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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13
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4answers
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Groups of order $pq$ without using Sylow theorems

If $|G| = pq$, $p,q$ primes, $p \gt q, q \nmid p-1 $, then how do I prove $G$ is cyclic without using Sylow's theorems?
5
votes
5answers
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Subgroups of a cyclic group and their order.

Lemma $1.92$ in Rotman's textbook (Advanced Modern Algebra, second edition) states, Let $G = \langle a \rangle$ be a cyclic group. (i) Every subgroup $S$ of $G$ is cyclic. (ii) If ...
3
votes
1answer
332 views

G is group of order pq, pq are primes

Problem. Let $G$ be a group of order $pq$ such that $p$ and $ q$ are prime integers. I am to show that every proper subgroup of $G$ is cyclic. My attempt. What I know: Any element $a$ divides $pq$ ...
2
votes
2answers
1k views

If a cyclic group has an element of infinite order, how many elements of finite order does it have?

If a cyclic group has an element of infinite order, how many elements of finite order does it have? I know that the order of the entire group must be infinite, for an element of the group must have ...
10
votes
2answers
268 views

Can we conclude that this group is cyclic? [duplicate]

Let $G$ be a finite group. If, for each positive integer $m$, the number of solutions of the equation $x^m = e$ in $G$, where $e$ is the identity element, is at most $m$, then can we conclude that $G$ ...
14
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7answers
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Prove that $\mathbb{R^*}$, the set of all real numbers except $0$, is not a cyclic group

Prove that $\mathbb{R^*}$ is not a cyclic group. (Here $\mathbb{R^*}$ means all the elements of $\mathbb{R}$ except $0$.) I know from the definition of a cyclic group that a group is cyclic if ...
6
votes
3answers
197 views

For what $n$ is $U_n$ is cyclic?

When can we say a multiplicative group of integers modulo $n$, i.e., $U_n$ to be cyclic? $$U_n=\{a \in\mathbb Z_n \mid \gcd(a,n)=1 \}$$ I searched in internet but did not get clear idea.
6
votes
6answers
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Give an example of a noncyclic Abelian group all of whose proper subgroups are cyclic.

I've tried but I could not find a noncyclic Abelian group all of whose proper subgroups are cyclic. please help me.
4
votes
4answers
4k views

Product of two cyclic groups is cyclic iff their orders are co-prime

Say you have two groups $G = \langle g \rangle$ with order $n$ and $H = \langle h \rangle$ with order $m$. Then the product $G \times H$ is a cyclic group if and only if $gcd(n,m)=1$. I can't seem to ...
1
vote
1answer
153 views

What are the generators for $\mathbb{Z}_p^*$ with p a safe prime?

lets consider $\mathbb{Z}_p^*$ with $p = 2 \cdot q + 1$ a safe prime ($p$ and $q$ have to be prime). Then $\varphi\left(p\right) = 2 \cdot q$ is the order of $\mathbb{Z}_p^*$, and ...
13
votes
3answers
548 views

Show $\langle a^m \rangle \cap \langle a^n \rangle = \langle a^{\operatorname{lcm}(m, n)}\rangle$

So I want to show that $\langle a^m \rangle \cap \langle a^n \rangle = \langle a^{\operatorname{lcm}(m, n)}\rangle$. My approach to this problem was to show a double containment, i.e. to show that ...
3
votes
3answers
764 views

Showing that a group of order $21$ (with certain conditions) is cyclic

How can i show that if $o(G)=21$ and if $G$ has only one subgroup of order $3$ and only one subgroup of order $7$, then show that $G$ is cyclic.
8
votes
1answer
377 views

How to find all groups that have exactly 3 subgroups?

How to find all groups that have exactly 3 subgroups? Any group must have identity and itself as subgroups, so we just need to find all the groups that only have one proper subgroup. I think ...
1
vote
0answers
49 views

Divisors of the totient function and congruences

Based on the question Question about the totient function and congruence classes, I would like to ask two new questions: Question 1 Does it hold that if $k\mid\varphi(n)$ then the elements of the ...
3
votes
3answers
179 views

A persisting element in all subgroups.

Let $G$ be a finitely generated abelian group and $a$ be a nontrival element of $G$ contained in all nontrivial subgroups of $G$. Is $G$ necessarily cyclic?
3
votes
4answers
225 views

$(\mathbb{Z}/2^n \mathbb{Z})^*$ is not cyclic Group for $n\geq 3$

Question is to Prove that $(\mathbb{Z}/2^n \mathbb{Z})^*$ is not cyclic Group for $n\geq 3$. Hint : Find two subgroups of order $2$. I somehow feel that a cyclic group can not have two distinct ...
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 ...
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 ...
1
vote
3answers
51 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
3answers
63 views

Prove that if $k\mid m,$ then $Z_m$ has a subgroup of order $k.$

Prove that if $k\mid m$, then $Z_m$ has a subgroup of order $k.$ Ok, so this doesn't look like too hard of a problem. So do I just show that it is closed under multiplication and inverse? I just ...
0
votes
0answers
46 views

Let $G = \langle x\rangle$ be cyclic of order $n$. Prove that $\langle x^r\rangle\leq \langle x^s\rangle$ iff $r$ is a multiple of $s$ modulo $n$.

Let $G = \langle x\rangle$ be cyclic of order $n$. Prove that $\langle x^r\rangle \leq \langle x^s\rangle$ iff $r$ is a multiple of s modulo $n$. Give the complete subgroup lattice of $G = \langle ...
-1
votes
3answers
640 views

Intersection of cyclic subgroups: $(x^m) \cap (x^n) = (x^{lcm(m,n)})$ [duplicate]

This group theory problem has stumped me. I want to prove that if $G=(x)$ is a finite cyclic group that $(x^n) \cap (x^m) = (x^{\operatorname{lcm}(m,n)})$ for all integers $m$ and $n$, where $(x)$ is ...
-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 & ...
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votes
2answers
76 views

Prove that the subgroup of the quotient group is cycling and infinitely generated

$$M = \left\{\,\dfrac{m}{13^n}\biggm| m\in \mathbb{Z}, n\in\mathbb{N} \,\right\}, \quad G = M/\mathbb{Z}$$ Prove that any subgroup $H < G$, $H\neq G$ is cyclic and infinitely generated and that ...
14
votes
1answer
273 views

Fibonacci Sequence in $\mathbb Z_n$.

Consider a Fibonacci sequence, except in $\mathbb Z_n$ instead of $\mathbb Z$: $$F(1) = F(2) = 1$$ $$F(n+2) = F(n+1) + F(n)$$ It is easy to see that each of these sequences must cycle through some ...
5
votes
4answers
2k views

Homomorphism between cyclic groups

I have some confusion in relation to the following question. Let $\langle x\rangle = G$, $\langle y\rangle = H$ be finite cyclic groups of order $n$ and $m$ respectively. Let $f:G \mapsto H$ ...
10
votes
4answers
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Finite groups with exactly one maximal subgroup

I was recently reading a proof in which the following property is used (and left as an exercise that I could not prove so far). Here is exactly how it is stated. Let $G$ be a finite group. Suppose ...
9
votes
5answers
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Are cyclic groups always abelian?

If a group $C$ is cyclic, is it also abelian (commutative)? If so, is it possible to give an “easy” explanation of why this is? Thanks in advance!
9
votes
2answers
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Order of automorphism group of cyclic group

Let $G$ be a cyclic group of order $m$. What is the order of $\text{Aut}(G)$? I want to know the proof as well (elementary if possible). I would still accept the proof if one answers with $m = p$, a ...
4
votes
2answers
230 views

Cyclic Group Presentation

Show that the the group with presentation $$\langle x, y\ \mid\ x^2=y^2x^2y,\ (xy^2)^2=yx^2, \ yx^{-1}y^2=x^7\rangle $$ is cyclic of order 24. This presentation was obtained using the Todd-Coxeter ...
4
votes
2answers
2k views

A subgroup of a cyclic group is cyclic - Understanding Proof

I'm having some trouble understanding the proof of the following theorem A subgroup of a cyclic group is cyclic I will list each step of the proof in my textbook and indicate the places that I'm ...
9
votes
2answers
225 views

Generators of a cyclic group

In a paper there is a lemma: Let $G= \langle a,b \rangle$ be a finite cyclic group. Then $G=\langle ab^n \rangle$ for some integer $n$. The proof is omitted because it's "straightforward" but ...
7
votes
2answers
435 views

Abelian group admitting a surjective homomorphism onto an infinite cyclic group

I am working on the following problem: Let $G$ an Abelian group and $f: G \to \Bbb Z$ a surjective homomorphism. Prove that $G \cong \ker(f) \times \Bbb Z$ By means of the First Isomorphism ...
6
votes
1answer
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Find all proper nontrivial subgroups of Z2 x Z2 x Z2 - Fraleigh p. 110 Exercise 11.10

$\newcommand{\lcm}[0]{\mathrm{lcm}}$I tried to fill in the steps but I'm confounded by this solution. Here $i$ is the identity element, not $e$. Because $\lcm(2, 2, 2) = 2$ hence all non-identity ...
6
votes
3answers
512 views

How to find subgroups of $ \;\;\Bbb Z_2\times \Bbb Z_6$

I am reading a first course in algebra and there is an example saying that "find all the subgroups of $\Bbb{Z}_2\times\Bbb{Z}_6$ and decide which of them are cyclic. I know that ...
6
votes
3answers
617 views

On the Factor group $\Bbb Q/\Bbb Z$ [duplicate]

Possible Duplicate: $\mathbb{Q}/\mathbb{Z}$ has a unique subgroup of order $n$ for any positive integer $n$? I have the factor group $\Bbb Q/\Bbb Z$, where $\Bbb Q$ is group of rational ...
5
votes
4answers
358 views

Show that if $ab$ has finite order $n$, then $ba$ also has order $n$. - Fraleigh p. 47 6.46.

This solution is from here and yahoo. Given $a,b$ elements of $G$, and $ab$ has finite order $n$. Hence $\color{magenta}{|ab| = n} \iff (ab)^n = e$. Need to show $n$ is the smallest positive integer ...
5
votes
3answers
192 views

Intuition and Tricks - Hard Overcomplex Proof - Order of Subgroup of Cyclic Subgroup - Fraleigh p. 64 Theorem 6.14

Update Dec. 28 2013. See a stronger result and easier proof here. I didn't find it until after I posted this. This isn't a duplicate. Proof is based on ProofWiki. But I leave out the redundant $a$. ...
4
votes
5answers
416 views

Prove any subgroup of a cyclic group is cyclic.

was just wondering if this is a valid proof for the aforementioned question? I am quite confident that it isn't, but not exactly sure why. Maybe I am missing the point of proofs by induction ...
1
vote
2answers
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Show that the set of non zero rationals is not a cyclic group under multiplication.

Show that the set of non zero rationals is not a cyclic group under multiplication.I know the set of rationals is not a cyclic group under addition. In an exercise of my book it is given that the ...
1
vote
2answers
126 views

What is $\mathbb{Z}/m\mathbb{Z}/n (\mathbb{Z}/m\mathbb{Z})$?

Is $\mathbb{Z}/m\mathbb{Z}/n (\mathbb{Z}/m\mathbb{Z})=\mathbb{Z}/(m\mathbb{Z} +n \mathbb{Z})$? Thanks.
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 ...
0
votes
1answer
53 views

Meaninig of a symbol at the Circle Group

I have $U_{20}$ (at the meainig of the Circle Group), What is the meaning of $W_{20}^{8}$?? What is the 20 and what is the 8? Thank you!
8
votes
5answers
802 views

Is $\mathbb{Z}^2$ cyclic?

Is $\mathbb{Z}^2$ cyclic? What does it mean for a group to be cyclic? Is it just that it has one generator? Thanks
6
votes
1answer
106 views

Necessary and Sufficient Condition for $\phi(i) = g^i$ as a homomorphism - Fraleigh p. 135 13.55

Let $g \in \text{ group } G $ and $n \in N$. Let $\phi : \mathbb{Z_n} \rightarrow G$ be defined by $\phi(i) = g^i$ for $0 \le i \le n$. Give a necessary and sufficient condition (in terms of g and n) ...
6
votes
3answers
173 views

Intuition and Tricks - Crafty Short Proof - Generators, Order of a Cyclic Group - Fraleigh p. 64 Theorem 6.14

This stronger result and easier proof is based on p. 58. Hence it isn't a duplicate of this. Theorem 206 and 207. Let $G$ be a group, $k \in \mathbb{N}$ and $a \in G$ such that $|a| = n$. Then: 206. ...
5
votes
1answer
243 views

A group with two non trivial subgroups is cyclic

Let $G$ be a group. Suppose that $G$ has at most two nontrivial subgroups. Show that $G$ is cyclic. Can anyone help me please to solve the problem?
5
votes
2answers
221 views

Determining whether two groups are isomorphic

I am reading "A First Course in Algebra", and there, I am trying to solve the exercises, but there is something i don't understand. How do we understand whether two groups are isomorphic or not? For ...
3
votes
2answers
85 views

Proof blueprint - If $G/Z(G)$ cyclic then $G$ Abelian - Fraleigh p. 153 15.37

(1.) Why didn't Fraleigh state the result in the direct form like in my title? Why state it with the negations and then prove the contrapositive? Isn't this extra unnecessary work? (2.) How do you ...
3
votes
3answers
263 views

Is there a simple way to distinguish between group homomorphisms?

More precisely, I am given a function $f:G\to H$ with the promise that it is a homomorphism. Is there an easy way to determine which homomorphism it is without looking through all of its values? For ...