with Schur's Lemma but I'm having problems without it. Let G be an abelian group. Let V be an irreducible ...
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33 views

Writing a group as a product of cyclic groups

How do I go about expressing a group as a product of cyclic groups? For example, express: $$O^*_K = \{\pm (24 + 5\sqrt{23})^r : r\in \mathbb{Z} \}$$ as a product of cyclic groups ($O^*_K$ is the ...
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22 views

Multiple maximal cyclic subgroups of a symmetry group

If a symmetry group T has a maximal cyclic subgroup Cn because of a projection I1, then it means it will have a rotational symmetry order of n. If we have another projection (of the same object) with ...
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50 views

How to tell if a directed graph has a cycle?

If I have the directed graph here: I am confused whether or not this is a cycle or not. Because in the underlying graph, this is a 3-cycle for sure, but in the directed graph, there is no cycle if ...
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70 views

How to find all ring homomorphisms from $\mathbb Z_{12} \to \mathbb Z_{30}$ ?

How to find all ring homomorphisms from $\mathbb Z_{12} \to \mathbb Z_{30}$ ? I know that it is enough to determine $f([1]_{12})$ ; moreover $f([1]_{12}$ should be an idempotent element of $\mathbb ...
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63 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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134 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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61 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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262 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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47 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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539 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 ...