# Questions tagged [cyclic-groups]

Questions about the branch of algebra that deals with cyclic groups.

13
questions

**4**

votes

**2**answers

203 views

### Do there exist general conditions for cyclicity of unit groups of quotient rings (generalizations of the primitive root theorem)?

Let $R$ by a commutative ring with $1$, and $I \subset R$ a non-zero integral ideal in $R$. When $R$ has finite quotients, and $I = P$ is prime in $R$, the group of units $(R/P)^{\times}$ of the ...

**8**

votes

**1**answer

344 views

### Classification of the functors on the category of cyclic groups

Let $\mathsf{Grp}$ be the category of groups and let $\mathsf{Cyc}$ be the subcategory of cyclic groups.
As seen in the posts here and there (and their answers), a functor $F: \mathsf{Cyc} \to \mathsf{...

**4**

votes

**0**answers

73 views

### Finite groups of cyclicality index $3$

Suppose $G$ is a group. Let’s define the cyclicality index of $G$ using the following recurrent relation:
$$CI(G) = \begin{cases} 1 & \quad G \text{ is cyclic} \\ \max_{H < G} CI(H) + 1 & \...

**0**

votes

**1**answer

124 views

### Number of cycles under a certain action on Z/nZ [closed]

Computer scientist here looking at a question that came about from in-place matrix transposition, but rusty on my abstract algebra and number theory...
Suppose we have the multiplicative group $\...

**1**

vote

**0**answers

100 views

### Structure of the group $ (\mathbb{F}_{2}[x]/(Q ^ { e }))^{*}$, where $ Q $ ie an irreducible polynomial over $\mathbb{F}_{2}$

Let $ Q $ be an irreducible polynomial over $\mathbb{F}_{2}$, can we find a decomposition of the group $ (\mathbb{F}_{2}[x]/(Q ^ { e }))^{*}$ into a direct product of cyclic groups ?.
We know (from $...

**3**

votes

**1**answer

186 views

### $P\in \mathbb{F}_{2}[x]$ for which $(\mathbb{F}_{2}[x]/(P))^{*}$ is a cyclic group

For $n \in \mathbb{N}$, we know that $(\mathbb{Z}/n\mathbb{Z})^{*}$ is a cyclic group if and only if $ n=2$, 4, $p^{k}$, or $2p^{k}$ for an odd prime number $p$.
Is there any known similar result for ...

**8**

votes

**1**answer

901 views

### Finite groups in which all proper subgroups are cyclic

Is there any classification of finite group in which all proper subgroups are cyclic?
Would you please tell me a reference?

**6**

votes

**1**answer

189 views

### Maximal cyclic quotient of a $p$-group

Let $G$ be a finite abelian $p$-group, $p$ a prime. I say that a pair $(G',\varphi)$ is a maximal cyclic quotient (please excuse me if this definition already exists and refers to a different concept) ...

**2**

votes

**0**answers

286 views

### Finitely generated subgroups are cyclic, and a generalization

Is there a name for groups $G$ which satisfy the property that for any $a$ and $b$ in $G$, there is a $c\in G$ and integers $n$ and $m$ such that $a=c^n$ and $b=c^m$? Such a group has to be abelian, ...

**6**

votes

**2**answers

783 views

### Cyclically symmetric functions

Where can I learn about the invariant theory associated with actions of cyclic groups (as opposed to symmetric groups)?
E.g., do the functions $x+y+z$, $xy+yz+zx$, and $x^2y+y^2z+z^2x$ generate the ...

**9**

votes

**1**answer

472 views

### Is the dual of the product of infinite cyclic groups a free abelian group ?

By a theorem of Specker, the group $\mathrm{Hom}(\prod_{\aleph_0} \mathbb{Z},\mathbb{Z})$ is isomorphic to $\bigoplus_{\aleph_0}\mathbb{Z}$ and is in particular a free abelian group. I wonder, if this ...

**6**

votes

**2**answers

3k views

### Example of an infinite abelian but non-cyclic group whose automorphism group is cyclic

Can anyone give me an example of:
An infinite abelian but non-cyclic group whose automorphism group is cyclic.

**4**

votes

**1**answer

352 views

### Fibered products of cyclic groups

Background
Let $m,n$ be positive integers and consider the cyclic group $\mathbb{Z}_{mn}$.
We have a natural epimorphism $\mathbb{Z}_{mn} \to \mathbb{Z}_n$ coming from the exact sequence
$$ 0 \to \...