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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1answer
43 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 $D_8$ [on hold]

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
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4answers
88 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 ...
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1answer
22 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= ...
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2answers
38 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 ...
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0answers
27 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
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3answers
37 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
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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
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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. ...
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2answers
60 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 ...
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2answers
50 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?
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1answer
39 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?
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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 ?
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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
24 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
72 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 ...
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0answers
34 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 ?
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1answer
23 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
598 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
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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 $o(\bar x^2)=2$ ...
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1answer
62 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 ...
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1answer
34 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 ...
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2answers
37 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 ...
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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 ...
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2answers
50 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
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1answer
34 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
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2answers
27 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 $m$. ...
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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 ...
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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
39 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 ...
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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 ...
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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 ...
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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 ...
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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
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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
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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
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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 ...
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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 ...
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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 ...
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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 ...
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2answers
42 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
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1answer
23 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 ...
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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
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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
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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 ...
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1answer
98 views

Suppose $G$ is a group with $|G|=35$. Prove that if $H$ is a subgroup of $G$ with order 7, then $H$ is a normal subgroup of $G.$

(1) Suppose $G$ is a group with $|G|=35$. Prove that if $H$ is a subgroup of $G$ with order $7$, then $H$ is a normal subgroup of $G.$ (2) Suppose that G is a group with $|G|=35.$ Prove that if $G$ ...
0
votes
1answer
22 views

Given a group is finite and non-abelian, why is the left coset with the centre of the group non-cyclic?

Assume $T$ is finite and non-abelian then why is $T/Z(T)$ non-cyclic? Where $Z(T)$ is the centre of the group $T$. I've shown $Z(T)$ is a normal subgroup of T, but not sure what to do next or if ...
1
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3answers
58 views

$G$ is a commutative group of order 72, which is a product of cyclic groups. What is max order of element?

I'm trying to understand the following practice question that has the given answer. Can someone help? Here are some specific questions: I presume the notation $(\mathbf{Z}/2)$ refers to some cyclic ...