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
44 views

Order and generators of intersection of cyclic groups

Let $\sigma$, $\tau$ be two permutations of $S_n$. We know that $\langle \sigma \rangle \cap \langle \tau \rangle$ is a cyclic subgroup and (from Lagrange's Theorem) that its order divides ...
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2answers
14 views

Multiplying elements of a multiplicative cyclic group

This is about correct operation in a multiplicative cyclic group. Say we have the group $\mathbb Z_4^{\times} = \{1,2,3\}$, if we multiply the elements, say $3\cdot 2\cdot 2 $ we get $12 \mod 4 = 0$ ...
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1answer
29 views

Subgroups of $S_{10}$ that contain $\langle \sigma = (123)(456)\rangle$

In $S_{10}$ let us consider the permutation $\sigma = (123)(456)$ and the cyclic subgroup $ G = \langle \sigma \rangle$. I can fairly easily find that $$\langle \alpha = (1,4,2,5,3,6)(7,8) \rangle ...
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1answer
34 views

$a^{(p-1)/n}=1$ implies $b^n=a$ for some $b\in\mathbb F_p$? [on hold]

Let $p$ be a prime. Let $n$ be a positive integer dividing $p-1$. Suppose that $a^{(p-1)/n}=1$ in the finite field $\mathbb F_p$ with exactly $p$ elements. Then does there exist $b$ with ...
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0answers
20 views

How to parametrize a cyclic quadrilateral

In hopes of parametrizing some variables (either lengths, or angles, or both) of a cyclic quadrilateral, I was looking for a rule, or a set of rules defining a cyclic quadrilateral in terms of these ...
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1answer
40 views

Can anyone explain how primitive roots work?

Right now I'm studying out of Audrey Terras' book Fourier Analysis on Finite Groups and Applications and we're on the section where we're talking about $(\mathbb{Z}/n\mathbb{Z})^*$ and when this group ...
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1answer
54 views

Alternative proof that every subgroup of a cyclic group is cyclic

I've seen the 'standard' proof where we essentially construct a generator and use the division method. I was thinking through the problem attempting to do this using isomorphism's and was looking for ...
3
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1answer
47 views

If the number of elements in a group $G$ of order $289$ is $n\geq 273$ then $G$ is not cyclic.

Let $G$ a finite group and $n$ the number of elements in $G$ of order $289$. Show that if $n\geq 273$ then $G$ is not cyclic.
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2answers
31 views

Is there a cyclic group such that is isomorphic to Z∗16? [closed]

How do I find an additive group modulo $n$ $Z_n$ which is isomorphic to $Z^*_{16}$ (group with multiplication modulo 16)?
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1answer
98 views

How to partition $nk$ objects $\frac{1}{n}\binom{nk}{k}$ times, each time making subsets of size $k$, so that no combination of $k$ is repeated.

What is an algorithm to partition $nk$ objects a total of $\frac{1}{n}\binom{nk}{k}$ times, each time making subsets of size exactly $k$, so that no subset of size $k$ is ever repeated? For example, ...
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2answers
79 views

Show that there is precisely one cyclic code C of length 4 and dimension 2. Write down all the codewords in C.

I have shown there is one cyclic code, put not sure how to calculate the codewords in C. I think that the generator matrix is \begin{bmatrix}1&0&1&0\\0&1&0&1\end{bmatrix} but ...
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0answers
31 views

Show that there is precisely one cyclic code C of length 4 and dimension 2.

So far I have calculated the unique factorization of $x^4-1$ to be $(x-1)(x+1)(x^2+1)$ but am unsure of where to go next. Would the generator matrix then be of the form ...
2
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1answer
34 views

Find the structure of $\mathbb{Z}_{120}^*$

How to find the structure (in term of cyclic groups) of $\mathbb{Z}_{120}^*$? I know that the number of elements of $\mathbb{Z}_{120}^*= \phi(120) = 32 = 2^5$ But then, any hints?
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1answer
26 views

Questions of a completely reducible module

Please help to deal with the tasks of: $1)$ Which cyclic groups are completely reducible as a $\mathbb Z$-modules? $2)$ Which cyclic modules are completely reducible over the ring $\mathbb F[x]$, ...
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1answer
29 views

a cyclic nonsimple module over $\mathbb{Z}$

Let $M$ be a module over $\mathbb{Z}$. Prove that: a.$M$ is a cyclic module if and only if it's a cyclic group. b.$M$ is simple if and only if it's cyclic of prime order. I tried ...
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1answer
11 views

Invariants of a finite abelian group written as a central extension of a cyclic group by a finite abelian group.

Notation : If $A$ is a finite abelian group then $(d_r,...,d_1)$ are the invariants of $A$ if $d_r>1$ : $$A\text{ is isomorphic to } \mathbb{Z}/d_r\times...\times \mathbb{Z}/d_1 \text{ and } ...
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0answers
17 views

Distribution of numbers from a cyclic group $Z_p$

Let assume, that I have a group of prime order $p-1$ denoted as $Z_p$. From this group I generate randomly two sets of numbers: $\{x_1, x_2, x_3, x_4, x_5\}$ and $\{y_1, y_2, y_3, y_4, y_5\}$ and ...
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1answer
17 views

Any subgroup $H$ (or ideal) in $\mathbb{Z}$ is of the form $(m) = m\mathbb{Z}$ for some $m\in\mathbb{Z}$

Any subgroup $H$ (or ideal) in $\mathbb{Z}$ is of the form $(m) = m\mathbb{Z}$ for some $m\in\mathbb{Z}$. Proof: Suppose $H=\{0\}$, then $H=(0)$. Now suppose $H\not=\{0\}$. Then there are ...
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3answers
47 views

Let G be a cyclic group of order 24 then what is the total number of isomorphism ofG onto itself ?? [closed]

Let G be a cyclic group of order 24 then what is the total number of isomorphism ofG onto itself ??
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1answer
39 views

group of order 27 must have a subgroup of order 3 [closed]

How to prove a group of order 27 must have a subgroup of order 3
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6answers
93 views

Show that any cyclic group of even order has exactly one element of order $2$

Show that any cyclic group of even order has exactly one element of order $2$ Attempt: Let $G$ be a finite group of even order. Assuming a set $A=\lbrace g \in G \vert g \neq g^{-1} \rbrace$, I ...
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1answer
29 views

Number of colorings under cyclic permutation.

Given $\lambda\vdash n$. How many ways to color $n$ beads of chaplet into $l$ colors, such that $\lambda_1$ of $1^{st}$ color, $\lambda_2$ of $2^{nd}$ color, etc. For, examples if $\lambda=(3,2)$, ...
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1answer
24 views

Prove that the presentation of a cyclic group of infinite order is $\langle a\mid \rangle$

Essentially, i need this for a third year mathematics project and originally i thought i just needed to have something like this: The group with this presentation is explicitly realized by the set of ...
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0answers
30 views

If $\alpha\approx\sqrt p$ and $\beta\approx\log p$, is $\alpha\beta^{-1}\bmod p\approx p$ with probability $1-o(1)$?

Given $p$ a prime and a random $\alpha\in\Bbb Z_p$ with $\alpha\approx\sqrt{p}$ suppose we pick a random $\beta\approx\log p$ then what is the probability that remainder $\alpha\beta^{-1}\bmod p$ is ...
3
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3answers
51 views

elementary proof on cyclic group

Let $\xi,\gamma \in S_n$ Let $$\xi=(a_1 a_2 ... a_k)(b_1 b_2... b_l)...(g_1 g_2... g_q)$$ (disjunct cycle). Prove that $$\gamma\xi\gamma^{-1}=\big((\gamma(a_1) \gamma(a_2) ... ...
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0answers
23 views

Isomorphism between matrix groups and multiplicative cyclic group of finite field

Suppose $\mathbb{F}_{p^n}$ is a finite field and $\mathbb{F}^*_{p^n}$ is the set of all non zero elements in $\mathbb{F}_{p^n}$. The set $\mathbb{F}^*_{p^n}$ is a cyclic multiplicative group. In ...
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1answer
29 views

A problem in Abelian p-group being indecomposable

I have recently met this very interesting problem in my Group theory course: Let $ p $ be a prime number and $ 1 \leq n $ is a natural number such that G is the Abelian p-group $ G = ...
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2answers
82 views

A finite abelian group $A$ is cyclic iff for each $n \in \Bbb{N}$, $\#\{a \in A : na = 0\}\le n$

Let $A$ be a finite abelian group. Prove that $A$ is cyclic iff for each $n \in \Bbb{N}$ $$\#\{a \in A : na = 0\}\le n.$$ Any help or hint will be appreciated.
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3answers
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Disproof of the Reversion of the Fundamental Theorem of Cyclic Groups

So the Fundamental Theorem of Cyclic Groups states that Every subgroup of a cyclic group is cyclic. In symbolic logic, we can state this as $A: G\text{ is cyclic}$ $P: \forall ...
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0answers
46 views

Book Proof Problem - nth roots of unity are of the form $z=\text{cis}(\frac{2k\pi}{n})$

So I found this theorem and proof in my abstract algebra book in the section about cyclic groups. Here it goes: Theorem: If $z^n=1$, then the nth roots of unity are ...
5
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2answers
58 views

How to do this without knowledge of cyclic groups?

I did the following exercise: Suppose $n$ is an even positive integer and $H$ is a subgroup of $\mathbb{Z}_n$ (integers mod n with addition). Prove that either every member of $H$ is even or ...
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2answers
36 views

Does this mean that there are only 2 possible groups structures for groups of order 6?

Does saying that "up to isomorphism, the groups of order 6 are the cyclic group $C_{6}$ and the dihedral group $D_{3}$" mean that there are only 2 possible groups structures for groups of order 6?
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21 views

Count pairwise integer modulo

Assume $N$ and $M=(N-1)/2$ are both prime. For any $L(L\leq N)$ different integers $1\leq i_1<i_2<\ldots<i_L \leq N$, denote $A_m(i_1,\ldots,i_L)$, $1\leq m \leq M$ to be the number of ...
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4answers
72 views

An infinite cyclic group has two generators. Does the cardinality of the infinite set matter?

I was reading in a book that an infinite cyclic group has exactly 2 generators. Now my question is, does the cardinality of the infinite set matter? If the set of the group is the natural numbers, or ...
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2answers
56 views

What is the no. of injective group homomorphisms from $\mathbb Z_m$ to $\mathbb Z_n$ ( when $m \le n$)?

What is the no. of injective group homomorphisms from $\mathbb Z_m$ to $\mathbb Z_n$ ( when $m \le n$) ? I know it should not exceed $\gcd(m,n)$ , but I cannot calculate the exact number . Please ...
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0answers
35 views

Right cosets of the cyclic subgroup in $A_5$

Here's a homework question in Abstract Algebra that I have some troubles solving. Given that permutation $\sigma=(1 2 4)(1 5 2)$, and $\sigma\subset A_5$. Find the number of right cosets of the ...
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1answer
47 views

A Proof Check for A Theorem About Cyclic Groups, A.A. Albert

I was working with A.A. Albert's Fundamental Concepts of Higher Algebra and noticed I was able to complete one of the exercises without actually using all the assumptions. I imagine there must be ...
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56 views

Cyclic Group permutations

I am working on the following exercise question: Consider the following construction of a “keyed” hash function from Katz & Lindell (ex. 7.22 (1st ed.)/ 8.21(2nd ed.)). Gen : On input 1n , ...
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0answers
22 views

How to prove $G$ is cyclic? [duplicate]

Given that $G$ is a finite abelian group, and for every prime $p$ that divides the order of $G$, there is a unique subgroup of order $p$. How can I prove that $G$ is cyclic?
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30 views

Colored beads on a loop

Suppose we have $p$ beads of $n$ different colors on a loop. $p$ is a prime number and we consider the loop to be the same if one is a rotation of the other. Then how many distinct beads are there? By ...
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1answer
32 views

How to write dihedral group in cycle notation?

Since each symmetry can be thought of as a permutation of the vertices, the elements of $D_n$ can be thought of as elements of $S_n$. So I'm wondering if there's a systematic way that we can always ...
0
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1answer
28 views

Number of cycles and sign

For an even $n \in \mathbb{N}$, the sign of $\tau \in S_n$ is 1 if and only if the number of disjoint cycles in $\tau$ is even? And it is 1 if and only if the number of disjoint cycles in $\tau$ is ...
2
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1answer
84 views

$G$ is a finite abelian group. For every prime $p$ that divides $|G|$, there is a unique subgroup of order $p$.

$G$ is a finite abelian group. Assume that for every prime $p$ that divides $|G|$, there is a unique subgroup of order $p$. I'd like to prove that $G$ is cyclic. I'm thinking about the approach of ...
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2answers
44 views

Cyclic group given a group action

I have been solving some old exam questions to prepare for my own exam, but I have been unable to solve the following question. I am unsure of where to start and would therefore like some hints on ...
0
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1answer
26 views

nth root of unity in a cyclic group $\mathbb{Z}_p^*$

Is there a specific set of steps that should be taken in order to the $n$-th root of unity in a cyclic group. To be more specific, I am trying to find the $8$th root of unity for $\mathbb{Z}_{17}^*$. ...
1
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1answer
47 views

units of group ring $\mathbb{Q}(G)$ when $G$ is infinite and cyclic

How would I be able to describe all units of the group ring $\mathbb{Q}(G)$ where $G$ is specifically an infinite cyclic group?
5
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3answers
189 views

Why group of order 6 has to have just two elements of order 3

In an attempt to prove that every group $G$ of order 6 is isomorphic to either $\mathbb{Z}_6$ or $S_3$, I stumbled upon one peculiar issue. We can use Cauchy's Theorem to argue that since ...
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2answers
60 views

Zero homomorphisms of cyclic groups [closed]

My question is the following : Let n ≥ 1 be an integer. Every homomorphism of groups $f$: ℤ/nℤ → ℤ is the zero homomorphism. a) Yes. b) No. c) Depends on n. I don't really know how to go ...
2
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1answer
23 views

Prove the group transitivity of alternating group $A_n \quad n>2$?

Does it not suffice to point out that $$(i, k)(i, j)\in A_n$$ The element at location $i$ is mapped to the element at location $j$ and and the element at location $j$ is mapped to some third ...
2
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2answers
112 views

Group of order $255$ is cyclic

Let $G$ a group and its order is $255$. Prove that $G$ is cyclic. I easily demonstrated that the group has only one $17$-Sylow subgroup $P$ that is normal in $G$ and it's cyclic since it is of ...