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

1
vote
1answer
15 views

Rational group algebras and maximal orders

Let $G$ be a finite group, and $\mathbb{Q}[G]$ be the rational group algebra. Then the group ring $\mathbb{Z}[G]$ is an order in $\mathbb{Q}[G]$, but is not in general a maximal order. What can we ...
0
votes
0answers
22 views

Prove identity involving alternating groups

Prove the following identity: where $I_{A_n}(x_1,...,x_n)$ is a cyclic index the natural action of the alternating group $A_n$ on the set ${1,...,N}$ (assuming that $I_{A_0} = 1$).
0
votes
2answers
31 views

Cyclic groups: find power

Given the group $\mathbb Z_7$ and the generator $3$, we know the values generated are $a^0=3^0\pmod{7}=1$ $a^1=3^1\pmod{7}=3$ $a^2=3^2\pmod{7}=2$ $a^3=3^3\pmod{7}=6$ $a^4=3^4\pmod{7}=4$ ...
0
votes
0answers
30 views

If the automorphism group of a group is cyclic, then the group is commutative [duplicate]

Let $G$ be a group and the $Aut(G)$ group is cyclic $\Rightarrow$ the group $G$ is commutative. I looked at the homomorphism $\varphi : G \rightarrow Aut(G) \ g \mapsto (x \mapsto gxg^{-1})$. Let ...
1
vote
2answers
82 views

Show that $Z\times Z$ is not cyclic… [duplicate]

The full problem is as stated in the title. I am here to check if this is a valid proof. I thought it would be easiest using Linear Algebra. Recall that an infinite cyclic group is isomorphic to ...
1
vote
1answer
40 views

Show whether the following groups are cyclic or not…

Here's the full problem: Show whether the groups $G_{10},G_{7}$ are cyclic or not. If so, find their generators. $G_{10}$ and $G_7$ are the sets of invertible elements mod 10 and mod 7. So, I ...
1
vote
0answers
42 views

Prove that a group is non-cyclic

I have some problems with an exercise of group theory. It sounds like this: Let $G$ be a finite group of order $p^2q^2$ with $p$, $q$ prime numbers, $p$ is odd, $p<q$. I know that $G$ doesn't ...
2
votes
2answers
70 views

Cyclic subgroups of semidirect products

Let $H$ and $N$ be two finite groups and $\phi:H\to Aut(N)$ a homomorphism. Let $G=N\rtimes_\phi H$ and let $\pi:G\to G/N=H$ be the quotient map. Let $p$ be a prime which does not divide $|N|$ but ...
1
vote
2answers
66 views

What exactly does $\langle a,b \rangle$ mean? Where a and b are elements of a group G?

Does it mean either a or b will generate the whole of $\langle a,b \rangle$ or does it mean that some of the elements will be generated by a, some by b, and some by $a^rb^q$? The book I'm reading ...
2
votes
2answers
56 views

Are the only generators of a Cyclic Group $G=\langle g\rangle$, where $|g| = \infty$, $g$ and $g^{-1}$?

I'm self studying group theory, and this is a question in the textbook I've taken out, there is no answer given so I'm assuming that's because it's too simple to require one. I'm almost certain that ...
4
votes
3answers
99 views

Finding kernel of homomorphism $f:\mathbb Z \to S_8$ such that $f(1)=(1426)(257)$

Let $f:\mathbb Z \to S_8$ be a homomorphism such that $f(1)=(1426)(257)$ , then how to compute $\ker(f)$ and $f(20)$? I know that $f(n)=f^n(1)$ but this seems too tedious; please help
0
votes
2answers
74 views

Prove that if G is a cyclic group with more than 2 elements, then there always exists an isomorphism ϕ:G→G that is not the identity mapping [duplicate]

The full question is Prove that if G is a cyclic group with more than 2 elements, then there always exists an isomorphism ϕ:G→G that is not the identity mapping.I have no idea where to start?
3
votes
3answers
149 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?
-1
votes
3answers
57 views

$G$ is an abelian group of order a product of distinct primes $\implies G$ is cyclic?

If $G$ is an abelian group of order $p_1p_2...p_k$ , where $p_1,p_2,...,p_k$ are distinct primes , then is it true that $G$ is cyclic ?
4
votes
1answer
65 views

Another Presentation of Certain Cyclic Groups

Show that the the group with presentation $$\langle x, y\ \mid\ x^2=y^2x^2y,\ (xy^2)^2=yx^2, \ yx^{-1}y^2=x^n\rangle $$ is cyclic of order $3(n+1)$, for $n=0 \mod 3$ or $n= 1 \mod 3$, $n\ge 0$. This ...
-1
votes
1answer
21 views

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

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.
4
votes
2answers
204 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 ...
-1
votes
1answer
44 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
vote
1answer
68 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
66 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
votes
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.
0
votes
1answer
33 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 ...
0
votes
0answers
25 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 ...
2
votes
6answers
71 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 ...
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
330 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
47 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
votes
1answer
26 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
vote
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
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 ...
1
vote
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
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 ...
2
votes
6answers
97 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 ...
0
votes
0answers
14 views

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 ...
1
vote
0answers
42 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 ...
1
vote
2answers
76 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
55 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.
0
votes
1answer
35 views

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 ...
1
vote
1answer
55 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
votes
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 ...
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 ...