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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1answer
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Find a coset $f (= f + 22\Bbb Z)$ so that $(\Bbb Z/22\Bbb Z)^\times=U_{22} = \langle f\rangle$.

Find a coset $[f]=f+22\Bbb Z\in \Bbb Z/22\Bbb Z$ such that the units, $(\Bbb Z/22\Bbb Z)^\times)=U_{22} = \langle f\rangle$ are generated by $f$. I am unsure of how to go about this.
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2answers
157 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
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1answer
41 views

Prove that the cyclic group of order 3 is a group with a proof and justification [closed]

I have the Cayley table. Just need help proving why it is cyclic. The operation on G = {e; x; x2} (which I'll denote as o) is a binary operation, which is to say for a, b ∈ G , a o b ∈ G . The ...
1
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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
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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 $G$?
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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 ...
2
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0answers
69 views

Let $F$ be a field. If $F^*$ is a cyclic group, must $F$ be a finite field? [duplicate]

Let $F$ be a field. If $F^*$ is a cyclic group, must $F$ be a finite field? (It's well-known that if $F$ is a finite field, $F^*$ is a cyclic group). Thank you in advanced.
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1answer
31 views

Discrete math Group - Isomorphism and Automorphism

Let G be a Cyclic group Prove or disprove: A.let $ a,b \in G \quad $ so the function $ f:G \to G,f(a^k) = b^k$ is Automorphism of G(which means G is Isomorphism to herself) B.let a,b generators ...
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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 ...
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6answers
70 views

A question on cyclic group

I have no trouble proving the following statement. Let $G$ be a cyclic group of order $n$ and let $k$ be an integer relatively prime to $n$. Prove the map $x \mapsto x^k$ is surjective. It is clear by ...
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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 ...
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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?
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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 ...
0
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1answer
25 views

Why is G a cycle? Graph G with n vertices, m edges. “(∀v ∈ V(G) : δ(v) = 2 ⇒ G is Hamiltonian) is true because G is a cycle”.

I'm reviewing an earlier exam with solutions by the professor. I found this: Problem: Let G denote a simple, connected graph with n vertices and m edges. Translate the following statement to ...
1
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1answer
21 views

for cyclic permutation $g=(i_1i_2\cdots i_p)$, prove $\operatorname{sign}(g)=(-1)^{(p-1)}$

I'm reading "A course in algebra" by E. B. Vinberg for a basic understanding. Now I met a problem in Exercise 4.99: Deduce the following formula for the sign of a cyclic permutation: ...
1
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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 ...
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1answer
43 views

Rapid easy question on cyclic groups

If a cyclic group $C=\langle c\rangle$ has an involution $z$, then $C$ is finite $C$ has even order the involution is unique If $C$ was not finite, then it must be isomorphic to $(\mathbb Z,+)$ ...
2
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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} ...
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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 ...
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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 ...
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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 ...
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6answers
85 views

how to find a generator of a cyclic group

A cyclic group is a group that is generated by a single element. That means that there exists an element g, say, such that every other element of the group can be written as a power of g. This ...
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0answers
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Cyclic Factor group abelian proof [duplicate]

Show that if G is nonabelian, then the factor group G/Z(G) is not cyclic. I started to prove this via contrapositive. If G/Z(G) is cyclic, then G is abelian. I'm messing around with elements and ...
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0answers
40 views

Describe up to isomorphism the semidirect product

I have the following problem: I have to describe up to isomorphism the semidirect product $C_6 \rtimes C_2$, where $C_6$ denotes cyclic group of order six. I think I have to use external semidirect ...
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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 ...
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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 ...
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3answers
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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 ...
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2answers
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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 ...
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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.
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1answer
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Abstract Algebra: prove it is cyclic

I have question in referring to below link. Question. Suppose if I have [M:K]=2 and I know that K is subset of M. M:=$\mathbb{Z}_2[x]/(f(x))$ where f(x)=x$^4$+x+1. Then how this will be cyclic? I ...
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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 ...
0
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1answer
34 views

Direct product of groups is cyclic or not?

Let $\Bbb Z$ be the additive group of integers and $S = \{-1,1\}$ be a group under multiplication. Is the product $\Bbb Z \times S$ cyclic? Why or why not? I am really confused on this question ...
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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 ...
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1answer
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$n^2$ 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 ...
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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 ...
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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 ...
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2answers
50 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 ...
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2answers
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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 = ...
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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} ...
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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
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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
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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
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1answer
34 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
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1answer
69 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
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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 ...
2
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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
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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 $d=\gcd(m,n)$ ...
0
votes
1answer
22 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 ...