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 ...
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
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
43 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
0answers
24 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 ...
0
votes
1answer
40 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 $d=\gcd(m,n)$ ...
2
votes
2answers
82 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
36 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
81 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
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 ...
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
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
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
127 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?
2
votes
2answers
67 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. ...
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 ...
6
votes
0answers
47 views

Find all subgroups of $\mathbb{Z_2} \times \mathbb{Z_2} \times \mathbb{Z_4}$ isomorphic to the Klein $4$-group - Fraleigh p. 110 Exercise 11.12

I tried to fill in the steps but I'm still confounded by this solution. $|\mathbb{Z_2} \times \mathbb{Z_2} \times \mathbb{Z_4}| = 2\cdot2\cdot4$. $order(\mathbb{Z_2} \times \mathbb{Z_2} \times ...
4
votes
1answer
81 views

Find all subgroups of $\mathbb Z_2\times\mathbb Z_4$ - Fraleigh p. 110 Exercise 1.11

I tried to fill in the steps but I'm still confounded by this solution. Any other subgroup must have order 4, since by Lagrange's Theorem, the order of any subgroup must divide 8 and: The subgroup ...
6
votes
1answer
346 views

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 ...
2
votes
2answers
64 views

Fast way of calculating the order of an element in $\mathbb{Z}_n$?

Is there a fast way of calculating the order of an element in $\mathbb{Z}_n$? If i'm asked to calculate the order of $12 \in \mathbb{Z}_{22}$ I just sit there adding $12$ to itself and seeing if the ...
1
vote
0answers
41 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 ...
0
votes
1answer
32 views

If $\sigma\in S_n$ is of length $n$, then $\sigma$ generates its centralizer

Let $S_n$ and $C_G(\sigma)$ denote the symmetric goup and the centralizer of $\sigma\in S_n$, respectively. I want to show: If $\sigma$ is of length $n$, then $C_G(\sigma)=\langle\sigma\rangle$, i.e. ...
1
vote
3answers
59 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 ...
1
vote
2answers
41 views

Simple question regarding number of elements in cyclic subgroups

Let $G$ be a cyclic group with $N$ elements. Then it follows that $$N=\sum_{d|N} \sum_{g\in G,\text{ord}(g)=d} 1.$$ I simply can not understand this equality. I know that for every divisor $d|N$ ...
-1
votes
4answers
122 views

Cyclic Group Problem [closed]

If G is cyclic group of 24 order, then how many element of 4 order in G? I can't understand how to find it, step by step.
0
votes
1answer
56 views

If $\chi(a)=1$ for all $\chi\in\hat G$ then $a=0$.

Let $G$ be finite abelian group and $\hat G$ be its character group. I need hint proving that if $a\in G$ and $\chi(a)=1$ for all $\chi\in\hat G$ then $a=0$ (the identity element). I can prove it ...
2
votes
1answer
57 views

subgroups of finite cyclic group

Let $G=(g)$ be a finite cyclic group generated by $g$ with $|G|=n$, and let $d \in \mathbb{N}$ with $d|n$, then an unique subgroup $H$ of $G$ with $|H|=d$ exists. Proof of existence: $\exists m \in ...
-3
votes
3answers
88 views

Let $G = \{1, a, b, c\}$ be a group of order 4…Exist two groups of order $4$.

Let $ G = \{1, a, b, c\}$ be a group of order 4. Show that, if $G$ is cyclic $G \cong \mathbb Z_4,$ and if $G$ is not cyclic then $G \cong K_4.$ It now follows that there are only 2 groups of order ...
2
votes
3answers
81 views

Number of elements which are cubes/higher powers in a finite field.

This question is a slight generalization of This Question. How many elements are there in a finite field of order $q$ which are : Squares. Cubes. Higher powers. I mean : How many elements are ...
0
votes
2answers
72 views

quotient group Z/3Z equated to 0,1,2

This a followup question on the first answer to this post: Why the term and the concept of quotient group? In the first answer, Lahtonen says that for the quotient group Z/10Z, one can "equate 9 ...
1
vote
1answer
64 views

Counting Elements of Order $2$ in a Direct Product of Cyclic Groups

I am not sure if I am oversimplifying this question or not, but here it is: Suppose we have the following direct product of groups: $$G=\mathbb Z/60\mathbb Z \times \mathbb Z/45\mathbb Z ...
1
vote
4answers
120 views

Semi-direct product of groups with one of them cyclic

Let $K$ be a cyclic group. Let $\phi,\psi: K\rightarrow Aut(H)$ be group homomorphisms such that there exists $\zeta\in Aut(H)$ satisfying $\phi(K)=\zeta \psi(K)\zeta^{-1}$. Then can we prove ...
0
votes
2answers
145 views

surjective homomorphisms between cyclic groups (wrong question)

This statement does not hold. Let $C, D$ be cyclic groups and $f_1,f_2:C\rightarrow D$ be surjective homomorphisms. Show that there exists $\xi: C\rightarrow C$ such that $f_2=f_1\circ \xi$. My ...
4
votes
2answers
82 views

Suppose a unique a generates a cyclic subgroup of order. Show ax = xa. - Fraleigh p. 67 6.50

(1.) I don't understand above. How do you magically envisage and envision to let $b = xax^{-1}$? What I did was to start from the answer and see if I can get a chain of equivalences. $ax = xa ...