Questions tagged [cyclic-groups]

Use with the (group-theory) tag. A group is cyclic if it can be generated by a single element. That is to say, every element in a cyclic group can be written as some specified element to a power.

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Groups of order $16$ with a cyclic quotient of order $4$

Question: I am interested in (a) listing the groups $G$ of order $16$ which have a cyclic quotient of order $4$; (b) in each case knowing in how many essentially different ways this occurs (...
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57 views

Asking for clarification in a group theory proof

The question is: If $G = \left<a\right>$ have order $rs$, where $\gcd(r,s)=1$, show that there are unique $b,c \in G$ with $b$ of order $r$, $c$ of order $s$, and $a=bc$. As shown here, the ...
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1answer
69 views

Question about "Quotient Group of Cyclic Group is Cyclic"

I found a proof of the fact that if $G$ is a cyclic group and $H$ is a subgroup of $G$, then $G/H$ is a cyclic subgroup. They don't mention that $H$ is a normal subgroup. But to define the quotient ...
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1answer
92 views

Does this contradict my definition of "cyclic"?

I would like to explain my question with an example. Let me first give you a definition of cyclic group. A group $G$ is called cyclic if $\exists x\in G$ s.t $G= \langle x \rangle$. In other words, ...
2
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2answers
69 views

Let $G$ be a abelian group such that $|G| = 2p$ and $p$ Is a odd prime number. Prove $G$ is a cyclic group. [duplicate]

I need to prove the following: Let $G$ be a abelian group such that $|G| = 2p$ and $p$ Is a odd prime number. Prove $G$ is a cyclic group. So far I was able to show that there must be atleast one ...
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1answer
57 views

Confusion about the last step of this proof of " Every subgroup of a cyclic group is cyclic":does not subcase $2.2$ contradict the desired conclusion

Source : Reversat & Bigonnet, Algèbre pour la licence ( Undergraduate abstract algebra), Dunod, 1997, p. 20. Let $G$ be a cyclic group of order $n$ and let $x$ be a generating element of $G$, ...
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37 views

How can this subgroup $H$ of a cyclic group $G=\langle x \rangle$ contain the identity element if $H= \{1x, 2x, 3x, .... \} $?

Source : Reversat & Bigonnet, Algèbre pour la licence ( Undergraduate abstract algebra), Dunod, 1997, p. 20. Let $G$ be a cyclic group of order $n$ and let $x$ be a generating element of $G$, ...
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3answers
233 views

Is the cyclic group $\langle x\rangle$ always a subgroup of $G$ for any $x\in G$?

I have been thinking about the following: If $G$ is a finite group and $x\in G$ an element of order $n$ is then $\langle x\rangle$ always a subgroup of $G$? I have the definition that $\langle x\...
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1answer
177 views

How many cyclic subgroups are in $\Bbb Z_8\oplus\Bbb Z_4\oplus\Bbb Z_2$?

Consider the group $\Bbb Z_8\oplus\Bbb Z_4\oplus\Bbb Z_2$. How many nontrivial cyclic groups does it contain? I know the answer is $27$, so my question is how to get it effectively. I was trying to ...
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1answer
48 views

Proof that all subgroups of $(\Bbb Z_{15},+)$ are cyclic and list all of its distinct groups [closed]

Two questions: Prove that all subgroups of $\Bbb Z_{15}$ are cyclic List all distinct groups of $\Bbb Z_{15}$. For part 1) I've done this much: $$\gcd(r,15) = 1$$ The generators are $1,2,4,7,8,11,13,...
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Is this statement about an element in $\mathbb{Z}/n\mathbb{Z}$ correct?

I have the following question. We have $l\geq 1$ an integer and we consider the cyclic group $(\mathbb{Z}/l\mathbb{Z},\times)$. Given $n\in \mathbb{Z}$ we write $[n]=n+l\mathbb{Z}$ for the class of n....
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1answer
36 views

cyclic representation of $L^∞(X, Ω, µ)$

Let $(X, Ω, µ)$ be a measure space. For $ϕ ∈ L^∞(X, Ω, µ)$, set $π(ϕ)f := ϕf$ for $f ∈ L^2(X, Ω, µ)$. Let $π(ϕ) ∈ B(L^2(X, Ω, µ))$, and that the map $ϕ → π(ϕ)$ yields a representation $π: L^∞(X, Ω, ...
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1answer
58 views

Frattini subgroup of a cyclic group which is not a p group for some prime p

What can you say about the frattini subgroup of a finite cyclic group which is not a p-group? I am just wandering do I need to check for each individual case or is there any general result for the ...
4
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1answer
60 views

What are the values of $a$ and $b$ so that $\Bbb Z_2\times\Bbb Z_3\times\Bbb Z_4\times\Bbb Z_9$ is isomorphic with $\Bbb Z_a\times\Bbb Z_b$?

I'm studying group theory, (at basic level), and i got this problem: Find all pairs $(a,b)$ of positive integers so that: $$\Bbb Z_2\times\Bbb Z_3\times\Bbb Z_4\times\Bbb Z_9$$ is isomorphic with $$\...
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1answer
32 views

How do I prove that the image of a cyclic group has the following property?

I have the following problem: Let $C_m$ and $C_n$ be two finite cyclic groups of cardinality $m$ and $n$. Let $d=\gcd(m,n)$ and $f:C_m\rightarrow C_n$ be a morphism of groups. Let $C_d\subset C_n$ be ...
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2answers
87 views

Proving Schur-Zassenhaus Theorem, with added assumption that $G/H$ is cyclic

Schur-Zassenhaus Theorem: If there exists normal Hall-subgroup $H$ of finite group $G$, then there exists complement $K$ of $H$ in $G$. So if $\exists$ H $\unlhd$ G s.t. |H| is coprime to [G:H] then ...
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2answers
86 views

Showing that for $G = \langle x \rangle$ if $|G|$ is odd, then $\phi$ is an injection.

I'm given that $G$ is a finite cyclic group and the map $\phi: G \rightarrow G$ defined by $\phi(g) = g^2$, and asked to show that $\phi$ is a homomorphism which is straightforward but also that $\phi$...
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45 views

How to show that the group $S_3 \times C_2$ contains three element of order 3?

The question is, Which group of order $12$ is isomorphic to $S_3 \times C_2$ ? I know that this group is a non-abelian group so it's not isomorphic to $C_4 \times C_3$ and $C_2 \times C_2 \times C_3$...
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Describe all of the homomorphisms from $\Bbb Z_{24}$ to $\Bbb Z_{18}$. [duplicate]

I am trying to figure out all of the homomorphisms between these groups. First by homomorphisms properties, the identity of $\Bbb Z_{24}$ is mapped to $\Bbb Z_{18}$ (i.e $\phi(0) = 0$). After that, I ...
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Generator/generators for cyclic group $\mathbb Z[i]/(11-8i)$, given the following isomorphism: [duplicate]

I need to find a generator for $V$, being $V$ the $\mathbb Z[i]$-module generated by $v_1,v_2$ where: $(1+i)v_1 + (2-i)v_2 = 0$ $3v_1 + 5iv_2 = 0$ [EDIT] To clarify $V = \frac{\mathbb Z[i]^2}{A\mathbb ...
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Let $P \in{\rm Syl}_p(G)$ and suppose $P$ is metacyclic. Suppose $(|G|, p^2 - 1) = 1$. Show that $G$ has a normal $p$ complement.

Let $P \in{\rm Syl}_p(G)$ and suppose $P$ is metacyclic. Suppose $(|G|, p^2 - 1) = 1$. Show that $G$ has a normal $p$ complement. Since $P$ is metacycilc, let $N \unlhd P$ so that $P/N$ is cyclic. ...
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Working out cyclic subgroups of $D_{8}$ [duplicate]

I've been trying to work out the cyclic subgroups of $D_{8}$ out by hand and thought I could do it but I've run into something I don't understand. $D_{8}$ = $\{1,r,r^{2},r^{3},s,sr,sr^{2},sr^{3}\}$ I ...
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1answer
50 views

Cyclicness of a quotient of subgroups of infinite cyclic group

Let $A$ be an abelian group and let $G(A) = A \times S_3$ be the direct product of $A$ and the symmetric group of $3$ elements $S_3$. We define the following operation on $G(A)$: For each $(x,\sigma),(...
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4answers
117 views

How to prove that $\mathbb{C}_n$ is a subgroup of $(\Bbb C\setminus\{0\},\cdot)$. [closed]

$\forall n \in \mathbb{N} \setminus \{0\}$ prove that $\mathbb{C}_n=\{z \in \mathbb{C}\;|\;z^n=1\}$ is a subgroup of $(\mathbb{C}\backslash\{0\},\cdotp)$ and that is a cyclic group of order $n$. I ...
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1answer
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Help with proof on finitely generated abelian groups and its product of cyclic groups [duplicate]

In my lecture notes, there is a corollary stating that if $G$ is a finitely generated abelian group, then there is a surjective homomorphism $f: \mathbb{Z} ^n\rightarrow G$ for some integer $n$. It ...
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1answer
56 views

Is $(\mathbb{Q^+},\cdot)$ cyclic?

the case of the group $(\mathbb{Q^+},\cdot)$, is it cyclic (that is, a group that is generated by a single element)? If each element is multiplied by another in a pair, then the only way to reverse ...
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Given $G=\langle a\rangle$, $|G|=n$ and $d\mid n$, show $G$ has a unique subgroup of order $d$.

Given $G=\langle a\rangle$, $|G|=n$ and $d\mid n$, show $G$ has a unique subgroup of order $d$. Proof: (Existence) : $|\langle a\rangle|=|a|=n$ and $ |a^\frac{n}{d}|=d$. Then, $H_d=\langle a^\frac{n}{...
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2answers
66 views

if $G$ only has one subgroup of order $p$ and one subgroup of order $q$, then $G$ is a cyclic group

I'm having some trouble proving the following: Let $G$ be a group with $|G|=pq$ where $p$ and $q$ are two distinct prime numbers. Show that if $G$ only has one subgroup of order $p$ and one subgroup ...
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38 views

Find all homomorphisms $\varphi : \Bbb Z_6 \to S_3$. [duplicate]

Find all homomorphisms $\varphi : \Bbb Z_6 \to S_3$. Since $|\varphi(x)|$ divides $|x|$ we have that $|\varphi(1)|$ divides $|1|$, but the order of $|1|$ in $\Bbb Z_6$ is $6$ so $|\varphi(x)|$ must ...
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0answers
22 views

Let $n$ be a positive integer and $k \in \Bbb Z$. Show that $\langle [k]_n \rangle$ is an ideal for $(\Bbb Z_n, +, \cdot)$.

Let $n$ be a positive integer and $k \in \Bbb Z$. Show that $\langle [k]_n \rangle$ is an ideal for $(\Bbb Z_n, +, \cdot)$. I'm always confused when we have two binary operations. How am I to know if ...
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57 views

if $x$ and $y$ are commuting and have coprime orders then $\langle x,y\rangle $ is cyclic

Let $G$ be a group, let $x,u \in G$ show that if $x$ and $y$ commute and have coprime orders then $\langle x,y\rangle$ is cyclic. Knowing that $$\langle x,y\rangle= \{u_1 u_2, \ldots, u_k: u_i \in \{x,...
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156 views

A proof attempt of the cyclicity of $(\Bbb Z/p\Bbb Z)^\times$

Let $(\Bbb Z/p\Bbb Z)^\times$ be the multiplicative group of the integers modulo $p$, for $p$ a prime. It is well known that this group is cyclic (for a survey of the many proofs so far, see here). I'...
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54 views

Show by giving a counterexample that $H \cap N$ is not necessarily a normal subgroup of $G$.

Let $G$ be a group and $H$ a subgroup of $G$ and $N$ a normal subgroup of $G$. Show that $H \cap N$ is a normal subgroup of $H$ and show by giving a counterexample that $H \cap N$ is not necessarily a ...
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1answer
70 views

Let $G$ be a finite Abelian group that has exactly one subgroup for each divisor of $|G|$. How does this imply that $G$ is cyclic? [duplicate]

Suppose $G$ is a finite Abelian group that has exactly one subgroup for each divisor of $|G|$. How does this imply that $G$ is cyclic? By the Fundamental Theorem of Abelian Groups, $G \cong \mathbb{Z}...
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1answer
44 views

After using Sylow Theorems, how can we say how many elements of order 5 might be there in a group of order 20? [duplicate]

I know this question is asked here. I get the first half which says that from the Sylow Theorems (3rd one), we can say that a group of 20, must have a unique subgroup which has order 5. But from here ...
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1answer
58 views

Quotient of the direct product of cyclic groups

It occurs to me the following is true: $$(\mathbb{Z}_n \times \mathbb{Z}_m) / \mathbb{Z}_k \cong \mathbb{Z}_{n/k} \times \mathbb{Z}_m$$ when $k \mid n$. But I fail to see the way to prove that. The ...
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2answers
76 views

Prove that any group of order 15 is cyclic? [duplicate]

I have multiple questions regarding: https://math.stackexchange.com/a/864985/997899 Q: Prove that any group of order 15 is cyclic. A: Let $G$ be a group such that $|G| = 15$. Show that the group ...
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0answers
24 views

How to prove if subgroups are Cyclic then Group is Cyclic too? [duplicate]

The duplicate question doesn't really answer my problem. Given $G$ with order of 15. $G_3$ and $G_5$ are the only sub-groups of $G$ with order of 3 and 5 respectively. (In other words, there could be ...
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1answer
67 views

If Cylic subgroup implies abelian implies normal then how A5 is simple group [closed]

I am facing problem $A_5$ is simple group, but $A_5$ had 10 cyclic subgroup of order 3, from cyclic $\Rightarrow$ abelian $\Rightarrow$ normal we can say $A_5$ has 10 normal subgroups, but $A_5$ is ...
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0answers
21 views

Order of element of a cyclic group proof [duplicate]

Let $G$ be a cyclic group of order $n$ and supposed that $a \in G$ is a generator of the group. If $b = a^{k}$, then the order of $b$ is $n/d$, where $d$ = gcd$(k,n)$ I came across the above statement ...
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29 views

If $G$ nilpotent and $G/G'$ is cyclic then $G$ is cyclic? [duplicate]

If $G$ nilpotent and $G/G'$ is cyclic then $G$ is cyclic? It is very easy to see that this is true when $G$ is finite: If $G$ is finite nilpotent then all maximal subgroups are normal, so $G/N$ has to ...
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35 views

Homomorphism between cyclic groups is determined by an integer

Suppose $G$ is a cyclic group and $f \colon G \rightarrow G$ is any homomorphism. Show there is an integer $n$ such that $f(\gamma) = \gamma^n$ for all $\gamma \in G$. Show that if $G$ is infinite, ...
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3answers
94 views

Let $f: (\Bbb Z_{28}, +)\to(\Bbb Z_{16}, +)$ be a group homomorphism such that $f(1)=12$. Find $\ker f$.

Let $f: (\Bbb Z_{28}, +)\to(\Bbb Z_{16}, +)$ be a group homomorphism such that $f(1)=12$. Find $\ker f $. 1- $\langle 2\rangle $ 2-$\langle 4\rangle $ 3-$\langle 7\rangle $ 4-$\langle 1\rangle $ I ...
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1answer
32 views

Identity element in a finite cyclic group $G$.

I was reading the definition of a cyclic group $G$ which states: A group $Q$ is cyclic if there is an element $a \in Q$ such that the subgroup generated by $a$ is the whole of $Q$. ...
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2answers
45 views

A Question about a non trivial morphism

Define $\phi : A \to \mathbb{C}^*$ a non trivial morphism. Note that a finite subgroup of $\mathbb{C}^*$ will be cyclic. Suppose that $A$ is cyclic. Then there exists $ x \in A$ such that $A = \langle ...
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1answer
55 views

Prove the order of each nonindentity element of $G$ is $p$. [closed]

Sorry to disturbed, I have an abstract algebra coming and I'm unsure of how to prove this. If $p$ is prime and $G$ is noncyclic group of order $p^2$, prove the order of each nonindentity element of $G$...
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2answers
58 views

Find the order $\bar{5} + H$

Let $G=\mathbb{Z}/18\mathbb{Z}$ and $H=\langle 6+ 18\mathbb{Z}\rangle= \langle \bar{6}\rangle $ a subgroup of $G$. Consider the group $G/H$. What is the order of $(5 + 18 \mathbb{Z}) + H \in G/H$? ...
3
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2answers
122 views

How many subgroups does $\mathbb{Z}_{13}\times\mathbb{Z}_{13}$ have?

How many subgroups does $\mathbb{Z}_{13}\times\mathbb{Z}_{13}$ have ? My attempt: Firstly, we observe that the possible orders for an element of this group are $1$ and $13$ only. So, we would need to ...
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0answers
19 views

Count the subgroups of order $25$ in $C_{75} \times C_{10}$ [duplicate]

Count the subgroups of order $25$ in $G=C_{75} \times C_{10}$ Is there a general way to tackle this kind of question? That is, find the number of subgroups of a direct product of cyclic groups?. In ...
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1answer
38 views

Every abelian group of order $9p^2$ where $p\equiv 2\bmod 3$, can be written as the direct product of two cyclic subgroups.

Let $G$ an abelian group of order $9p^2$, where $p$ is an odd prime such that $p\equiv 2\bmod 3$, I have to show that $G$ can be written as the direct product of two cyclic subgroups. In this case, I ...

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