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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2answers
47 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
33 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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2answers
42 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
25 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 ...
2
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3answers
60 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
28 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
65 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
34 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
39 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
113 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
37 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}/ ...
3
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3answers
117 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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0answers
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
19 views

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 ...
2
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2answers
49 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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2answers
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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0answers
35 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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2answers
42 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
60 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
37 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
32 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 ...
1
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0answers
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 ...
2
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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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2answers
60 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
26 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
22 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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0answers
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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25 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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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 \langle ...
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1answer
23 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" ...
1
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0answers
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 \\ ...
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 ...
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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 ...
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1answer
10 views

Suppose that $H$ is a (cyclic) subgroup of order $m$ of a cyclic (abelian) group $G$ of order $n$. What is $G/H$?

Suppose that $H$ is a (cyclic) subgroup of order $m$ of a cyclic (abelian) group $G$ of order $n$. What is $G/H$? This is taken from an exercise at the end of a section that I must have read 6 or ...
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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}, \frac{\mathbb{Z}/12\...
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0answers
21 views

G is cyclic if it is a finite p-group and has only one maximal subgroup [duplicate]

How can I show that for a finite $p$-group $G$,$G$ is cyclic if it has just one non trivial maximal subgroup?
1
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1answer
25 views

Proof groups equality: $<x^{t_1},x^{t_2}> = <x^{gcd(t_1,t_2,n)}>$

Let $C_n$ be a cyclic group of order n. I need to proof $<x^{t_1},x^{t_2}> = <x^{gcd(t_1,t_2,n)}>$ where gcd is the greatest common divisor of $t_1,t_2,n$ and $<>$ denotes the ...
1
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1answer
50 views

$\sum_{d|n} \varphi(d)=n$

I want to solve $\sum_{d|n} \varphi(d)=n$ using Group theory. Here, $\varphi(d)$ is Euler's totient function. I think I should use $\Bbb Z_n$ and fundamental theorem of cyclic group. Then I use $\...
2
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1answer
481 views

Structure of $Gal(\mathbb{Q}(\zeta_{15})/\mathbb{Q})$?

$\zeta_{15}$ is $15$th primitive $n$th root of unity. Question: Find the structure of the group $Gal(\mathbb{Q}(\zeta_{15})/\mathbb{Q})$ I know that if $p$ is prime then $G=Gal(\mathbb{Q}(\zeta_{p})...
0
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1answer
29 views

Cyclic extension of degree $2$ of $\mathbb{F_2}$

I want to find a cyclic extension of degree $2$ of $\mathbb{F_2}$ The degree of its minimal polynomial is $2$, i.e. a quadratic polynomial. The only cyclic field of degree $2$ which I can think of ...
1
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1answer
32 views

List all subgroups of $\mathbb Z_6$ and $\mathbb Z_8$

List all the subgroups of $\mathbb Z_6$ and $\mathbb Z_8$. I think this implies that the operation is addition because that makes the sets above groups. I was thinking that for $\mathbb Z_6$, the ...
0
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1answer
50 views

Sum of two cyclic codes is a cyclic code.

My text made a small comment about cyclic codes: The sum of two cyclic codes $C_1$ and $C_2$, defined by: $C_1+C_2=\{c_1+c_2: c_1\in C_1, c_2\in C_2 \}$ The sum of two cyclic codes is also a cyclic ...
2
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1answer
59 views

How to prove for a finite group that $| \langle g \rangle|=o(g)$? [duplicate]

How to prove for a finite group that $|\langle g \rangle|=o(g)$? I really don't know how to show that this is true I tried to say let $o(g)=m$ and then show that $|\langle g \rangle|$ has exactly $m$ ...
0
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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....
2
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1answer
36 views

Is there a strategy for expressing finitely generated abelian group as the direct sum of cyclic groups?

I know that every finitely generated abelian group can be expressed as a direct sum of cyclic groups. I wondering how easily we can find the cyclic groups given an abelian group. Specifically, one of ...
0
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2answers
36 views

Proving a subgroup is cyclic

Let $G=Z_n$ for $n\gt1$ and let $a,b \in G$ where $a,b$ are two integers (with at least one nonzero). Prove that the subgroup of G generated by $a$ and $b$ is indeed cyclic and is generated by $c \in ...