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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18 views

The generator of a cyclic quotient group

(I'll use (a) to denote the subgroup of G generated by a) Let G be a finite abelian group of order n. Let a be an element of G of order k. We can easily see that (a) is a normal subgroup of G. If we ...
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1answer
24 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 $\gcd(k,r)=1$.
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0answers
12 views

Cyclic group generator and Multicative identity iof 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 (Z, +) into ring with unity ...
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0answers
54 views

question regarding group theory proof

Can someone please explain the sentence in red?, how does it follow?
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3answers
55 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
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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
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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 ...
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1answer
54 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
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1answer
54 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
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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 ...
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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
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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), ...
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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 $G$ ?
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1answer
39 views

Find Cyclic Group [closed]

Given the following : $$ <\Bbb Z^{*} _{27}, *> $$ $$ <\Bbb Z^{*} _{35}, *> $$ $$ <\Bbb Z^{*} _{91}, *> $$ $$ < \Bbb Q^{*}, *> $$ $$ <\{ \begin{bmatrix}1 & ...
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1answer
28 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.
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1answer
84 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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
45 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 ...
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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
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2answers
59 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 ...
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3answers
59 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
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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 $(\mathbb{Z}\times \mathbb{Z})/\langle(a,b)\rangle$ ...
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2answers
35 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 ...
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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 ...
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1answer
34 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 ...
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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: $G=(123)(456)(789)\ \text{and} \ S=(147)(258)(369)$ ...
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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 ...
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1answer
34 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
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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 ...
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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 ...
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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.
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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
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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 )
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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 ...
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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 ...
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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
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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
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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 ...
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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
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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
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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 ...
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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
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4answers
60 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 ...