Questions tagged [cyclic-groups]

Use with the (group-theory) tag. A group is cyclic if it can be generated by a single element. That is to say, every element in a cyclic group can be written as some specified element to a power.

Filter by
Sorted by
Tagged with
2
votes
1answer
23 views

Transforming a permutation into a rotation

Let $n\geq 3$ and $G=C_{n}=\{1,r,...,r^{n-1}\}$ be the cyclic group of $n$ elements where $r$ is the rotation of $360/n$ degrees. Here, let us consider a vector $x\in\mathbb{R}^{n}$ as consisting of ...
0
votes
0answers
29 views

Quotient C8 by C2 group

I attempt to divide a C8 by C2. My reasoning: C2 is a normal subgroup of C8. C2 forms 2 cosets: {0,2,4,6} and 1+{0,2,4,6}. C8/C2 isomorphic to C4. But I know that C2*C4 is an Abelian group. It is ...
1
vote
1answer
44 views

Automorphism of commutative groups.

For every group G there is a natural group homomorphism G → Aut(G) whose image is the group Inn(G) of inner automorphisms and whose kernel is the center of G. Thus, if G has trivial center it can ...
-1
votes
1answer
26 views

Cardinality of the kernel of multiplication map

Let $m$ and $n$ be positive integers and let $F : \mathbb{Z}_n \to \mathbb{Z}_n$ be given by $F(\overline{x}) = m \overline{x} = \overline{mx}$. I am struggling to show the fact that $\ker F$ has ...
-1
votes
2answers
32 views

Working out subgroups of cyclic groups [closed]

If I have cyclic groups: $X = C_3 = \langle x\rangle$ and $Y = C_2 =\langle y \rangle$ , with $x$ being of order 3 and $y$ being of order 2. How would I work out the subgroups of $X \times Y$ And if ...
0
votes
1answer
25 views

Groups of units of cyclic groups of order $2^n$ [duplicate]

In Dummit & Foote's Abstract Algebra (2nd ed.), Corollary 20(2) states the following: $(\mathbb{Z}/2^n\mathbb{Z})^\times$ is the direct product of a cyclic group of order 2 and a cyclic group ...
0
votes
0answers
21 views

Inverse of modulo sum of elements from a cyclical multiplicative group without descrete log

So the question: Given a hash function $$h(x) = \sum_{i=1}^n g^{x_i} \mod p$$ where p is a prime number and g is a generator of the multiplicative cyclic group $\mathbb{Z}_p^×$. n is the length of the ...
3
votes
1answer
47 views

If $G$ is an Abelian group and contains cyclic subgroups of order 4 and 6, what other sizes of cyclic group G must contain?

Let $G$ be an abelian group. Let $a,b \in G$ such that $|a|=4, |b|=6$. $(ab)^{24}=a^{24}b^{24}=e$, identity of G. Hence $|ab|$ can be $1,2,3,4,6,8,12,24$ If $|ab|=8$ then $(ab)^8=e\implies e=a^8b^8=...
1
vote
4answers
60 views

Let p be a prime. If a group has more than $p-1$ elements of order $p$, then prove that the group can't be cyclic.

Result: Let G be a group which has more than $p-1$ elements of order $p$. I need to prove that such a group can't be cylic. $p$ is a prime number. Let's consider the case when G is finite. I want ...
3
votes
1answer
52 views

Unit roots group is isomorphic to $\Bbb{Q}/\Bbb{Z}\left[\frac{1}{p}\right]$ in a field of characteristic $p\ge0$

Let $K$ be a field so that the group of all unit roots of all orders $\mu_\infty=\bigcup_n {\mu_n}$ (where $\mu_n=\{x\in K\mid x^n=1\}$) splits on $K$. If $K$ is of characteristic $0$, take $p=1$; ...
0
votes
3answers
28 views

Finding all the maximal ideals of $\mathbb{Z}_{63}$

How do I go about finding all the maximal ideals of this ring ? I realise that all ideals are subgroups with respect to addition. Therefore, since $\mathbb{Z}_{63}$ is cyclic then every subgroup, and ...
1
vote
0answers
39 views

Let $G$ be a non-nilpotent group where all the non-normal abelian subgroups of $G$ are cyclic. Then $G$ has cyclic center.

Theorem : Let $G$ be a non-nilpotent group such that all the non-normal abelian subgroups of $G$ are cyclic. Then $G$ has cyclic center. Proof. Suppose that $Z(G)$ is non-cyclic. since $G$ is non-...
0
votes
1answer
30 views

Number of elements in the cyclic group

I recently came across this question: Find the number of elements in the following cyclic group: The cyclic sub-group of $C^{*}$ generated by 1+i The solution says the answer is: O(Z), where ...
-1
votes
1answer
21 views

If teo subgroups are cyclic, the groups is cyclic? [closed]

i've got a group $G$, and its subgroup $L$, L is normal in G. If L and G/L are cyclic, is G cyclic? I know that in the other direction is true but what about this? i have no idea
2
votes
2answers
83 views

Element of a cyclic group of even order has two square roots?

The problem Im struggling with is "Let $G$ be a cyclic group of order $d$. If $d$ is even, then each element of $G$ has either two or zero square roots in $G$" Does this mean I need to find a cyclic ...
0
votes
0answers
27 views

Intuition behind Order of an element of finite cyclic group.

Let $\color{brown}{C}$ be a finite cyclic group, and $\color{brown}{a\in C}$ , such that $\color{brown}{|a|=n}$, then $$\color{green}{|a^k|=\frac{n}{gcd(k,n)}=\frac{lcm(k,n)}{k}}$$How to intuitively ...
0
votes
1answer
34 views

$G$ a group of order 210. If $G$ is commutative show it is cyclic. If $G$ is any group show it contains a subgroup of index 2.

$G$ a group of order 210. Part 1 If $G$ is commutative show it is cyclic. Consider the prime factorization $210=(2)(3)(5)(7)$. By Sylow's theory we know that Sylow subgroups exist of order 2, 3, 5, ...
0
votes
1answer
61 views

Characterizing Generators of $\mathbb{Z}_p^*$

I have started to study cyclic groups and generator. I can't prove that $g\in \mathbb{Z}_{p}^{*}$ is a generator for cyclic group $\mathbb{Z}_{p}^{*}$ if and only if $g^{\frac{p-1}{q_{i}}}\neq 1\;\...
0
votes
1answer
29 views

Prove that $M_{16} / Z(M_{16}) \cong C_2 \times C_2$

I am trying to answer the following question: Prove that $M_{16} / Z(M_{16}) \cong C_2 \times C_2$. Where $M_{16}=\langle x, y | x^{8}=1, y^{2}=1, yx=x^{5}y\rangle$ is the modular group of order ...
0
votes
1answer
22 views

all possible combinations of $a_j$s for $\sum_{j=1}^na_j\epsilon^{j}=0$ where $\epsilon=e^{\frac{2\pi i}{n}}$

Let $n\geq 3$ and $d$ be a divisor of $n$. Let us consider the set $\{1,\epsilon,\dots,\epsilon^{n-1}\}$ where $\epsilon=e^{\frac{2\pi i}{n}}$. Can we classify all possible cases for $d$ distinct ...
0
votes
2answers
48 views

What, if anything, do permutations have to do with order?

My course notes have: $$ \text{"A bijective function } f:X\rightarrow X \text{ is called a } permutation \text{ of } X" \tag{0}. $$ So let $$A=\{1,2,3\} \tag{1.1},$$ $$f:A\rightarrow A \text{ be a ...
0
votes
0answers
9 views

How to prove that square of a permutation in a cyclic form is even [duplicate]

I want the proof for the above result I tried to prove by using proof by cases where the cycle is an n-cycle and what if n is even or odd
1
vote
2answers
32 views

Using Definition of Cyclic Group to prove B is a Subgroup

Given the Dihedral group $ D_4 $ (that is where $ D_4 = $ { $ id, R, R^{2}, R^{3}, F, RF, R^{2}F, R^{3}F $} ); Let $B =$ {$id, RF$} I now wish to prove that $B$ is a subgroup of $D_4$: Note that $B =...
0
votes
1answer
50 views

Can I write any element from a cyclic group as a power of other any element from that cyclic group?

I've got G, a cyclic group, and x and y two element from G. Let's suppose that o(x)|o(y). Can we make sure that x is a power of y? (I mean, can we make sure that we can write x=y^(something) ?) I need ...
1
vote
1answer
17 views

Elements of an specific order in a non cyclic non abelian group

I've got a non cyclic non abelian group G = < (1 2 3 4 5),(2 5)(3 4) > which is a subgroups of S_5 and the order of G is 10. And my question is: Is there any element of order 4 in G? What I have ...
2
votes
2answers
54 views

The subgroups of a cyclic group

We've got $G=U(\mathbb Z/(27)\mathbb Z)=\langle 2 \rangle$ a cyclic group, and $H=\langle -8, -1 \rangle$ a subgroup of $G$. I've calculated all the subgroups of $G$. Now I have to indentify $H$ with ...
1
vote
1answer
53 views

Proof regarding a subgroup of a cyclic group.

I am practicing Abstract Algebra using course resources from when it was taught at my university last year. I am confused, in part (b) as to how $x^a=e$. Could someone explain this?
1
vote
3answers
55 views

How do I show that $G$ of order $p^n$ is cyclic if and only if it is abelian with a unique subgroup of order $p$?

I spent quite some time thinking about it but I don't see why must be $x^p\in\left\langle a\right\rangle$ in the given conditions. Can someone please explain?
0
votes
0answers
29 views

Showing that there exists an automorphism of a cyclic group mapping a generator to another.

I'm currently going through Lang's Algebra, I've only recently started it, and I'm trying to do every proof left as an exercise. One such question is the following : Let $G$ be a cyclic group, and ...
5
votes
1answer
93 views

Group Isomorphism Question

In this pdf (https://kconrad.math.uconn.edu/blurbs/gradnumthy/classgroupKronecker.pdf) the author claims that the ideal class group of the ring of integers of $\mathbb{Q[\sqrt{-199}]}$ is the cyclic ...
0
votes
1answer
33 views

Number of ring automorhism of $\mathbb{Z}_4$?

What are Ring automorphism of $\mathbb{Z}_4 ?$ As per the theory, since it's a cyclic group, it depends on the image of the generating element 1. so there are two possiblities, $\phi(1)=1, \phi(1)=3$ ...
1
vote
1answer
45 views

Which of the following are true about the group $\Bbb{Z}_6×\Bbb{Z}_9×\Bbb{Z}_{15}/\langle(5,5,3)\rangle$?

Let, $\displaystyle{G=\frac{\Bbb{Z}_6\times\Bbb{Z}_9\times\Bbb{Z}_{15}}{\langle(5,5,3)\rangle}}$ Which of the following are ture about $G$? The given options are- (a) $G$ is cyclic. (b) $G$ is Abelian....
0
votes
2answers
26 views

What's the relationship of $e_G$ to $G \times G$?

Context: 1st year Mathematics BSc. Let \begin{align} & G = \langle g \rangle \text{ be a cyclic group of order } n, \tag{0.1}\\ & e_1 \text{ be the neutral element of } G, \tag{0.2}\\ & ...
0
votes
0answers
11 views

One-dimensional representations of a cyclic group over a finite field

I need to describe all the representation of $\mathbb{Z}/ n \mathbb{Z} $ over $\mathbb{F}_q$ of the dimension 1. It is clear that the generator $1 \in \mathbb{Z}/ n \mathbb{Z} $ acts on $\mathbb{F}...
0
votes
2answers
61 views

Work out all the automorphisms of $C_7$. Now using a semidirect product, construct a nonabelian group of order 21

I know that ${\rm Aut}(C_7$) is $C_6$, and ${\rm Aut}(C_6$)= $C_4 \times C_2$ but I do not know if that will help. Any hints or tips welcome!
-1
votes
1answer
42 views

symmetric groups and cyclic groups [closed]

Are all the symmetric groups cyclic groups? I know that by definition a cyclic group is a group that is generated by a single element. But if I've got a symmetric group like for example $H=\langle(2\ ...
1
vote
1answer
52 views

Prove if there is a group whose order is $p^2$,and it is non-abelian, then it is cyclic [closed]

Suppose the non-abelian group G whose order is $p^2$ where p is a prime number, prove it is a cyclic group. My work: there is a $\tau\not=e$ in the group and the order of $\tau$ is either $p$ or $p^...
1
vote
2answers
43 views

Question on Lagrange theorem

Find order of $H$ if $H$ is subgroup of some group of order $100$ and $H$ contains no element of order $2$, with $H$ non cyclic The ans given in a book is $25$. I can understand that $\...
2
votes
3answers
36 views

$H$ prime order implies $G/H$ cyclic?

I am revising some concepts of group theory, but I must admit that it's been a while and I have forgotten quite a bit. I want to show that if $H$ is a normal subgroup of $G$, where $|H| = p$ ($p$ ...
-1
votes
1answer
28 views

How do I find out the orders of $G^k$ and $G_k$? [closed]

Suppose that $k\in\mathbb Z,$ how do I find out the orders of $G^k=\{x^k\in G\,|\,x\in G \}$ and $G_k=\{x\in G\,|\,x^k=1\}$ where G is a cyclic group of order $n$?
2
votes
2answers
56 views

Preimage of cyclic subgroup under projection is Abelian

Let $G$ be a finite non-Abelian group, let $Z(G)$ denote its centre, let $\pi: G \to G/Z(G)$ be the canonical projection, and let $H \vartriangleleft G/Z(G)$ be a cyclic normal subgroup of the ...
1
vote
3answers
48 views

Fundamental Theorem of Cyclic Groups

When proving that every subgroup of a cyclic group is cyclic. Let $G = \langle a \rangle$ and suppose that $H$ is a subgroup of $G$ and assume that $H \ne \{e\} $. The author begins with the ...
2
votes
2answers
57 views

Infinite Cyclic Group of Integers

I am new to group theory. While reading about cyclic groups, according to my understanding, A Cyclic group has a generator that generates all other elements by several copies of it. Now coming to set ...
0
votes
1answer
36 views

I need to find all subgroups of $ G $= {1, -1, i, -i}, with $G \leq$ ($ \mathbb {C}$- {0}, $\cdot$)

If you can guide me with this exercise, to be able to do other similar ones, it would be of great help to me.
2
votes
2answers
39 views

Structure of ideals in $\mathbb{F}_q[G]$

Let $\mathbb{F}$ be a finite field of characteristic $p$ and let $G$ be a cyclic group of order $p^n$. I read in a paper that all ideals of the group ring $\mathbb{F}[G]$ are of the form $I_n^j$ where ...
1
vote
1answer
66 views

subgroup of cyclic and their order

I understand how any subgroups are cyclic and there is a subgroup for each divisor of $d$ Let's say for example Let $d$ be a divisor of $n=|G|$. Consider $H=\{ x \in G : x^d =e\}$. Then $H$ is a ...
-3
votes
1answer
41 views

Help me to prove that $(\mathbb R,+)$ is not cyclic? [closed]

Can you help me to prove $(\mathbb R,+)$ is not cyclic?
1
vote
1answer
38 views

Order of cyclic subgroups in $\mathbb{Z}/n\mathbb{Z}$

I'm interested in efficiently computing minimal $x$, s.t. $a^x \equiv 1 \pmod{n}$, where $\gcd(a, n) = 1$. Let's denote order of cyclic multiplicative subgroup $\langle a\rangle$ in $\mathbb{Z}/n\...
1
vote
2answers
87 views

Number of elements of order $2$ in a group of order $10$.

Consider a group $G$ of order $10$. Then $G$ can be abelian as well non-abelian. What is the number of non-trivial elements of $G$ of order $2$? Answer: If $G$ is abelian, $G$ can be cyclic as ...
1
vote
1answer
35 views

Irreducible representation of finite Abelian group

I have seen that every finite Abelian group $G$ is isomorphic to a product of cyclic groups of prime power order, that is $$G = \mathbb{Z}_{p_1} \times \mathbb{Z}_{p_2} \times ... \times \mathbb{Z}_{...

1
2 3 4 5
33