3
votes
2answers
163 views

Cyclic Group Presentation [on hold]

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 ...
1
vote
1answer
53 views

Artin Algebra 2.8.3 “Does every group whose order is a power of a prime $p$ contains an element of order $p$?”

I'm not sure whether or not my answer and proof for this question are valid. Could you point out any flaw? Let $G$ be an arbitrary group, an arbitrary element of $G$ be $g$ and $|G|=p^n$. Since a ...
1
vote
1answer
51 views

What are all the automorphisms of a group of order $9$ generated by two elements?

Let $G$ be a group of order $9$ generated by two elements $a$ and $b$ such that $a^3 = b^3 =e$. How to determine all possible automorphisms of $G$?
0
votes
3answers
64 views

Cosets/ Cyclic group

Let $G=\langle a\rangle$ and $H=\langle a^2\rangle$. Find all the right cosets of $H$ in $G$. Additional info: I understand that a right coset of $H$ in $G$ is of the form $Ha=\{ha:h \in H\}$. But I ...
0
votes
0answers
24 views

Be $G$ a group of order $|G|=2p$ where $p$ is a prime number odd, prove that either $ G $ is cyclic or dihedral $G\simeq D_p$ the group of order 2$p$. [duplicate]

Be $G$ a group of order $|G|=2p$ where $p$ is a prime number odd, prove that either $ G $ is cyclic or dihedral $G\simeq D_p$ the group of order 2$p$. I thought the question a little difficult ...
0
votes
0answers
58 views

Free groups vs. free abelian groups

I'm trying to solve this question in page 74 of Hungerford's book: A free abelian group is a free group (Section I.9) if and only if it is cyclic. I have no idea how to proceed, a solution or a ...
4
votes
5answers
322 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
37 views

Why $(\mathbb{Z}/N\mathbb{Z})^{\times}[2]$ is of order $2$?

Why if $(\mathbb{Z}/N\mathbb{Z})^{\times}$ is cyclic, the group of his elements of order dividing $2$ is of order 2?
2
votes
3answers
45 views

Find a generator of the multiplicative group of $\mathbb{Z}/23\mathbb{Z}$ as a cyclic group

I need to find a generator of the multiplicative group of $\mathbb{Z}/23\mathbb{Z}$ as a cyclic group. Since $\mathbb{Z}/23\mathbb{Z}$ only has $23$ elements and ord$(x)$ where $x$ is a generator ...
1
vote
1answer
55 views

Show that a subset of the $2 \times 2$ matrices is an infinite cyclic group

Let $M$ Denote the set of 2x2-matrices of the form $$A=\pmatrix{1&m\\0&1}$$ where the entries are integers. Show that $M$, with respect to matrix multiplicaation, is an infinite cyclic ...
2
votes
1answer
32 views

Matrix Groups in Abstract Algebra

QUESTION: Let $h= \begin{pmatrix} -1 & 1\\-1&0 \end{pmatrix} \in GL_2(\Bbb R)$. Find $\langle h\rangle$. I'm stuck on the solution, but here is what I have: Let $h=\begin{pmatrix} ...
0
votes
3answers
45 views

Algebra I: Cyclic Generators

The direct product $\mathbb{Z}_{45} \times \mathbb{Z}_{98}$ is cyclic and isomorphic to $\mathbb{Z}_{4410}$ because $gcd(45,98)=1$; furthermore the element $n=([1]_{45},[1]_{98})$ is a cyclic ...
1
vote
1answer
30 views

computing the order of an element in a semidirect product and the center

$\newcommand{\lcm}{\text{lcm}}\newcommand{\Aut}{\text{Aut}}$Let $G=C_9\rtimes_\phi C_6$, where $C_9=\langle x:x^9=1\rangle$ and $C_6=\langle y:y^6=1\rangle$, and $\phi:C_6\mapsto \Aut(C_9)$ is defined ...
4
votes
2answers
41 views

What is this notation? Cyclic group $\mathbb{Z}^*_8$

$\mathbb{Z}^*_8$ As I understand it - $\mathbb{Z}_8$ is the group of integers under addition modulo 8. So am I correct in thinking its elements are: $\{0,1,2,3,4,5,6,7\}$? I thought the $*$ meant ...
1
vote
2answers
71 views

Why define $\gcd(r,s)$ as a positive generator $d$ of the cyclic group $H=\{nr+ms|n,m\in\mathbb{Z}\}$?

This is in regards to definition 6.8, p. 62 from Fraleigh's "A first course in abstract algebra". 6.8 Definition Let $r$ and $s$ be two positive integers. The positive generator $d$ of the ...
0
votes
1answer
34 views

Question about 3-cycles?

Assume $n \ge 5$ and let $ H \triangleleft S_n$ be a normal subgroup. If $H$ contains at least one $3$-cycle, prove $H = S_n$ or $H = A_n$. I have no idea how to do this? I think i need to use the ...
2
votes
3answers
85 views

Any group of prime order is cyclic - Proof blueprint [Fraleigh p. 100 Cory 10.11] [closed]

Not querying the proof or formality. I include only part of the proof. The order of the group is a prime number. Call it p. Hence by means of the definition of prime number, $p > 1$. Since the ...
1
vote
2answers
22 views

if $f$ is of prime order, then can the orbit of $s$ under $f$ have one element?

if $f\in A(S)$ has order $p$, $p$ a prime, show that for every $s\in S$ the orbit of $s$ under $f$ has one or $p$ elements.* Since the cyclic group generated by $f$ has $p$ elements, therefore ...
1
vote
1answer
54 views

How to count generators for a cyclic group

Show that there are $\varphi (n)$ generators for a cyclic group $G$ of order $n$. Give their form explicitly. Here $\varphi (n)$ is the Euler's function. I don't know what to do, please help.
1
vote
1answer
54 views

Finding homomorphisms from $\mathbb Z_{12}$ to $\mathbb Z_{6}$.

Find all homomorphisms from $\mathbb Z_{12}$, the cyclic group of order $12$, to $\mathbb Z_6$. For each homomorphism $f\colon \mathbb Z_{12}\to \mathbb Z_6$, determine the kernel $\ker(f)$ and the ...
4
votes
0answers
32 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
34 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
89 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
73 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 = ...
2
votes
2answers
91 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
24 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
47 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
53 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
31 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
47 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
98 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
37 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 ...
2
votes
1answer
121 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
42 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
32 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 $2$?
3
votes
2answers
68 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
56 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
59 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
60 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 ...
8
votes
5answers
243 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
70 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
107 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
181 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
140 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
54 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
169 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?
4
votes
1answer
47 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
43 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
46 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 ...