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.

1,971 questions
Filter by
Sorted by
Tagged with
38 views

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 (...
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 ...
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 ...
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, ...
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 ...
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 $G$, ...
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 $G$, ...
233 views

37 views

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....
36 views

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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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, ...
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 ...
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 $Q$. ...