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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2
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1answer
29 views

Is there any cyclic subgroup of order 6 in in $ S_6$?

Is there any cyclic subgroup of order 6 in $ S_6$? Attempt: $|S_6|=6!=720$ Let $H$ be a subgroup of $S_6$ ,$H$ cyclic $\iff\langle H \rangle=\{e,h,h^2,...,h^{n-1}\}=S_6$
0
votes
2answers
68 views

Abelian group which is not one of these

Im struggling to find a finite abelian (commutative , associative) group $(G,\circ)$ with some specific conditions: $a\circ b$ isn't naive addition $a+b$ for $a,b\in G$ $G$ is a subset of ...
2
votes
1answer
25 views

Cyclic consecutive zeros of binary sequence with prime length

I found a feature that if $N>5$ is a prime, and $M \triangleq \frac{N-1}{2}$ is also a prime, then we will always have a binary sequence $x_1,\ldots, x_N$ with $L=\frac{N-1}{2}$(or ...
1
vote
1answer
44 views

Let $G$ be a non-abelian group. Prove that if $G/Z(G)$ is isomorphic to $\Bbb Z_2 × \Bbb Z_2$ then $G$ is isoclinic to dihedral group $D_8$ [closed]

prove that if $G/Z(G)$ is isomorphic to $\Bbb Z_2 \times \Bbb Z_2$ then $G$ is isoclinic to dihedral group $D_8$.
7
votes
4answers
95 views

If $G$ is cyclic then $G/H$ is cyclic?

If $G$ is cyclic, then $G/H$ is cyclic? The proof I got goes like this: $G$ is cyclic, so $G=<g>$ for some $g\in G$. So any coset in $G/H$ would be of the form $Hg'=Hg^n$ for some $n$. So ...
1
vote
1answer
23 views

Find the Existence and Uniqueness of a Cyclic subgroup

Giving that $|H|$ is cyclic. If $|H|=n$ then for each $a>0$ such that $a|n$ there exist a unique subgroup of $H$ of order $a$. This subgroup is Cyclic subgroup $\langle x^d\rangle$ where $d= ...
0
votes
2answers
39 views

Cyclic groups and other groups

Ok So I know that a cyclic group is a group that is generated by a single element like $\large{(Z_n,+)}$. Now I was wondering that if every group has a generator , and I found that the answer is yes ...
0
votes
0answers
28 views

Writing a group as a product of cyclic groups

How do I go about expressing a group as a product of cyclic groups? For example, express: $$O^*_K = \{\pm (24 + 5\sqrt{23})^r : r\in \mathbb{Z} \}$$ as a product of cyclic groups ($O^*_K$ is the ...
2
votes
3answers
39 views

Isomorphism of a product $C_n \times C_m$ of cyclic groups with the cyclic group $C_{mn}$

Given that $C_n$ is a cyclic group of order $n$, what conditions must integers $n$ and $m$ satisfy such that the group $C_n \times C_m$ is isomorphic to C$_{mn}$? So I attempted to investigate a ...
0
votes
1answer
63 views

$g$ has order $n$, then $\langle g\rangle=\langle g^2\rangle=\cdots=\langle g^{n-1}\rangle$

If $g$ has order $n$, then $\langle g\rangle=\langle g^2\rangle=\cdots=\langle g^{n-1}\rangle$. This should be fairly easy but somehow I just couldn't prove it. I only managed to prove the case ...
1
vote
1answer
32 views

Groups and Subgroups elements

Let $G$ be a cyclic group order $12$ with $G=\left<a\right>$. Let $H=\left<a^3\right>$. List the elements of $H$ and find the cosets. I am lost as to what the elements of $H$ would be. ...
1
vote
2answers
62 views

$|G|=p_1p_2p_3$ distinct primes with $p_i \nmid p_j-1$ then $G$ is cyclic

Problem Let $p_1,p_2,p_3$ be three distinct primes with $p_i \nmid p_j-1$ for all $1\leq i,j \leq 3$ and let $G$ be a group of order $p_1p_2p_3$. Show that $G$ is cyclic. I've tried to come up with ...
0
votes
2answers
52 views

Is the direct product $\mathbb{Z}_4 \times \mathbb{Z}_2$ cyclic? [closed]

Is the direct product $\mathbb{Z}_4 \times \mathbb{Z}_2$ cyclic?. How would you check this?
1
vote
1answer
40 views

Is the direct product $\Bbb Z \times \Bbb Z$ with operation $(n,m)+(p,q):=(n+p,m+q)$ a cyclic group?

Is the direct product $\Bbb Z \times \Bbb Z$ with operation $(n,m)+(p,q):=(n+p,m+q)$ a cyclic group? I know its not a cyclic group but how would i show this in a formal way?
-1
votes
1answer
44 views

What are all the subgroups of $\Bbb{Z}/10\Bbb{Z}$?

I thought they would be $\Bbb{Z}/n\Bbb{Z}$, where $1 \leq n \leq 10 $. What is wrong in that ? And, how is $\Bbb{Z}/n\Bbb{Z}$ a cyclic group. Is it's generator the element 1 ?
1
vote
0answers
34 views

Clarification for question: Homomorphisms from $\Bbb Z / n \to \Bbb C_{\ne 0}$

Please don't solve the problem - it is for an assignment - this is just a question for clarification purposes. Let $G$ be a group and $\hat G$ be the set of homomorphisms from $G$ to the group of ...
3
votes
1answer
27 views

Is a finite cyclic group a Poincare duality group?

I am trying to understand whether the finite cyclic group of order $n$, $C_n$ is a Poincare duality group, i.e. whether it's classifying space $K(C_n,\,1)$ is a Poincare complex. I know that the ...
5
votes
2answers
73 views

Help to prove that $ U_{p} $ is a cyclic group.

As part of my study of Abstract Algebra, I’m trying to prove that $ U_{p} $ is cyclic for $ p $ a prime number. It’s a classical result, but I’m trying to prove it following four steps stated as ...
-2
votes
0answers
35 views

Cyclic subgroups of $\mathbb{Z} /100\mathbb{Z} \oplus\mathbb{Z}/25\mathbb{Z}$

$\mathbb{Z} /100\mathbb{Z} \oplus\mathbb{Z}/25\mathbb{Z}$ has 24 elements of order 10. Why each cyclic subgroup of order 10 has four elements of order 10 ?
1
vote
1answer
24 views

Group theory disjoint cycles

Let $a=(1 3 5)(1 5 6)(1 3 5)$ I had to write this as a product of disjoint cycles and got $(1 5)(3 6)$ which I believe is correct. Then figure out $a^{24}$ and $a^{25}$. Now $a^{24}$ is the ...
7
votes
2answers
600 views

Why is the set of integers with the operation of addition considered a cyclic group?

The first sentence in the Wikipedia article entitled "Cyclic Groups" states that "In algebra, a cyclic group is a group that is generated by a single element". How is this consistent with addition on ...
2
votes
1answer
25 views

Looking for different proofs for $-1$ is a quadratic residue of primes of the form $4k+1$ and related facts

Suppose $p$ is a prime of the form $4k+1$ , then $4|p-1=|\mathbb Z_p^*|$ , as $\mathbb Z^*_p$ is a cyclic group , so there is $\bar x \in \mathbb Z_p^*$ such that $o(\bar x)=4$ , then $o(\bar x^2)=2$ ...
1
vote
1answer
63 views

Quotient group(Factor group)

Prove that the quotient group $\frac{Z\times Z\times Z}{<(1,1,1)>}$ is an infinite, non-cyclic group. Here Z is the group of integers with operation of addition, $<(1,1,1)$> is the ...
-3
votes
1answer
35 views

Subgroup of a cyclic finite group [closed]

Let $G$ be a cyclic group of order $n$ and let $m$ be a positive integer dividing $n$. Show that $G$ contains one and only subgroup of order $m$. I have started the proof by stating the smallest ...
1
vote
2answers
42 views

Is $\mathbb{Z}\times\mathbb{Z}/((6,5),(3,4))$ is finitely generated?

Let $A$ be the quotient of the free abelian group $\mathbb{Z}^2$ by the subgroup generated by $(6,5)$ and $(3,4)$. the question is $A$ is finitely generated? and if Yes. Can we Decompose it into a ...
1
vote
0answers
18 views

Question about exponents of groups

Okay, I'm trying to understand exponents of groups. I will start with the Set Z_3, where Z_3 is the integers mod3 under addition. Now, I want to set out to find the exponent of this group, but ...
1
vote
2answers
52 views

Can a cyclic group have more than two generators? [duplicate]

Can a cyclic group have more than two generators? for example the group $\mathrm{Z}$ has two generators $-1$ and $1$, but can a group have more than two generators?
1
vote
1answer
35 views

Generating elements of large group of units of a field.

Suppose $F$ is a large but finite field, and $F^\text{x}$ is the group of non-zero elements of the field. It is known that this group is always cyclic, and we could ask a natural question "what are ...
0
votes
2answers
28 views

Do Equal Sets Have the Same Enumerations?

One of the first proofs in group theory is to show that in a finite $G$, the order of the element $g$ is the same as the order of the subgroup $\langle g \rangle$. Let the order of $g$ equal $m$. ...
0
votes
0answers
18 views

Multiple maximal cyclic subgroups of a symmetry group

If a symmetry group T has a maximal cyclic subgroup Cn because of a projection I1, then it means it will have a rotational symmetry order of n. If we have another projection (of the same object) with ...
1
vote
1answer
37 views

How many cycles $A$ and $B$ can form this cycle

How many cycles $A$ and $B$ can form this cycle: $AB=(axyguimjrcwk)(bvqphsleofzt)(d)(n)$ I can see that $A$ and $B$ must share the cycle $(dn)$, and I believe due to ordering, both $A$ and $B$ must ...
0
votes
1answer
40 views

How do I prove that there exists a cyclic subgroup of order lcm of orders of cyclic subgrpups of an abelian group?

Before I start, please note that this post is not duplicate Let $G$ be an abelian group. Let $H,K$ be finite cyclic subgroups of $G$ such that $|H|=r,|K|=s$. Then, how do I prove that there exists ...
1
vote
1answer
41 views

Is $A_n$ isomorphic to $C_n$ in general

Let $A_n$ denote the alternating group of order $n$, and let $C_n$ be the cyclic group of order $n$. Correct me if I'm wrong, but I know that $A_3 = \{(),(1,2,3),(1,3,2)\}\cong \{0,1,2\}=C_3$. I'm not ...
3
votes
0answers
65 views

prove that a is of order m if and only if $a^m = e$ and $a^k $is not $e$ for all $0 < k < m$.

$a$ is of order $m$ if and only if $a^m = e$ and $a^k \neq e$ for all $0 < k < m$. It is about order in Algebra. I sketch the proof. Is it correct? I need your help. First ($\Rightarrow$) Let ...
0
votes
1answer
43 views

Finding subgroups of order $4$ of $\mathbb{Z}_2\times\mathbb{Z}_4$

Now, the question asks me what the subgroups of order $4$ are of this relation and then to give them as sets and identify the group of order $4$ that each of the subgroups is isomorphic to. How do I ...
0
votes
0answers
43 views

How to tell if a directed graph has a cycle?

If I have the directed graph here: I am confused whether or not this is a cycle or not. Because in the underlying graph, this is a 3-cycle for sure, but in the directed graph, there is no cycle if ...
0
votes
1answer
41 views

Group Theory - Cyclic Groups & Direct Product

Prove that if $G$ is an infinite group and $H$ is a group then $G \times H$ is cyclic if and only if $G$ is cyclic a $H =$ {${e_H}$}. Solution: I can see that this is true but I don't know how to ...
1
vote
1answer
64 views

Let G be an infinite cyclic group. Prove that G cannot have any non-identity elements of finite order.

SO I know that I'm suppose to prove it by contradiction and assume that the element has a positive power. I'm not really sure how to answer it though.
5
votes
1answer
66 views

List all the elements of order 3 in the group $\mathbb{Z_{18}}$

Task: Consider the group $\mathbb{Z_{\large18}}$ under the operation of addition modulo $18.\;$ List all the elements of order $3.$ My professor said the answer was $6$ and $12$. But isn't ...
0
votes
1answer
20 views

Cyclic sub-spaces, polynomials proof

In a vector space $V$ of finite dimension over the field $F$, not zero $v\in V$ and $T:V\to V$, prove: a) $Z(v,T)=\left<v,Tv,T^2v,...\right>$, (Span) is the intersection of the T-invariant ...
3
votes
1answer
45 views

Show that $(\mathbb{Z}/2^n \mathbb{Z})^{\times}$ is not cyclic for any $n > 3.$ [duplicate]

I need some help with the following question. Show that $(\mathbb{Z}/2^n \mathbb{Z})^{\times}$ is not cyclic for any $n > 3.$ There is the following hint in my book: find two distinct ...
1
vote
1answer
74 views

Prove that group $\mathbb{Q}\times Z_2$ is not isomorphic to $\mathbb{Q}$

I need some help with the following question. Prove that group $\mathbb{Q}\times Z_2$ is not isomorphic to $\mathbb{Q}.$ My proof: Let $a,b \in \mathbb{Q}$ and let $\phi$ be isomorphism.We have ...
0
votes
1answer
33 views

A well defined homomorphism from $\mathbb{Z}/48\mathbb{Z}$ into $\mathbb{Z}/{36}$

I read book of Dummit and Foot Abstract algebra. I need some help with the following question. Let $\mathbb{Z}/{36} = <x>.$ For which integers $a$ does the map $\psi_{a}$ defined by ...
1
vote
0answers
53 views

How to find all ring homomorphisms from $\mathbb Z_{12} \to \mathbb Z_{30}$ ?

How to find all ring homomorphisms from $\mathbb Z_{12} \to \mathbb Z_{30}$ ? I know that it is enough to determine $f([1]_{12})$ ; moreover $f([1]_{12}$ should be an idempotent element of $\mathbb ...
0
votes
2answers
43 views

Clarification on Lagrange's Theorem and the group $Z_n$ of integers modulo $n$

I want to show that $Z_3 \leq Z_6 \leq Z_{12}$. Since $|Z_3|$ and $|Z_6|$ divide $|Z_{12}|$, then they must be subgroups of $Z_{12}$. Similarly, $Z_3 $ is a subgroup of $Z_6$. I feel like here I am ...
0
votes
1answer
24 views

Express the following permutations as products of transpositions and identify them as even or odd.

So I am still getting the hang of cyclic notation. Express the following permutations as products of transpositions and identify them as even or odd. I think this is saying express the following in ...
0
votes
2answers
40 views

Compute cycle notation

So I am new to cycle notation and needless to say I am finding it a bit confusing. I know that when computing these, I need to work right to left=. Compute each of the following: a. $$(12)(1253)$$ ...
2
votes
1answer
46 views

Showing two groups are not isomorphic using the order of their elements.

I am trying to solve this question: "$\text{Prove that no two of the groups } C_2 \times C_2 \times C_2 , C_2 \times C_4 \text{ and } C_8 \text{ are isomorphic.} $" I understand that to show they ...
0
votes
2answers
30 views

suppose that $G$ is a finite group and $H$ is a subgroup of $G$ that is not cyclic,I want to prove that all conjugates of $H$ are not cyclic.

suppose that $G$ is a finite group and $H$ is a subgroup of $G$ that is not cyclic,I want to prove that all conjugates of $H$ are not cyclic. I supposed that $S$ is a conjugate of $H$ and it is ...
0
votes
1answer
25 views

List the elements of the cyclic subgroup $<(13, 3)>$ in $\mathbb Z_{26}$ x $\mathbb Z_{9}$

List the elements of the cyclic subgroup $<(13, 3)>$ in $\mathbb Z_{26}$ x $\mathbb Z_{9}$ I'm not entirely sure I am doing this problem right. Do I simply add (13,3) until I reach (0,0)? I ...