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

1
vote
0answers
52 views

question regarding group theory proof

Can someone please explain the sentence in red?, how does it follow?
1
vote
3answers
54 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
52 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
36 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
53 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
52 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.
3
votes
1answer
112 views

A Finite Group of Nonprime Order which is Unique up to Cyclic Group

Maybe I have to wait until I learn and study more, but I just became curious. I know that every finite group of prime order is cyclic, and hence unique up to isomorphism. I have 2 questions about ...
0
votes
1answer
30 views

Find Cyclic Group [duplicate]

Given the following : $$ \langle\Bbb Z^{*} _{91}, {*}\rangle,$$ where $*$ is multiplication, and $\mathbb{Z}^{*}_{91}$ is $\mathbb{Z}_{91}\setminus \{0\}$. ($\mathbb{Z}^{*} _{91}$ contains any ...
2
votes
1answer
43 views

Finite group with unique subgroup of each order.

Let $G$ finite group, and suppose $G$ has unique subgroup of each order (which divides $G$'s order) - Show that $G$ is cyclic. I reduced the problem to sylow subgroups of $G$ (they are all normal), ...
0
votes
2answers
34 views

If $G$ is a group and $y \in G$\ $\langle x \rangle$ , then can $\langle y \rangle$ be $G$?

If $G$ is a group and $e \ne x \in G$ and $y \in G$\ $\langle x \rangle$ , then can $\langle y \rangle$ be $G$ ?
-2
votes
1answer
39 views

Find Cyclic Group [closed]

Given the following : $$ <\Bbb Z^{*} _{27}, *> $$ $$ <\Bbb Z^{*} _{35}, *> $$ $$ <\Bbb Z^{*} _{91}, *> $$ $$ < \Bbb Q^{*}, *> $$ $$ <\{ \begin{bmatrix}1 & ...
0
votes
1answer
27 views

Dihedral group of order 2n

I would appreciate if someone could prove this for me: Let G be a dihedral group of order 2n and suppose H is a cyclic quotient group of G. Show that |H|is less than or equal 2.
1
vote
1answer
83 views

What is a generator of a finite cyclic group? (General)

I have asked a few questions about this but I am still confused. So, in general, what is a generator of a finite cyclic group and how is it found? I have seen in books and my notes a lot of ...
0
votes
3answers
45 views

Finding the generator of cyclic group $U(Z_{27})$

$U(\mathbb{Z}_{27})$ is a group of order $18$. $U(\mathbb{Z}_{27})=\{1,2,4,5,7,8,10,11,13,14,16,17,19,20,22,23,25,26\}$ How do I find the generators to prove that this group is cyclic? The final ...
0
votes
1answer
44 views

Is $U(\mathbb{Z}_{54})$ a cyclic group? [duplicate]

Is the group $U(\mathbb{Z}_{54}) = \{1,5,7,11,13,17,19,23,25,29,31,35,37,41,43,47,49,53\}$ a cyclic group? If so, how do I show this? This group was found as a group of units of the monoid ...
0
votes
0answers
57 views

A $p^n$th root of unity is in $N_{F/k}(F^\times)$ if and only if there is an extension of $F$ which is a cyclic extension of $k$ of degree $p^{n+r}$

Hello, I have to solve this homework but I have completely no idea. please help me, it is very difficult for me. any attempt will be welcomed and appreciated. Conditions : $n, r$ are fixed number ...
0
votes
0answers
38 views

let $G$ to be group such that $O(G)=p^2$ where $p$ is prime,prove that $G$ is cyclic or $G$ is Direct product of two cyclic subgrops of order $n$. [duplicate]

the only hint that i got is Sylow's first theorem, which implies that if $p^n$ is any prime power dividing $O(G)$, then $G$ has a subgroup of order $p^n$. in our case $p$ devides $p^2$, then we can ...
3
votes
4answers
68 views

$O(G)=p^2 $ ,and p is prime, it is also known that $|Z(G)|>1 $. proof that G is abelian

We know that $Z(G)<G,\;$ then $O(Z(G)) \mid O(G). $ If $\;O(Z(G))= p^2, $ then $\;Z(G)=G$ and we are done. Now, if $O(Z(G))= p,\,$ how can I prove that $G$ is abelian ? Is it by proving that ...
0
votes
1answer
48 views

$(\mathbb Z/6\mathbb Z)/(2\mathbb Z/6\mathbb Z)$

$$(\mathbb Z/6\mathbb Z)/(2\mathbb Z/6\mathbb Z)=\left\{ \left\{ 6\mathbb Z,2 + 6\mathbb Z, 4 + 6\mathbb Z\right\} ,\left\{ 1+6\mathbb Z,3 + 6\mathbb Z, 5 + 6\mathbb Z\right\} \right\} $$ Is that ...
3
votes
1answer
43 views

Number of unique cycle paths on an octahedron

I am looking at the number of unique cyclic paths along the edges of an octahedron. A cycle starts and ends on the same vertex and an edge can only be walked once. They are also invariant under ...
1
vote
2answers
102 views

Show that the group is cyclic.

I'm trying to show that the group $U(Z_{54})$ is cyclic. To start, I found the divisors of 54 = {1, 2, 3, 6, 9, 18, 27, 54} Then I started to find the elements using the powers of a. Where ...
3
votes
2answers
58 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
57 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
34 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
27 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
33 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
1answer
33 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
26 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
33 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
47 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
37 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
29 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
84 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
60 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
37 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
vote
0answers
23 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
24 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
85 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
29 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
54 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
83 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
231 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
83 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
45 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
2
votes
4answers
57 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
52 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
65 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
36 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
89 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 ...