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Questions tagged [cyclic-groups]

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 a cyclic group is abelian.

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1answer
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Find the smallest $n \in \mathbb{N}$ such that the group is isomorphic to the direct product of $n$ cyclic groups

Find the smallest $n \in \mathbb{N}$ such that the group $\mathbb{Z}_{6} \times \mathbb{Z}_{20} \times \mathbb{Z}_{45}$ is isomorphic to the direct product of $n$ cyclic groups. I'm not sure but if I ...
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0answers
20 views

Group of order $105$ with a normal Sylow $3$-subgroup is cyclic [duplicate]

Let $G$ be a group of order $105 = 3 \times 5 \times 7$ with a normal Sylow $3$-subgroup, prove that $G$ is cyclic. My attempt I've been able to show that $G = NP$, where $N$ is the normal Sylow $3$-...
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1answer
22 views

Group as direct sum of cyclic groups

What are necessary conditions for a cyclic group $G$ to be a direct sum of cyclic groups? I saw somewhere that $G$ must be a non $p$-group. But I couldn't prove it. Thank you for your hints/help
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1answer
47 views

Generators for $\mathbb{Z_n}$

I would like to show that $K$ is a generator for $\mathbb{Z}_n$ $\iff$ $\gcd(K,n)=1$ and $1 \leq K <n$. My Attempt: Assume $\gcd(K,n)=1$ and $1 \leq K <n$. That means $K \in \mathbb{Z}_n$ and ...
3
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0answers
36 views

Does there exist an infinite non-abelian group, such that all its nontrivial proper subgroups are isomorphic to $C_\infty$? [duplicate]

Does there exist an infinite non-abelian group, such that all its nontrivial proper subgroups are isomorphic to $C_\infty$? It is rater obvious, that if such group exists then it is generated by any ...
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1answer
65 views

Let $G= \langle a \rangle$ be a cyclic group and $H$ a subgroup of $G$. Show $(G/H,*)$ is a cyclic group with generator $aH$.

Let $G=\langle a \rangle$ be a cyclic group and $H$ a subgroup of $G$. Show $(G/H,*)$ is a cyclic group with generator $aH$. Also find a group $K$ with normal subgroup $L$ such that $(K/L,*)$ is ...
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1answer
29 views

Finding the left cosets for cyclic group $G=\langle a\rangle>$ of order $12$ and the cyclic subgroup $H=\langle a^4\rangle$

I am trying to check my work mostly and if not then could I get some hints, First I am asked to find the left cosets of $H$ in $G$. So I need to find the elements in $H$. To do this I use a theorem ...
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0answers
29 views

Prove that a permutation is a $p$-cycle

I have difficulties proving the following theorem: Let $σ ∈ S_n$ satisfy $σ \neq (1)$, and $σ^p = (1)$, where p is a prime number such that $\frac{n}{2} < p ≤ n$. Prove that σ is a p-cycle. ...
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1answer
22 views

Find the number of generators of a cyclic group with a given order n [duplicate]

Let us say that $n = 5$. The number of generators of a finite cyclic group would be the number of numbers that are relatively prime to $n$ and the identity element. Here, when $n=5$, the number of ...
2
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1answer
44 views

There is an element with order $p$ [duplicate]

Let $p$ is an odd prime and $G$ is a group which has $2p$ elements. Show that there exists at least one element with order $p$. I tried showing in 2 parts as $G$ is cyclic and not cyclic, but I ...
2
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2answers
38 views

$K$-rational points on an elliptic curve

Let $E$ be an elliptic curve defined over a number field $K$. Denote with $E(K)$ the set of $K$-rational points. Is $E(K)$ always a cyclic group? My attempt: I think this is not true and I am ...
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2answers
18 views

Use Langrange's Theorem to see if you can conclude that a proper subgroup H of G is cyclic

Let $(G,•)$ be a group of order n. Let $H$ be a proper subgroup of $G$ where $H$ does not equal $G$. Use Langrange's theorem to see if you can conclude that $H$ must be cyclic. $n=12$ Divisors: $1, 2,...
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2answers
49 views

What is the factor group $C_{12}/C_{6}$?

I know this factor group is isomorphic to $C_2$, but I have tried calculating it and I only get one coset.
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0answers
45 views

Is 2 always a generator of the unit group Z_{3^m} for any nonzero positive integer m?

Is $2$ always a generator of the unit group of $Z_{3^m}$ for any nonzero positive integer m? I know that $|\langle 2\rangle | > m \;Log_2(3)$, and by Lagrange $|\langle 2\rangle|\>\lVert\> ...
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0answers
61 views

Given a pair of primes $g < p$, is there an efficient test if $g$ is a generator of the group $\mathbb{Z}_p^*$?

There are posts around on how to find generators, but I want to start from a supposed guess, a prime candidate $g$ for a generator of a for the multiplicative group of units for the prime-sized field $...
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2answers
78 views

Finding quadratic residues without Legendre symbols

I ran into two very similar problems concerning quadratic residues, and I'm having a bit of trouble working through them. These problems are supposed to rely exclusively on the theory of cyclic groups,...
2
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1answer
42 views

Galois Group of Q(6th roots of unity) over Q

I'm trying to justify that $Gal(\mathbb{Q}(e^{2\pi i/6}):\mathbb{Q})$ is isomorphic to $C_2$, the cyclic group of order $2$. So far, I've easily shown that the minimal polynomial of $r=e^{2\pi i/6}\ $ ...
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0answers
33 views

How to find nodes that are in any cycle in a UNdirected graph?

Let's say the adjacency matrix of a undirected graph $A$ is given by $$A = \begin{bmatrix}0 & 1& 0& 1 \\ 1& 0& 1& 1 \\ 0& 1& 0& 1 \\ 1& 1& 1& 0\end{...
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0answers
90 views

How many elements in the cyclic group $C_{54}$ have order $6$? How many have order $8$?

I'm new to the algebra game and I'm learning about group theory at Uni. I understand a cyclic group is a group that can be generated by a single element, but I'm not sure how I would answer the above ...
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0answers
28 views

Give three non-isomorphic groups of order 24, and explain why they are not isomorphic. [duplicate]

I am not sure how to do this question, any help would be very much appreciated!
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1answer
43 views

How to derive the relation between $k$ and $l$ given $\langle g^k \rangle = \langle g^l \rangle$ in a cyclic group $C_n = \langle g \rangle$?

It is known that For a cyclic group $C_n = \langle g \rangle$ of order $n$, we have $\langle g^k \rangle = \langle g^{(k, n)} \rangle$, where $k \in \mathbb{Z}$. I am able to verify this result. ...
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2answers
41 views

prove that for $G$ a multiplicative nonabelian group of order $pq$, where $p$ and $q$ are prime numbers, any proper subgroup of $G$ is abelian

I need to prove that for $G$ a multiplicative nonabelian group of order $pq$, where $p$ and $q$ are prime numbers, any proper subgroup of $G$ is abelian. I use that, from Lagrange's theorem, the order ...
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3answers
46 views

Are generators of finite cyclic groups unique? [closed]

Are generators of finite cyclic groups unique? Can someone explain to me why they are unique or why they are not?
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1answer
46 views

Let $G$ have exactly $24$ elements of order $6$. Why are there $12$ cyclic subgroups of $G$ of order $6$?

This Exercise 8.47 of Gallian's "Contemporary Abstract Algebra". Answers that use material from the textbook prior to the question are preferred. Let $G$ be a group with exactly $24$ elements of ...
2
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0answers
48 views

What is the order of the element $10 \in (\mathbb Z/p \mathbb Z)^*$?

Let $p \geq 23$ be a prime number such that the decimal expansion (base 10) of $\frac{1}{p}$ is periodic with period $p-1$ . Let $(\mathbb Z/p \mathbb Z)^*$ denote the multiplicative group of integers ...
0
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1answer
72 views

In a cyclic group with subgroup $H$ and $N$, list the elements in $H + N$ and $H \cap N$ [closed]

In the Group $G= \mathbb{Z}/24\mathbb{Z}$, let $H = \langle4 + 24\mathbb{Z}\rangle$ and $N = \langle 6 + 24\mathbb{Z}\rangle$. List the elements in $H + N$ and $H \cap N$ I am aware that $G$ is the ...
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0answers
50 views

Group theory problem - the order of elements

Given a group $G$, $a,b\in G$, $ab=ba$, $o(a)=n$, $o(b)=m$. If $\gcd(n,m)\ne 1$, and $(a)\cap (b) = \{e\}$, prove that $o(ab)=\operatorname{lcm}(n,m)$. P.S. $(a)$ denotes the cyclic group ...
2
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1answer
35 views

$G/H$ cyclic and $H$ subgroup of $Z(G)$, then $G$ cyclic.

Gallian, in his Contemporary Abstract Algebra, first proves this theorem: Theorem: Let $G$ be a group and let $Z(G)$ be the center of $G$. If $G/Z(G)$ is cyclic, then $G$ is Abelian. Proof: ...
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0answers
22 views

$Gal(f)$ is cyclic of order $m$ for irreducible $f \in K[X]$ of degree $m$, where $K$ is a finite field

Let $K$ be a finite field, $f \in K[X]$ irreducible with degree $m$. Show that $Gal(f)$ is cyclic of order $m$. I have shown that $f$ is separable over $K$ by using that $K$ is finite and thus ...
2
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1answer
52 views

Find the subgroup of $\mathbb{Z}_{12}$ generated by the subset $\{4,6\}$. Also draw the digraph of this subgroup $\langle\{4,6\} \rangle$.

I have the following problem: Find the subgroup of $\mathbb{Z}_{12}$ generated by the subset $\{4,6\}$. Also draw the digraph of this subgroup $\langle\{4,6\} \rangle$. I've done the first part ...
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0answers
11 views

Abstract Algebra cyclic groups digraph question

Find the subgroup of Z_12 generated by the subset { 4 , 6 } . Also draw the digraph of this subgroup ⟨ { 4 , 6 } ⟩ . I have the subgroup of z_12 generated by {4.6} as < gcd {4,6} > = <2>= {0,2,...
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1answer
46 views

Degree of splitting field of $X^n-1$ over some finite field

Let $k$ be a finite field of order $q$ in characteristic $p$, let $n$ be a positive integer not divisible by $p$, and let $K$ be the splitting field of $X^n-1$ over $k$. Prove that $[K:k]$ equals the ...
4
votes
1answer
69 views

Elements and cyclic subgroups of order $15$ in $\Bbb Z_{30}\times \Bbb Z_{20}.$

This is Exercise 8.22 of Gallian's, "Contemporary Abstract Algebra". Please use only methods from this book prior to the exercise. This is an alternative-proof question. Find the number of ...
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0answers
27 views

Describing the behaviour of subgroups of the direct product of finite cyclic groups based on the gcd of their orders

I am trying to understand the following proof: I understand most of it just fine, except for the last line. I don't really understand why the projection will show us that $\langle xy \rangle \neq \...
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0answers
19 views

Distribution of elements of a particular order in $(\mathbb{Z}/m\mathbb{Z})^*$

Consider the group $G = (\mathbb{Z}/m\mathbb{Z})^*$, where $m$ is such that $G$ is cyclic. Let $g\in G$ be some fixed generator, and let $a_1,\dots,a_{\varphi(m)}$ be an enumeration of the elements of ...
2
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2answers
34 views

Prove that $S_3/C_3$ is a (quotient) group

Consider $S_3$ to be the symmetries of a triangle, and let $C_3$ be a subgroup that cycles the three corners, so generated by: $(1 \ 2 \ 3 )$. Using Lagrange's theorem, compute $k = |S_3/C_3|$. ...
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1answer
24 views

Injective and surjective homomorphisms between non cyclic group of order $4 $ to $Z_8$

Let $G$ be a non cyclic group of order $4$. Consider the following statements: $I:$ There is no one-one map homomorphism from $G$ to $Z_8$ $II:$ There is no onto homomorphism from $Z_8$ to ...
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3answers
910 views

Prove that a cyclic group with only one generator can have at most 2 elements

Prove that a cyclic group that has only one generator has at most $2$ elements. I want to know if my proof would be valid: Suppose $G$ is a cyclic group and $g$ is its only generator. Let $|G|=n$ ...
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3answers
191 views

Visualized group tables for $\mathbb{Z}$ and $\mathbb{Z}/n\mathbb{Z}$

Let me share a kind of visualization of group tables which is especially well suited for cyclic groups like $\mathbb{Z}$ and $\mathbb{Z}/n\mathbb{Z}$. In these groups you can easily give colors to ...
2
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1answer
95 views

Is the Quotient Group Cyclic?

I'm just wondering how to show that a quotient group $H = (G/N)$ is cyclic? Let $G= \mathbb{Z}/4\mathbb{Z} \times \mathbb{Z}/6\mathbb{Z}$ Let $N = \left<(2,3)\right>$ , where N is a cyclic ...
2
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0answers
62 views

Order of a cyclic subgroup in G

Let G be a group where G = Z/4Z x Z/6Z. I calculated the order of G to be 24. Given the cyclic subgroup <(2,3)> in G. Can I say the order of this subgroup is also 24 given that a cyclic subgroup ...
1
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1answer
25 views

Set of Rotations Cyclic?

For the dihedral group $D_{n}$ of order $2n$, is the group $R$ formed by its $n$ rotations cyclic in general? Or is the factor group $D_{n}/R$ cyclic? I am trying to show the series $D_{n}>R>(1)$...
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1answer
33 views

Homomorphism between cyclic groups of orders 7 and 25

I am not sure how to start solving this question for my modern algebra course. I understand Lagrange's theorem, but I am not sure if I should use it here. I just need help on where to start. Let G ...
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2answers
66 views
3
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1answer
37 views

Determine if exists a subgroup of order $3$ of $H=\langle\sigma^{8440}\rangle$

Consider the following permutation of $S_{13}$. $\sigma=(1\;3\;13\;5\;11\;8)(2\;10\;4\;6\;12\;7\;9)$ Determine if exists a subgroup of order 3 of $H=\langle\sigma^{8440}\rangle$. If yes, exhibit it ...
0
votes
1answer
45 views

Group Representations of $C_5$

If I have the cyclic group $C_5=\langle g\mid g^5=e\rangle$ and the left regular representation $V=\mathbb{C}C_5$. Would the matrices of this representation (in the standard basis) be defined by $\...
1
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1answer
53 views

Derived subgroup of $\langle{x,y\,|\, x^p=y^{p^{n-1}}=1,\,{{x^{-1}}{yx}}={y^{1+p^{n-2}}}}\rangle$. [closed]

I would like to prove that if $M_n(p)=\langle{x,y\,|\, x^p=y^{p^{n-1}}=1,\,{{x^{-1}}{yx}}={y^{1+p^{n-2}}}}\rangle$, then $M'_n(p)$ is a cyclic group of order $p$. I was wondering if someone could ...
1
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0answers
64 views

Abelian Group In The Middle Of A Short Exact Sequence

Let $p$ be a prime number. Determine all isomorphism classes of abelian groups $A$ that can appear as the middle term of a short exact sequence: $$0 \rightarrow \mathbb{Z}/(p^a) \rightarrow A \...
1
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1answer
53 views

Understanding the group product of $\mathbb{Z_n}$ and $\mathbb{Z_m}$, $G=\mathbb{Z_m} \mathbb{Z_n}$

$G=\mathbb{Z_m} \mathbb{Z_n} = \{[a][b] : [a] \in \mathbb{Z_m}, [b] \in \mathbb{Z_n}\}$ Specifically when $gcd(m,n)=1$. Can somebody show me what $G$ will equal as a set, and if you could go to far ...
0
votes
1answer
30 views

Remembering how to find the elements of a certain order in a group

I am working on a math problem and am stuck on remembering the correct way to solve it. I basically need to figure out how much elements of order 8 are in Z (of 23432). (Where Z is a cyclic group ...