# Questions tagged [cyclic-groups]

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

4
questions

**6**

votes

**2**answers

782 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 ...

**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

**2**answers

199 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 ...

**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 ...