Questions tagged [cyclic-groups]

Question 1

For background, say that a centrifuge has $n$ slots arranged in a circle and $k$ tubes are placed within it. This is equivalent to choosing $k$ distinct $n$-th roots of unity. The centrifuge is ...

Question 2

I would like to know if it's possibile that a countable direct product of countable cyclic group could be even cyclic: here is proved that this could be for finite product (and so a finite direct sum, ...

Question 3

Consider an alphabet $\Sigma$ of size $k=|\Sigma|$ and the set of words $\Sigma^L$ of length $L$, with the natural action of the cyclic group $\mathbb{Z}_L$ with generator $\tau$ acting naturally as $\...

Question 4

Let $S_n$ be the symmetric group on $n$ elements, and let $n=r\times s$. I've come across a subgroup $H\le S_n$ whose (non-identity) elements are all the product of $s$ disjoint $r$-cycles. I want to ...

Question 5

Problem: Prove that all the roots of an irreducible polynomial over $\mathbb{Q}$ of degree 3 with a cyclic Galois group are real. I know this question has already been asked here, however, I want to ...

Question 6

Consider a subgroup $G$ of the permutation group $S_n$ generated by cycles $G=\langle c_1,\ldots, c_n\rangle$. I draw the points of the group as points in the plane and each cycle is drawn in a ...

Question 7

In Visual Group Theory by Nathan Carter (Exercise 10.30), the following question is posed: "Why can the group $C_4$ under addition not be made into a finite field by overlaying a multiplicative ...

Question 8

Consider the set $$ X = \{(x_1,\ldots,x_n,x_1+1,\ldots,x_n+1) : x_i \in \mathbb{F}_2 \text{ for all } 1 \leq i \leq n\} $$ so that the cardinality $|X| = 2^n$. Consider an action of $\mathbb{Z}/2n\...

Question 9

Describe the elements of order $2$ and $4$ in $Aut(\Bbb Z_{17})$. First we have that $Aut(\Bbb Z_{17})\cong U(17)=\{1,\ldots,16\}=\Bbb Z_{17}^{\ast}$. Then is same find elements of same order in that ...

Question 10

Let $p$ be a prime. The group $(Z/p\mathbb Z)^\times$ acts by automorphisms on $\mathbb Z/p\mathbb Z$, with orbits $\{0\}$ and $X:=\{1,\dots,p-1\}$. Let $\varphi$ be the action. This means that, for ...

Question 11

Let $p$ be a prime. Quite tautologically, $G=\operatorname{Aut}(C_p)$ acts by automorphisms on $\mathbb Z/p\mathbb Z$, with orbits $\{0\}$ and $X:=\{1,\dots,p-1\}$. Let $\varphi$ be the action. This ...

Question 12

Fix $n\in\mathbb{N}.$ Let $Z_n$ be the cyclic group of order $n.$ Say $Z_n=\langle x\rangle.$ Let $H≤Z_n.$ It is known that $H$ is cyclic. What are the generators of $H?$ There are two approaches I ...

Question 13

Let $G$ be a group of order $1443$. I want to show that $G'$ (the commutator subgroup of $G$) is cyclic. I attempted to analyze the structure of $G$ using its Sylow subgroups. Since $1443 = 3 \cdot 13 ...

Question 14

I'm a bit confused on a question from a previous exam. It goes as follows: Let $G$ be a finite group such as there is an epimorphism $\phi: G\to \mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_3$. ...

Question 15

I was reading Abstract Algebra by Dummit and Foote. There was a portion of text in the book( on Section 5.2, p-163) which I didn't understand. It went on like this: Proposition 6. Let $m, n \in \...

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Questions tagged [cyclic-groups]

For which $n$ and $k$ does there exist a "cursed" centrifuge arrangement?

Could a countable direct product of countable cyclic groups be cyclic? [closed]

Orbits of a cyclic group on words of length $L$ with alphabet of size $k$ [duplicate]

Subgroup of permutations whose elements are products of $k$-cycles

Prove that all the roots of an irreducible polynomial over $\mathbb{Q}$ of degree 3 with a cyclic Galois group are real

Determine visually if a finite group generated by cycles is cyclic. [closed]

Why can’t $C_4$, $C_6$, or $C_{15}$ be made into finite fields? [duplicate]

Calculating orbits of a $C_{2n}$ action.

Find elements of order $2$ and $4$ in $Aut(\Bbb Z_{17})$

Is there a graph-theoretic proof that $(\mathbb Z/5\mathbb Z)^\times$ is cyclic?

Inconsistency between a regular action by automorphisms and non-cyclicality

Generators of Subgroups of Cyclic Groups: $\gcd(n,b)=\gcd(n,a)$ vs $\gcd(t,d)=1.$

If $G$ is a group of order $1443$, must $G'$ be cyclic

Epimorphisms and Sylow groups

Help needed in understanding the proof of Proposition 6 in the book, "Abstract Algebra " by Dummit and Foote

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