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.

learn more… | top users | synonyms

2
votes
1answer
17 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
51 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
32 views

Subgroup of a cyclic finite group [on hold]

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
35 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 ...
-2
votes
0answers
29 views

problem about p-cycles [on hold]

Let a be a cycle in $S_n$ so that $a\neq(1)$ And $a^p=(1)$ with $n/2<p\leq n$ and $p$ being prime. Prove that $a$ is a p-cycle.
1
vote
0answers
16 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
43 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
29 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
26 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
36 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
34 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
39 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
64 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
42 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
33 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
37 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
2answers
46 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
61 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
43 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
32 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
46 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
39 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
21 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
37 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. $$(12)(1253)$$ ...
1
vote
1answer
32 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
29 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 ...
1
vote
1answer
94 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
21 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
vote
3answers
53 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 ...
6
votes
3answers
75 views

In a group of order $pq^2$, if p divides $|Z(G)|$, then G is abelian

I want to prove that in a group of order $pq^2$, if $p$ divides $|Z(G)|$, then $G$ is abelian. I know that in this case, the order of $G/Z(G)$ must be $q^2$. If $G/Z(G)\simeq ...
0
votes
1answer
20 views

M an R-module, where M is a commutative ring, if $M≅R/I$ for some ideal I of R, then $M$ is a cyclic R-module

M is an R-module, where M is a commutative ring, if $M≅R/I$ for some ideal I of R, then show $M$ is a cyclic R-module. Note: $M$ is a cyclic R-module means $M=<m>$ for some $m\in M$. Please ...
1
vote
2answers
40 views

A cyclic group of order $n$ can be generated by $a^k$ if $(k,n)=1$

I am trying to provide a solution to the following exercise. Please point out anything that you find wrong and/or bad. Show that a cyclic group $G$ of order $n$ generated by an element $a$ can ...
1
vote
1answer
52 views

Aut($C_p) = C_{p-1}$when $p$ is prime, why?

Aut($C_p) = C_{p-1}$ when $p$ is prime, why? I don't see why this result follows, I'm sure it's obvious though. I appreciate that Aut($C_p)$ will be of order p-1
1
vote
2answers
104 views

$F$ a field and $G$ finite subset of $F \setminus \{0\}$ with 1 & satisfying $a, b ∈ G$ then $ab^{−1} ∈ G$. Show that $G$ is cyclic

Let $F$ be a field and let $G$ be a finite subset of $F \setminus \{0\}$ containing $1$ and satisfying the condition that if $a, b ∈ G$ then $ab^{−1} ∈ G$. Show that there exists an element $c ∈ G$ ...
0
votes
2answers
31 views

Non-Cyclic Subgroup of $GL(2, \mathbb Z)$

I'm trying to find an infinite, non-cyclic, subgroup of: H =$\left \{ \begin{bmatrix} a & b\\ c & d\end{bmatrix} : a,b,c,d \in \mathbb Z \right\}$ My current thought is to look at the ...
0
votes
3answers
41 views

If |G|>1 is not prime, there exists a subgroup of G which is not G or {e}

Question: Prove the following: if|G|>1 is not prime, then $G$ has a subgroup other than $G$ and {e} We know that $\langle g \rangle$ is a subgroup of $G$ by previous part of the question and $G$ is ...
1
vote
1answer
58 views

Show that a group that has only a finite number of subgroups must be a finite group

Show that a group that has only a finite number of subgroups must be a finite group. I started by assuming that group is infinite. But, don't understand how I should go from this assumption
2
votes
2answers
60 views

give an example of a cyclic group with 6 generators.

Give an example of a cyclic group with 6 generators. Give the generators, explain how you know that these are generators and that they are the only generators. I don't even know how to begin this ...
2
votes
1answer
40 views

$r$-cycle to a power $k$ is also an $r$-cycle if and only if $\gcd(k, r) = 1$

Let $\sigma$ be an $r$-cycle in $S_n$ and let $k\in\Bbb Z$. Show that $\sigma^k$ is also an $r$-cycle if and only if $\gcd(k,r)=1$.
3
votes
1answer
97 views

Cyclic group generator and multiplicative identity of correspondng ring

Can cyclic groups made into ring with unity such that multiplicative identity is not any generator? (Or does there exist example of one such cyclic group?) Can we make $(\mathbb{Z}, +)$ into ring ...
1
vote
0answers
59 views

question regarding group theory proof

Can someone please explain the sentence in red?, how does it follow?
1
vote
3answers
58 views

The elements of $\Bbb{Z}_{20}^{\times}$

The elements of $\Bbb{Z}_{20}^{\times}$, as I understand, are all the number from 1 to 20 included that are relatively prime to 20? I am having troubles finding a coherent definition of this kind of ...
2
votes
2answers
55 views

Subgroup of $\Bbb {Z}_m \oplus \Bbb {Z}_n$ where $(m,n)=1$.

Let $m,n>1$, $(m,n)=1$. Prove that every subgroup $H$ of $\Bbb {Z}_m \oplus \Bbb {Z}_n$ is $H=A\oplus B$ where $A=H\cap \Bbb {Z}_n$ and $B=H\cap \Bbb {Z}_m$. First attempt: $G=\Bbb {Z}_m \oplus ...
2
votes
0answers
39 views

Some subgroup of $GL_2(\mathbb{Q})$

Let's consider $GL_2(\mathbb{Q})$ and $C_2\times C_2 \times C_2$, $C_2$ - cyclic group of order 2. I can't show, that group $C_2\times C_2 \times C_2$ is not a subgroup of $GL_2(\mathbb{Q})$.I don't ...
4
votes
1answer
68 views

Prove any group of order $185$ is cyclic.

This is my attempt. I am not sure as for its plausibility. $Attempt$: Let $G$ be a group of order $185$. Then $G=185=5\cdot 37$. The $Sylow-p$ subgroups are unique and normal and therefore $G$ is ...
3
votes
1answer
67 views

Is the group $G =\{a+b\sqrt{2}|a,b \in \mathbb{Z}\}$ cyclic?

$G = \{a+b\sqrt{2}|a,b \in \mathbb{Z}\}$ under addition: I am going to say it's not cyclic because a,b can be distinct. I tried finding a generator.