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

4
votes
0answers
25 views

Subsets of cyclic group with distinct pairwise differences

Given a number $m\in\mathbb N$, let $\mathbb Z_m=\{0,1,\dots,m-1\}$ denote the ring of integers modulo $m$ (although we won't need multiplication, so any cyclic group of order $m$ will do). Given a ...
6
votes
1answer
69 views

$n^2$ divides $\phi(a^n-1)$ whenever $n$ is even and $a>2$

My problem is to show that $n^2$ divides $\phi(a^n-1)$ whenever $n$ is even and $a>2$. I have thought a solution but it is quite long and tedious. I wonder if anyone has a nice and clear ...
1
vote
1answer
33 views

Number of elements in a group

The group $G$ consists of the binary strings of length $5$ under addition $\mod 2$ in each component. (It is isomorphic to $(\mathbb Z_2)^5$, the direct product of $5$ copies of $\mathbb Z_2$.) Let ...
2
votes
2answers
77 views

How many subgroups of $\Bbb{Z}_4 \times \Bbb{Z}_6$?

I have been trying to calculate the number of subgroups of the direct cross product $\Bbb{Z}_4 \times \Bbb{Z}_6.$ Using Goursat's Theorem, I can calculate 16. Here's the info: Goursat's Theorem: Let ...
1
vote
2answers
41 views

Quotient ring is cyclic group implies every ideal is generated by 2 elements

I'm trying to solve the following exercise: Let $R$ be a commutative ring with identity. If for every ideal $\mathfrak{a} \neq 0$ of $R$ we have ($R/\mathfrak{a}$,+) is a cyclic group then ...
1
vote
2answers
54 views

Is this subgroup normal?

Let $T$ be a cyclic subgroup of a group $G$ such that $T$ is normal in $G$. Let $S$ be a subgroup of $T$. What can we say about whether or not $S$ is normal in $G$? My work: Let $T \colon = ...
4
votes
2answers
88 views

group theory proof

Let $G=\langle t \rangle$ be a cyclic group with $\text{ord}(t)=n$. I want to show that for all $d|n$ it holds that $$\left\lbrace s \in G ;\text{ord}(s)=d \right\rbrace=\left\lbrace t^{\frac{n}{d}k} ...
1
vote
1answer
23 views

Does this condition gaurantee the cyclicity of a finite abelian group?

Let $G$ be a finite abelian group in which there are at most $n$ solutions of the equation $x^n = e$ for each posivite integer $n$. How to determine if $G$ is cyclic or not?
0
votes
1answer
42 views

Which elements of this cyclic group would generate it?

Let $n$ be a given arbitrary positive integer, and let $U_n$ denote the group of all the positive integers less than $n$ and relatively prime to $n$ under multiplication mod $n$. Then for which values ...
0
votes
1answer
33 views

Concerning a Cyclic Galois Group

Why is it that: $\forall K \supseteq \mathbb{Q}(\mathbb{i}), G=Gal(K / \mathbb{Q}) = \langle \sigma \rangle \implies \sigma (\mathbb{i}) = - \mathbb{i}$? (Note: I am guessing that $\sigma ≠ ...
0
votes
1answer
23 views

cyclic repetitions in a string

Not really sure if this is purely a maths question.. I am looking if there is a faster way to look for cyclic repetitions in an input string. Say for example the input string is abcabcdabcabcd, it has ...
4
votes
1answer
60 views

Why is $\left(\mathbb{Z}_{51}\right)^* \cong \mathbb{Z}_2 \times \mathbb{Z}_{16}$?

I have to show that $\left(\mathbb{Z}_{51}\right)^* \cong \mathbb{Z}_2 \times \mathbb{Z}_{16}$. I know that $\mathbb{Z}_{51}\cong\mathbb{Z}_3 \times \mathbb{Z}_{17}$ and that $(\mathbb{Z}_p)^*\cong ...
0
votes
0answers
23 views

Is the subgroup of orthogonal matrices in $GL_3(\mathbb R)$ cyclic? What's generating it?

Is the subgroup of $GL_3(\mathbb R)$ denoted by $Q=\{A \in GL_3(\mathbb R)\mid AA^t=I_3\}$ cyclic? What's generating it? I showed $Q$ is a group, and realized it's the group of all orthogonal ...
2
votes
0answers
32 views

What primes can ramify and decompose in $k(\mu_{p^m}) \mid k$?

Let $k$ be a number field and $p$ be a rational prime. Then consider the extension $k(\mu_{p^m}) \mid k$ of adjoining all roots of unity of degree $p^m$ to $k$. Assuming that $\mu_{p^m} \cap k$ is ...
0
votes
1answer
39 views

Finite cyclic group

Can anyone give me a specific example of this: Let $G=\langle a\rangle$ be a finite cyclic group of order n. If $m\in \mathbb{Z}$, then $\langle a^m\rangle =\langle a^d\rangle$, where $d=\gcd(m,n)$ ...
0
votes
1answer
21 views

Cycles “converging” to an infinite cycle?

I recently had as an assignment, to find cycles $\sigma,\tau\in S_{\mathbb{N}}$ (i.e. permutations over the naturals) such that $ord(\sigma)=ord(\tau)=2$ and $\tau\circ\sigma$ has order infinity. This ...
2
votes
2answers
80 views

Suppose $G$ is a group of order 4. Show either $G$ is cyclic or $x^2=e$.

I've figured out that if I know $G$ is not cyclic, then it for any $a \in G, o(a) \neq 4$ (or the order of any element in group $G$ is not 4). I know ahead of time that the elements in the group ...
2
votes
1answer
26 views

Proper Subgroup is Cyclic Proof

Let G be a group with |G| = p^2, where p is prime. Show that every proper subgroup of G is cyclic. Proof: Let H be a proper subgroup of G where |G| = p^2 and p is prime. Then |H| divides |G| by ...
4
votes
2answers
29 views

Lagrange's Theorem Proof Help

Let H and K be subgroups of a group G with |H| = n and |K| = m, where gcd(n,m) = 1. Show that the intersection of H and K equals <1>. Proof: Let H and K be subgroups of a group G with |H| = n and ...
2
votes
1answer
35 views

Abstract Algebra: Cosets

Find all of the distinct left cosets of <4> in Z18 and all the cosets of <4> in the subgroup <2> of Z18. So The distinct left cosets of <4> in Z18 are 0 + <4> and 1 + <4>. Do I ...
1
vote
1answer
80 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 $pq$ ...
1
vote
0answers
41 views

Finding a permutation $ \alpha $ given $ \alpha^4 $ [duplicate]

I have the following question: Find a permutation $\alpha ∈ S_7 $ such that $\alpha^4 = (2 1 4 3 5 6 7)$. Is $\alpha$ unique? How should I go about this? I've tried a few different trial and error ...
1
vote
1answer
35 views

Order of ab when <a> and <b> are distinct

I have the following problem: Consider a finite group G with elements $a,b\in G$ such that $ab=ba$ and $\langle a\rangle\cap\langle b\rangle=\langle e \rangle$. Prove that $|ab|=lcm(|a|,|b|)$. I found ...
1
vote
1answer
27 views

Regarding the order of elements in a factor group

If I have understood it correctly, a factor group consists of all cosets of a subgroup $H$ of $G$. Since it is a requirement that $H$ is normal, left and right cosets are equivalent. I am asked to ...
0
votes
1answer
74 views

Symmetric group is not cyclic [closed]

Show that for $n > 2$ the group $S_n$ is not cyclic, but can be generated by two elements. Guys how can I show this? Now how can I prove my problem?
0
votes
1answer
51 views

Cyclic subgroup of $S_n$

Find the smallest natural number n ∈ ℕ such that $S_n$ contains a cyclic subgroup of order 101. Proof: We seek the smallest n such that Sn contains a permutation of order 101. Permutation can be ...
1
vote
1answer
59 views

If a group has only one element $a$ of order $n$, then $a$ belongs to $Z (G)$ and $n=2$.

I understand that $a \in Z (G)$ by this proof: Group question: only one element $x$ with order $n>1$, then $x\in Z(G)$ But I don't understand why $n$ must be equal to $2$?
2
votes
2answers
62 views

Morphisms between $\mathbb{Z}/n\mathbb{Z}$ and $\mathbb{Z}/m\mathbb{Z}$

I'm trying to determinate how many morphisms of groupes exist between $\mathbb{Z}/n\mathbb{Z}$ and $\mathbb{Z}/m\mathbb{Z}$ for $n,m\in\mathbb{N}$. I know a morphism is determinated by the image of ...
2
votes
2answers
53 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 ...
1
vote
2answers
47 views

Prove that $\sigma_a \circ \sigma_b = \sigma_{ab}$.

Let $\mathbb Z_n$ be a cyclic group of order $n$ and for each integer $a$ define $$\sigma_a : \mathbb Z_n \mapsto \mathbb Z_n, \qquad \sigma_a(x) = x^a, \quad\forall \; x \in \mathbb Z_n.$$ Prove ...
1
vote
1answer
46 views

Subgroup of a cyclic groups and are isomorphic

I don't know whether I am right or wrong. Can anyone help me to clear below problem ? Question 1: Let C2 be the cyclic group of order 2 and C202 be the cyclic group of order 202. Find all subgroups ...
2
votes
1answer
48 views

Find homomorphisms from $C_8$ to $S_3$ and from $C_6$ to $S_3$, and are there any isomorphisms?

For $C_8$ to $S_3$: Let $C_8$ be generated by $x$. Then $C_8 = \langle x : x^8 = e \rangle = \{e,x,x^2,x^3,x^4,x^5,x^6,x^7\}$. But how do I show homormorphisms? I'm supposed to pick elements of $x$ ...
2
votes
1answer
44 views

Nontrivial Homomorphism(s) from $\mathbb{Z_3}$ to $S_3$ - Fraleigh p. 134 13.37

Reference: http://users.humboldt.edu/pgoetz/Homework%20Solutions/Math%20343/hwsome number 1 to 17 that I forgotsolns.pdf There are exactly two nontrivial ...
3
votes
0answers
80 views

In a finite cyclic group of order n, number of solutions to $x^m = e$ - Fraleigh p. 68 6.53,54

(53.) Show that in a finite cyclic group G of order n, written multiplicatively, the equation $x^m = e$ has exactly m solutions $x$ in G for each $m \in \mathbb{N}$ that divides n. (54.) With ...
8
votes
5answers
177 views

Nonzero rationals under multiplication are not a cyclic group

Are nonzero rationals under multiplication cyclic? Here's my thinking. They are not. The generator must be a rational $q = a/b$, $a$, $b$ integers with no common factors. Assume $a/b$ generates ...
3
votes
3answers
67 views

What are the subgroups of $C_2\times C_{202}$?

Basically, if $C_2$ is a cyclic group of order two and $C_{202}$ is the same with order $202$, what are the subgroups of the product of the two? The farthest I've gotten is that if $C_2$ is addition ...
-2
votes
1answer
104 views

Why isn't $\langle a ; a^2 \rangle$ (or $\langle a;a^3, a^7\rangle$) a presentation of $C_4$?

I've just read the first few pages of Combinatorial Group Theory by Magnus, Karrass, and Solitar, and based on their definitions there, and more specifically, the reasoning given in the hint to ...
-4
votes
2answers
125 views

multiplicative group $\mathbb{R}^∗$ of non-zero real numbers is not cyclic. [duplicate]

How will I be able to show that multiplicative group $\mathbb{R}^∗$ of non-zero real numbers is not cyclic?
1
vote
1answer
138 views

Let $\sigma$ be the $m$-cycle (1 2 $\ldots m)$. Prove that $\sigma ^i$ is also an $m$-cycle if and only if $i$ is relatively prime to $m$

I have consulted some other sources, and think that I have some handle on the basic ideas behind the proof, but am having trouble articulating them. I understand that $\sigma$ sends $a_k$ to $a_{k} + ...
0
votes
1answer
42 views

Elementary Properties of Order

Need help with this question: Prove: Let $x = a_1, a_2...a_n$, and let $y$ be a product of the same factors, permuted cyclically. (That is, $y = a_k, a_{k+1}...a_na_1...a_{k-1}$. Then ...
2
votes
2answers
66 views

When is the automorphism group $\text{Aut }G$ cyclic?

Let $G$ be a finite group. Under which conditions on $G$ is the automorphism group $\text{Aut }G$ cyclic? More precisely, does $G$ is abelian or $G$ is cyclic implies $\text{Aut }G$ is cyclic?
1
vote
1answer
43 views

cyclic group - automorphism

If $G$ is a cyclic group where both $a$ and $b$ are generators. How would I prove that $f:G \rightarrow G$ given by $f(a^i)=b^i$ is an automorphism. I know that an automorphism is the identity map. ...
3
votes
3answers
55 views

Prove $\{x \mid x^2 \equiv 1 \pmod p\}=\{1, -1\}$ for all primes $p$ [duplicate]

One way to prove that $\{x \mid x^2 \equiv 1 \pmod p\}=\{1, -1\}$ is to use the fact that $\{1, -1\}$ is the only subgroup of the cyclic group of primitive residue classes modulo $p$ that has the ...
2
votes
1answer
42 views

$\rm{Aut}(G)$ is not cyclic when $G$ is not abelian

we have a $G$ which is not abelian and we need to prove that the group of all automorphisms is not cyclic. any ideas?
0
votes
2answers
34 views

Subgroups of Galois groups of finite fields

According to the notion of Galois group, for $E=GF(2^n)$ as an extension of the field $F=GF(2)$, the Galois group $Gal(E/F)$ is a cyclic group of order $n$. Now my question is: for finding the ...
0
votes
4answers
125 views

Prove that a cyclic group can have no more than one element of order two.

(1)Why can't a cyclic group have more than one element of order two? (2)Why does the group $U(n^2 -1)$ have to have more than one element of order two? Thank you so much for your time, ...
4
votes
1answer
43 views

Visualize $C_2 \times C_4$ is normal subgroup

Page 120 says: Given our recent work with subgroups, you may have noticed that $C_2$ is a subgroup of $C_2 \times C_4$; specifically, it is the subgroup $<(1,0)>$. Furthermore, the cosets of ...
2
votes
1answer
40 views

Order of $x^k$ in cyclic subgroup of order $n$ generated by $x$

$ord(x)=n$. Prove that $ord(x^k)= \frac nd$ where $d=gcd(k,n)$ This is what I have so far: Let $k=ad$ and $n=bd$. We need to find the smallest $p$ such that $(x^k)^p = 1$. $(x^{ad})^b = x^{abd} = ...
4
votes
0answers
40 views

Nontrivial homomorphism for $Z_a \times Z_b $to $Z_c \times Z_d$ - Fraleigh p. 134 13.35

This isn't a duplicate of this. Let $(A, B) \in \mathbb{Z_a \times Z_b}$. Hinging on p. 2, I guess homomorphism is $h(A,B) = (A \text{ mod } c, B \text{ mod } d)$. I'm unsettled. p. 2 sprang it up ...
6
votes
0answers
52 views

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

Let G be a group, g an element of G, and n a positive integer. 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 ...