# Questions tagged [cyclic-groups]

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

21
questions

4
votes

0
answers

126
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### Major indices of standard tableaux of shapes obtained from addable cells of a given Young diagram

I have a "very" indirect proof that the following fact is true for every Young diagram $\lambda \vdash n$ and every $r \in \{0,\dotsc,n\}$:
\begin{equation}
d_\lambda = \sum_{a \in \mathrm{...

3
votes

0
answers

92
views

### Conjugate actions and isomorphic Zappa–Szép products

Let $A$ and $G$ be two cyclic groups. Let $\alpha:G\rightarrow\operatorname{Bij}(A)$ and $\beta: A\rightarrow\operatorname{Bij}(G)$ be two group homomorphisms satisfying some conditions given in ...

9
votes

2
answers

522
views

### When are two semidirect products of two cyclic groups isomorphic

(I have posted this question in Math Stack Exchange, only to have received no answer.)
It is well known that a semidirect product of two cyclic groups $C_m$ and $C_n$ has the form
$$
C_m \rtimes_k C_n ...

0
votes

0
answers

68
views

### Fourier coefficient of close functions

Let $p$ be some prime. Let $\mathbb{Z}_p$ be the cyclic group of order $p$. Let $f, g \colon \mathbb{Z}_p \to \{\pm 1\}$ be two functions. Recall that the Fourier transform is defined as
$$ f(x) = \...

11
votes

2
answers

2k
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### Is Gauss's generalization of Wilson's theorem non-superficially related to the classification of moduli for which primitive roots exist?

Wilson's theorem (actually proven by Lagrange) from elementary number theory states that: If $n\ge 2$ is an integer, then
$$
(n-1)! \equiv
\begin{cases}
\hfill -1 \pmod {n} &\text{ if } n \...

4
votes

1
answer

253
views

### A question on a possible cyclic sieving phenomenon?

(This is an old MSE question from me, which did not get any answer, and when looking back seems interesting to post it here:)
Let $G$ be a finite group. Consider the set $X_G:=\cup_{H\le G} G/H$, ...

4
votes

0
answers

181
views

### An addition theorem for three functions similar to $\sin,\cos$ and $\sinh,\cosh$ and one / some questions?

Define the functions $t_k(x) = \sum_{n=0}^{\infty}{\frac{x^{3n+k}}{(3n+k)!}}$ for $k=0,1,2$. The functions then satisfy:
$
\begin{pmatrix}\exp(x) \\ \exp(\omega x) \\ \exp(\omega^2 x)\end{pmatrix} =
\...

2
votes

0
answers

63
views

### The number of elements with order less than $k$ in a larger cyclic group

I am working on a problem where it has become important to count (or at least bound from above and below) the number of elements of ${\bf Z}/n{\bf Z}$ that have order less than a given $k$, where $2\...

5
votes

2
answers

382
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

388
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

77
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

249
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

107
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

196
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

2k
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

290
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

432
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

1k
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 ...

10
votes

1
answer

740
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

4k
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### 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

430
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 \...