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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A kind of permutations and possible relation to cyclic groups.

Any permutation that moves $n$ elements in some fashion never revisiting the same until all others have been visited, in other words so that: $${\bf P}^n = {\bf I}, \text{ but no } 0<m<n \text{ ...
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$G$ a finite group with two non-trivial normal subgroups. $|G| = pq$. Why G cycle?

How can I prove that if $G$ is a finite group , and the order of $G$ is $pq$ while $p$ and $q$ are primes, and in addition , $G$ with two normalic subgroups , so --> $G$ is cycle? Ideas? Hwo can i ...
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39 views

A group $G$ is locally cyclic if and only if $G$ is a union of a chain of cyclic subgroups ?

Is it true that a group $G$ is locally cyclic if and only if $G$ is a union of a chain of cyclic subgroups ?
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23 views

Common generator of units in finite prime fields

It is well known that the unit group of a finite field is cyclic. What can we say about the generators? Specifically I am interested in the following question: Suppose we fix a positive integer $a$, ...
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27 views

Write $\mathbb{Z}^3/L$ as a direct sum of cyclic groups

Let $L\subset \mathbb{Z}$ be the subgroup of $\mathbb{Z}^3$ generated by the elements $(-1,-1,4),(2,4,0),(3,3,8)$. Write $\mathbb{Z}^3/L$ as a direct sum of cyclic groups. I've tried creating a ...
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2answers
36 views

Group $G$ cyclic as it coincides with the multiplicative group of a finite field

I have a group $G=(\mathbb{Z}_{251}^{*}, \cdot)$ with generator $g=71$ (so, having a generator, I'm given with the fact that is cyclic, right?) further in the example of my study notes I read: "$n = |...
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1answer
42 views

Group of order $pq$ with $p<q$ prime has $q-1$ elements of order $q$

If $p<q$ are primes and $G$ is a group of order $pq$, then $G$ has exactly $q-1$ elements of order $q$. My attempt was: Use Sylow theorem to show that $G$ has only one subgroup of order $q$, so ...
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factorising polynomials related proofs

$P(x)= a{x}^3+b{x}^2+c{x}+d$ where $a,d$ are not equal to zero. (All the coefficients are integer) Now $P(x)$ is divided by $x-r$. Here why r needs to be an integer to be a factor of d(constant term)...
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Factorising Cyclic expression .

What are ways for factorising cyclic expressions? Note: I am not saying about specific one. Just ways of factorising cyclic expressions.
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Multivaritate version of Fermat's little theorem

If $f$ is irreducible over $\mathbb{F}_p[x]$ of degree $d$, then $$g(x)^{p^d} \equiv g(x) \bmod f(x)$$ and $p^d - 1$ is the order of the cyclic group $\left( \mathbb{F}_p[x]/f(x) \right)^{\times}$. ...
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39 views

Help showing this is an element of $(\mathbb{Z}/p^n\mathbb{Z})^{\times}$

Let $p$ be an odd prime number and $n$ be a positive integer. Use the binomial theorem to show that $(1+p)^{p^{n-1}} \equiv 1 \mod p^n$ but $(1+p)^{p^{n-2}} \ne 1 \mod p^n$ Deduce that $(1+p)$ is an ...
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1answer
42 views

$G$ be an infinite abelian group such that every proper non-trivial subgroup of $G$ is infinite and cyclic ; then is $G$ cyclic?

Let $G$ be an infinite abelian group such that every proper non-trivial subgroup of $G$ is infinite cyclic ; then is $G$ cyclic ? ( The only characterization I know for infinite abelian groups to be ...
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1answer
38 views

generators of $\mathbb{Z}_p^*$ are all the elements in $\mathbb{Z}_p^*$?

I know that a finite group with a prime number of elements is cyclic and every element in the group is a generator for the group. Thinking about $(\mathbb{Z}_p^*, \cdot)$ I thought that the order of ...
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2answers
33 views

Why if $(\mathbf B, \cdot)$ is a finite order group with prime order then $(\mathbf B, \cdot)$ is cyclic? [duplicate]

In the notes I'm studying from ( again =) ) I read: If $(\mathbf B, \cdot)$ is a finite order group with prime order then $(\mathbf B, \cdot)$ is cyclic Could someone give me a justification for ...
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2answers
51 views

How can $ \langle -1 \rangle $ be the same as C2?

How can $ \langle -1 \rangle $ be the same as $ \text{C2} =\{-1, 1\} $? Why can we just write $ \langle -1 \rangle $? Why do we say that $ \{-1, 1 \} $ is generated by $ \langle -1 \rangle $? With $ \...
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1answer
38 views

Order of product of non-disjoint cycles

Let $a$ and $b$ be two non-disjoint cycles of order $m$ and $n$. Is there any general formula for the order of $a b$? I understand that we can convert any non-disjoint cycles into disjoint cycles and ...
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44 views

cyclic groups homomorphism

I have the following task: "Determine the homomorphism between two cyclic groups. Which are injective, surjective or bijective?" I already found this for the cyclic group of integers: http://users....
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2answers
26 views

Existence of subfields such that $[\mathbb{Q}(\zeta_{25}) : K_1]=4$ and $[\mathbb{Q}(\zeta_{25}) : K_2]=5$

I have the following two questions questions I am working on and am a little stuck : Let $L=\mathbb{Q}(\zeta_{25})$ where $\zeta_{25}$ is the primitive $n$-th root of unity. Prove that there are ...
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3answers
62 views

Describe the structure $\operatorname{Gal}(\mathbb{Q}(\zeta_4)/\mathbb{Q})$

I know that if $n$ is prime then $G=\operatorname{Gal}(\mathbb{Q}(\zeta_n)/\mathbb{Q}) \simeq \Bbb Z_{n-1}$ But I am unsure what $G$ is when $n$ is not prime. For example when $n=4$: $\operatorname{...
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1answer
30 views

Number of fields between $\mathbb{Q}$ and $\mathbb{Q}(\zeta_n)$

If $\zeta_n$ is the $n$-th primitive root of unity then $Gal(\mathbb{Q}(\zeta_n)/\mathbb{Q}) \simeq Z_n^*$ due to the following map $$\tau(\zeta_n)=\zeta^n$$ I was wondering if we could use this ...
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4answers
70 views

Number of homomorphisms between two cyclic groups.

Is it true that the number of homomorphisms between any two finite cyclic groups of order $m\,\&\,n$ is $\gcd(m,n)$? I have posted an answer which I believe is true, just wanted to know different ...
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2answers
42 views

Finding the order of an element

When we have permutation elements like $b=(12)(234)(1223)$ we can easily say that the order of each cycle is $2$, $3$ and $4$ respectively so the order of $b=\text{lcm}(2,3,4)$. When we have $C_n$, ...
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3answers
46 views

How to find the Permutation in S8 given as the product C= (1483)(12765)(34687)?

I don't understand how to answer this. Just by reading it off, "1 goes to 4, and 4 goes to 6, thus 1 goes to 6" but that logic doesn't match with the answer i've been giving: Answer: (127)(386)(45). ...
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1answer
119 views

Describe, as a direct sum of cyclic groups, given a map $\phi: \mathbb{Z}^{3} \longrightarrow \mathbb{Z}^{3}$

I'm trying to resolve the next one: Describe, as a direct sum of cyclic groups, the cokernel of the map $\phi: \mathbb{Z}^{3} \longrightarrow \mathbb{Z}^{3}$ given by left multiplication by the ...
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1answer
38 views

If $ p\neq q$ are odd prime integers then $(\mathbb{Z}/ pq\mathbb{ Z})^*$ is not cyclic

This is a question from Aluffi's Algebra Chapter 0, which I am self studying. Specifically this is from Chapter 2, Page 69, Question 4.10 Let $p\neq q$ be odd prime integers; show that $(\mathbb{Z}/ ...
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3answers
120 views

Is $\mathbb{Z}_4 \times \mathbb{Z}_ {12} \times \mathbb{Z}_9$ cyclic?

I find on internet this: $\mathbb{Z}_m \times \mathbb{Z}_n$ is cyclic if and only if $\gcd(m,n)=1$. Then I do the next steps: $\gcd(4,12,9)$ is 1. Then I assume that $\mathbb{Z}_4 \times \...
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27 views

Finite group $G$ satisfies that for all positive integer $m$ number of solutions of $x^m=e$ are at most $m$ is cyclic. [duplicate]

Finite group $G$ satisfies that for all positive integer $m$ number of solutions of $x^m=e$ are at most $m$ is cyclic. It is exercise in A First Course in Abstract Algebra by Fraleigh. The book ...
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1answer
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New to cyclic groups: A group $H$ is cyclic if every element of $H$ is an integral power of some single element of

$H=${${h^k}$|$k\in Z$} for some $h$ in $H$. (a) If $G$ is any group, and $g$ is a particular element of $G$, show that the set {${g^k}|k \in Z$} is a subgroup of $G$: The set of all integral powers ...
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2answers
51 views

Quotient of $G= \left\langle a, b \ \middle|\ a^4, b^2=a^4, aba=b \right\rangle$ by $\langle a^2 \rangle$

Let $G$ be finite group of order $8$ of the form: $G= \left\langle a, b \ \middle|\ a^4, b^2=a^4, aba=b \right\rangle$. The elements are $\left\lbrace 1, a, a^2, a^3, b, ab, a^2b, a^3b\right\rbrace$....
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29 views

How to show $2$ and $3$ are generators of the additive group $Z/(5)$.

This may sound like an easy question, but I never learned cyclic groups due to my time constraint of my class and that the professor ran out of time to teach it. However, he left me with a problem to ...
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36 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 ...
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44 views

The Group of Units in a Ring

Question Let $R$ be a subring of $\mathbb{C}$ and the group of units $\mathcal{U}(R)$ is finite. Show $\mathcal{U}(R)$ is cyclic. My Idea is to let an element of $R$ be $z\in\mathbb{C}$ so if $z\in\...
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3answers
63 views

Group of order $p^2$ [duplicate]

Let $G$ be a group of order $p^2$ where $p$ is a prime. I want to show that either $G$ is cyclic or is the product of two cyclic groups of order $p$. After some work, the problem reduces to showing ...
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1answer
32 views

Prove that if $H \cong \mathbb Z$ or $H \cong \mathbb Z_n$ then $H = ⟨g⟩$ for some $g ∈ G$.

I'm studying for my final and I came across this homework problem that I had previously done but I don't remember how to do it anymore. It is as follows ($G, H$ are groups): Suppose that $H \le G$. ...
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1answer
38 views

Generators of two groups with prime order $p$ already induce all the generators of the product group $G \times H$

Let $G = \langle g \rangle, H = \langle h \rangle$ be two cyclic groups (with $g \in G, h \in H$), both of them of order $p \in \mathbb{N}$, where $p$ is a prime number. I now want to show that $$\...
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2answers
33 views

can composite groups have primitive roots (be cyclic)?

Imagine Z/nZ with n not prime (n=pq). Can the multiplicative group be cyclic? I read the paper over here: http://math.uga.edu/~pete/4400primitiveroots.pdf At one point he dismiss the case where the ...
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25 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 ...
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1answer
30 views

Prove that for any n in $ \Bbb N $ there exists a unique cyclic subgroup $H_n$ $\subset$ G of order n. Where G is the group $\Bbb Q/$ $\Bbb Z$.

Prove that for any n in $ \Bbb N $ there exists a unique cyclic subgroup $H_n$ $\subset$ G of order n. Where G is the group $\Bbb Q/$$\Bbb Z$. Existence was not difficult to show: Let $H_n$ = < 1/...
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62 views

Can Lagrange's Theorem for algebraic structure apply here?

For a positive integer $n$ let $Φ(n)$ denote the number of elements $r∈\mathbb Z_n$ such that $\gcd(r,n)=1$. Show $Φ(mn)=Φ(m)Φ(n)$ for all $m, n∈\mathbb N$ such that $\gcd(m,n)=1$. The only thing I ...
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2answers
27 views

Question to the proof of: Let $A$ be a finite abelian group and let $g \in A$. Suppose that $\chi(g)=1$ for every $\chi \in \hat A$. Then $g=1$.

Good day, Currently I am working with the book "A First Course in Harmonic Analysis" by A. Deitmar and I am stuck in the beginning of Chapter 5 on the proof to Lemma 5.1.5. I am repeating the few ...
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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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1answer
23 views

Proving existence of automorphism

Let $G$ be a cyclic group, and $a,b$ two generators of $G$. Prove that exists automorphism $f$ such that $f(a)=b$ and that if $f$ is an automorphism then $f(a)$ generates $b$. $G$ is cyclic, thus ...
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61 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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1answer
40 views

Show that $|a^{k}|=|a^{n-k}|$

Let G be a group and let $a$ be an element of G of order $n$. For each Integer $k$ between $1$ and $n$, show that $\left | a^{k} \right |=\left | a^{n-k} \right |$ My attempt is as follows: $\left |...
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26 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 finite ...
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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 \langle ...
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1answer
26 views

Proof for generator of the group of integer under addition modulo

Theorem: An integer $k$ in $\mathbb{Z}_{n}$ is a generator of $\mathbb{Z}_{n}$ If and Only if $gcd\left ( n,k \right )=1$ My problem lies with proving the "If" condition and here is my attempt: ...
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0answers
25 views

Proof for generators of cyclic group [duplicate]

Theorem: Let $G=\left \langle a \right \rangle $ be a cyclic group of order n. Then $G = \left \langle a^{k} \right \rangle$ if and only if $gcd\left ( n,k \right )=1$ I've proven the "only if" ...
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24 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 \\ ...
4
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1answer
43 views

Subgroups of the Semi-Direct Product $\mathbb{Z}/\mathbb{7Z} \rtimes (\mathbb{Z}/\mathbb{7Z})^{\times}$

I want to list all the subgroups of the semi-direct product $\mathbb{Z}/\mathbb{7Z} \rtimes (\mathbb{Z}/\mathbb{7Z})^{x}$, under the homomorphism $\theta: (\mathbb{Z}/\mathbb{7Z})^{\times} \rightarrow ...