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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3
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2answers
51 views

Find the number of subgroups in $Z_p \times Z_p \times Z_p$

let $p$ be a prime number ; I want to find the number of subgroups in $G = Z_p \times Z_p \times Z_p$ $(Z_p = \mathbb{Z}/p\mathbb{Z})$. I know that there is $p^2 + p + 1$ copies of $Z_p$ in $G$ for ...
1
vote
3answers
53 views

Find the order of the intersection of two cyclic groups

Suppose $G=\langle a\rangle $ has order $140$. What is the order of the intersection $\langle a^{100}\rangle$ and $\langle a^{30}\rangle$? My attempt: I know that the order of $\langle ...
0
votes
1answer
39 views

show a quotient group is cyclic

Let a and b be nonzero integers. Let $\sigma:\mathbb{Z}\times \mathbb{Z}\to\mathbb{Z}$ and $\sigma((x,y))=ax+by.$ Suppose $\gcd(a,b)=1$. Show that $(\mathbb{Z}\times \mathbb{Z})/\langle(a,b)\rangle$ ...
2
votes
2answers
32 views

Let $G$ be any group and $x \in G$ , then $N_G(\langle x \rangle)/C_G(\langle x \rangle)$ is finite ?

Let $G$ be any group and $x \in G$ , then is it true that $N_G(\langle x \rangle)/C_G(\langle x \rangle)$ is finite ? I know that $N_G(\langle x \rangle)/C_G(\langle x \rangle)$ is isomorphic to ...
0
votes
2answers
26 views

Finding homomorphisms

Let $G$ be the dihedral group of order 14. In the following, justify your answer. 1. Let $A=C_2$ be a cyclic group of order 2. Find all homomorphisms $G→A$. 2. Let $B=C_7$ be a cyclic group of ...
1
vote
1answer
25 views

Existence of a generator over multiplication for integers modulo p

If we consider the integers modulo a prime $p$, then for every $x \not \equiv 0$ (mod $p$), we can get any $b \not \equiv 0$ by adding $x$ a number of times to itself. Is the same true for ...
0
votes
0answers
13 views

Cyclic subspaces and cyclic vectors

Prove that if an linear operator T^2 has a cyclic vector, then T has a cyclic vector. How do I get into this problem..? (Linear Algebra)
0
votes
1answer
30 views

Finding/Creating a Modern Algebra theorem

The question I'm trying to prove is this one: The subgroup $<G,S>$ generated by $G$ and $S$ is abelian and of order $9$. My Work: $G=(123)(456)(789)\ \text{and} \ S=(147)(258)(369)$ ...
0
votes
1answer
21 views

Symmetries of a $9$ puzzle (Rubik's Slide)

Consider this Rubik's slide. With these moves (and their inverses): $$\text{Vertical shift}\: v=(147)(258)(369)$$$$\text{Rotation}\: c=(12369874)$$$$\text{Horizontal shift}\: h=(123)(456)(789)$$ Also ...
0
votes
1answer
32 views

Homomorphism and kernel

My question is, for |G|=30, where G is a cyclic group of order $n$ where $G=<a>$. Consider the mapping from $\phi :G \rightarrow G'$ by $\phi(a^k)=b^k$. I have showed it is a homomorphism and ...
2
votes
2answers
44 views

Does (Z, +) have two generators but infinitely many generating sets?

We say the group of integers under addition Z has only two generators, namely 1 and -1. However, Z can also be generated by any set of 'relatively prime' integers. (Integers having gcd 1). I have ...
0
votes
1answer
34 views

Rubik's Slide Proof's and Symmetries in a Rubik's Slide

$\quad$In the February edition of The Mathematical Association of America Monthly there is a article called "$\mathit{Rubik's\ on\ the\ Torus}$". Where they are dealing with solving problems involving ...
1
vote
0answers
28 views

$\{m\in\mathbb Z_{2^n}:m\equiv 1 \mod 4\}$ is cyclic

How can I prove that the multiplicative group $G:=\{m\in\mathbb Z_{2^n}:m\equiv 1 \mod 4\}$ is cyclic? I tried claiming that $G=\langle5\rangle$ but failed.
1
vote
2answers
77 views

Cyclic finite groups and their direct product

Let $C$ be a cyclic group with three elements. prove that $C \times C \times C$ can not be generated by two elements. I was thinking that showing that $C ^{3}$ is isomorphic to ${\mathbb{Z}_{3}} ^{3}$ ...
4
votes
1answer
56 views

every group of order $105$ has a cyclic normal subgroup of index $3$ ?

Does every group of order $105$ has a cyclic normal subgroup of index $3$ ? (Please don't use Sylow theorems )
1
vote
1answer
25 views

Internal and Direct Product question, need help with explanation

I am asked to express a group G={1,7,17,23,49,55,65,71} under multiplication modulo 96 as an external and internal direct product of cyclic groups. However I also have an example to help with it, but ...
1
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0answers
22 views

Getting the least possible order of a group from existing cyclic subgroups or elements.

Let's say I have finite group $G$. Let it have have following subgroups: $<a> = \{e, a\}$, $<b> = \{e, b\}$, $<c> = \{e, c, c^2\}$, $<d> = \{e, d, d^2, d^3\}$ I can for sure ...
2
votes
0answers
22 views

How do I know the cyclic subgroups of G generated by a are equal?

I had a question of understanding that I was hoping you guys could help me with. Suppose there exists a group in the integers. For the sake of my understanding, let's say the group is, G = $Z_{60}$. ...
0
votes
1answer
72 views

Question about Z12≃Z4⊕Z3 generators and order

I apologize in advance for my messy language and questions; I've only been studying group theory for a month and thus these concepts aren't clearly locked in yet; hence my questions. :) Take ...
2
votes
2answers
27 views

Cyclicity of some special subgroup of $(\mathbb R,+)$

If $H$ is a subgroup of $(\mathbb R,+)$ such that $H\cap[-1,1]$ is finite and contains a positive element then is it true that $H$ is cyclic ?
5
votes
1answer
49 views

Example of infinite field $(F,+,.)$ such that $(F^*, . )$ is a cyclic group ? [duplicate]

It is known that any infinite cyclic group can never be a vector space , from this we can derive that if $(F,+,.)$ is an infinite field then $(F,+)$ cannot be cyclic . I am asking , is there any ...
1
vote
1answer
75 views

Union of all finite cyclic groups

Let $C_n$ be the cyclic group of order $n$. I'm trying to investigate properties of $\bigcup_{n=1}^{\infty} C_n$ It seems obvious that this is a group, but I don't really know much else about it. ...
2
votes
2answers
214 views

Are there any finite non-abelian group with one subgroup of each size ?

Let $G$ be a finite group with at most one subgroup of any size , then is it true that $G$ is cyclic ? I can prove that the answer is "yes" with the additional assumption of abelian ness on $G$ but ...
2
votes
2answers
79 views

Properties possessed by $H , G/H$ but not G

i) Does there exist a group $G$ with a normal subgroup $H$ such that $H , G/H$ are abelian but $G$ is not ? ii) Does there exist a group $G$ with a normal subgroup $H$ such that $H , G/H$ are cyclic ...
0
votes
1answer
40 views

If G is abelian and simple ,then G is cyclic

True /False .IF G is abelian and simple ,then G is cyclic Solution True If G is an abelian simple group then G is isomorphic to Zp for some prime p
1
vote
4answers
53 views

Can subgroups of subgroups be normal?

Prove that if $N$ is a normal subgroup of $G$ and $N$ is cyclic, then if $K<N$, $K$ is a normal subgroup of $G$. I understand that I have to show that $gKg^{-1}=K$ $\forall g \in G$. I also know ...
0
votes
1answer
50 views

Algebra: Cylic Groups

Let $G = \mathbb Z \times \mathbb Z = \mathbb Z^2 $ and let $H$ be the subgroup generated by $(1,3)$ and $(2,1)$, i.e., $$H = \{ m(1,3) + n(2,1) \, : \, m,n \in \mathbb Z\}.$$ This exercise will ...
0
votes
1answer
41 views

Isomorphism between Direct Products of Cyclic Groups

How would I be able to tell if two direct products of cyclic groups are isomorphic to one another? I'm working with two pieces of knowledge so far: 1) The order of the direct product is the product ...
0
votes
1answer
32 views

Character Table of $C_6$

$$\begin{array}{rrrrrrrrrrr} & 1 & g & g^2 & g^3 & g^4 & g^5\\ \phi_0 & 1 & 1 & 1 & 1 & 1 & 1 \\ \phi_1 & 1 & \zeta_6^1 & & & ...
0
votes
2answers
78 views

Find the subgroups of A4

I had a question I was hoping for some help on: Find all of the subgroups of $A_4$ Here is what I know: $A_4$ is the alternating group on 4 letters. That is it is the set of all even ...
1
vote
3answers
50 views

Prove that $\langle a^n \rangle \bigcap \langle a^k \rangle = \langle a^{lcm (n,k)} \rangle$

Let $G$ be a group. Let $a$ be an element. Let $n,k$ be pozitive integers. Let $m$ be least common multiple of $n$ and $k$. Prove $\langle a^n \rangle \bigcap \langle a^k \rangle = \langle a^{m} ...
0
votes
1answer
16 views

Help with finding cosets for cyclic subgroups

The question I'm working on is: Let $G=\mathbb{Z}_4\times\mathbb{Z}_3\times\mathbb{Z}_2$ and consider the subgroup $H=\langle\left(0,1,1\right)\rangle$ of G. Find all cosets of H. So I know that in ...
0
votes
1answer
24 views

If $G$ has only 2 non-trivial proper subgroups H, N, then H, N are cyclic subgroup of $G$.

If $G$ has only 2 non-trivial proper subgroups H, N , then H, N are cyclic subgroup of $G$. I searched essentially same problem at If $G$ has only 2 proper, non-trivial subgroups then $G$ ...
0
votes
1answer
24 views

List the elements of a subgroup

The question is to list the elements of $H \subset \mathbb{Z}_{1610}$ if $H = \langle 1035\rangle$. I know that there are 14 elements in the set, I am not sure how to find the elements. Is the ...
0
votes
2answers
45 views

Is this group cyclic?

Is this group cyclic, and what would be its generator? where H = {a+b(sqrt(2)) element of R | a, b element of Z} I know that in order for a group to be cyclic if the generator equals the group, but ...
2
votes
2answers
129 views

Group theory question for cyclic group

I met some problem during googling. The problem and its solution are next. and I'm wondering about 2nd YELLOW BOX $$ $$ $$ $$ Why $G$ has a unique element of order 2 in case of $H=G$ ? $$ $$ ...
3
votes
1answer
83 views

Describing all the homormorphisms from $\Bbb Z_{10}$ to $\Bbb Z_{15}$

I'm working on a problem that asks me to show all the homomorphisms from $\Bbb Z_{10}$ to $\Bbb Z_{15}$. So far, my attempt is as follows: Since $\Bbb Z_{10}$ and $\Bbb Z_{15}$ are both cyclic, I ...
0
votes
1answer
53 views

Abstract Algebra: Cyclic Groups (Lattice Diagram)

Example 4.2: Lets find all the subgroups of the given group and draw the lattice diagram for the subgroup. Z12 Z36 Z8 In the book finding the subgroups is explained well but it does not explain ...
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0answers
20 views

groups and symmetry

Which of the following may be false? A) Any non-trivial element of $C_n$ generates $C_n$. B) Any subgroup of $C_n$ is cyclic. C) if $m|n$ then $C_n$ has at least one subgroup of order $m$. D) if ...
2
votes
1answer
46 views

Cyclic group generators

My question is: Can you find a cyclic group with n generators? I know that zero (or any other identity element for that matter) is included, so there would be for $Z_n$ at most n-1 generators. ...
0
votes
1answer
92 views

$G$ infinite abelian group with $[G:H]$ finite for every non trivial subgroup $H$ , to prove $G$ is cyclic

Let $G$ be an infinite abelian group such that for any non-trivial subgroup $H$ of $G$ , $[G:H ]$ is finite ; then how to prove that $G$ is cyclic ? Please don't use any structure theorem of abelian ...
0
votes
3answers
47 views

Subgroups of $S_4$ generated by cycles

I am new with abstract algebra and I trying to find all the subgroups of $S_{4}$ generated by the cycles : a) $(13)$ and $(1234)$ b) all cycles of length $3$ I am not sure how to start so I would ...
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vote
1answer
58 views

Every subgroup is a union of cyclic subgroups of its group.

True or false? Every subgroup H of a group G is a union of cyclic subgroups of G. I think it is false,but cant think of counter example
0
votes
2answers
43 views

Prove that if $|a|=m $ and $|b|=n$ with gcd(m, n)=1 then $\langle a \rangle \cap \langle b \rangle = \{e\}$

$G$ is a group and $a, b, \in G$. So summarizing the question, if the order of $a$ and $b$ are relatively prime, then the cyclic group generated by $a$ and $b$ will only have the identity element in ...
1
vote
1answer
62 views

$\mathbb{Z}/\mathbb{2Z} \bigoplus \mathbb{Z}/\mathbb{2Z}$ not isomorphic to $\mathbb{Z}/\mathbb{4Z}$

Greetings fellow Mathematics Community. I am having some doubts about my solution to the following problem: Show that $\mathbb{Z}/ \mathbb{2Z} \oplus \mathbb{Z}/ \mathbb{2Z}$ is not isomorphic to ...
0
votes
1answer
69 views

Finite cyclic $\Bbb Z$-module and exact sequence

Suppose $M$ is $\Bbb Z$-module, cyclic, finite. How to prove $\require{AMScd}$ \begin{CD} 0 @>>> \Bbb Z @>>> \Bbb Z @>>> M @>>>0\\ \end{CD} is ...
1
vote
1answer
78 views

In S4, find all the even permutation and show that the set of odd permutations isn't stable for binary operations in S4.

I want to find the even permutations of $S_4$ so i am supposed to find the transpositions right? but of what permutation exactly do i find the transpositions? And how do i know which ones are even? ...
2
votes
3answers
29 views

GCD's and how they generate groups

I was reading something today an it was talking about $U_{15}$, all the integers relatively prime to 15, and how it was generated by the set {7,11}. I understood it all, but I thought that if the ...
2
votes
4answers
47 views

Concerning Cyclic Groups

I am new to group theory. I have a problem but I don't really understand what it is about, so I am asking somebody to explain what is the problem (I am not really seeking for solution). Here it is: ...
0
votes
4answers
56 views

How to prove that $\mathbb Z^n$ is not cyclic for $n > 1?$

The operation here is taken to be addition. Clearly $\mathbb Z$ is cyclic since $\mathbb Z = \langle 1 \rangle = \langle -1 \rangle.$ I was then looking at a question that asked if $\mathbb Z^4 \times ...