Questions tagged [cyclic-groups]
Questions about the branch of algebra that deals with cyclic groups.
21
questions
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votes
0
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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{...
- 133
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 ...
- 383
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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
$$
C_m \rtimes_k C_n ...
- 455
0
votes
0
answers
68
views
Fourier coefficient of close functions
Let $p$ be some prime. Let $\mathbb{Z}_p$ be the cyclic group of order $p$. Let $f, g \colon \mathbb{Z}_p \to \{\pm 1\}$ be two functions. Recall that the Fourier transform is defined as
$$ f(x) = \...
- 1
11
votes
2
answers
2k
views
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
$$
(n-1)! \equiv
\begin{cases}
\hfill -1 \pmod {n} &\text{ if } n \...
- 1,923
4
votes
1
answer
253
views
A question on a possible cyclic sieving phenomenon?
(This is an old MSE question from me, which did not get any answer, and when looking back seems interesting to post it here:)
Let $G$ be a finite group. Consider the set $X_G:=\cup_{H\le G} G/H$, ...
- 2,112
4
votes
0
answers
181
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An addition theorem for three functions similar to $\sin,\cos$ and $\sinh,\cosh$ and one / some questions?
Define the functions $t_k(x) = \sum_{n=0}^{\infty}{\frac{x^{3n+k}}{(3n+k)!}}$ for $k=0,1,2$. The functions then satisfy:
$
\begin{pmatrix}\exp(x) \\ \exp(\omega x) \\ \exp(\omega^2 x)\end{pmatrix} =
\...
- 2,112
2
votes
0
answers
63
views
The number of elements with order less than $k$ in a larger cyclic group
I am working on a problem where it has become important to count (or at least bound from above and below) the number of elements of ${\bf Z}/n{\bf Z}$ that have order less than a given $k$, where $2\...
- 787
5
votes
2
answers
382
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Do there exist general conditions for cyclicity of unit groups of quotient rings (generalizations of the primitive root theorem)?
Let $R$ by a commutative ring with $1$, and $I \subset R$ a non-zero integral ideal in $R$. When $R$ has finite quotients, and $I = P$ is prime in $R$, the group of units $(R/P)^{\times}$ of the ...
8
votes
1
answer
388
views
Classification of the functors on the category of cyclic groups
Let $\mathsf{Grp}$ be the category of groups and let $\mathsf{Cyc}$ be the subcategory of cyclic groups.
As seen in the posts here and there (and their answers), a functor $F: \mathsf{Cyc} \to \mathsf{...
- 25.3k
4
votes
0
answers
77
views
Finite groups of cyclicality index $3$
Suppose $G$ is a group. Let’s define the cyclicality index of $G$ using the following recurrent relation:
$$CI(G) = \begin{cases} 1 & \quad G \text{ is cyclic} \\ \max_{H < G} CI(H) + 1 & \...
- 2,618
0
votes
1
answer
249
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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
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 $...
- 11
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 ...
- 39
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?
- 81
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) ...
- 509
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, ...
- 345
6
votes
2
answers
1k
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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 ...
- 16k
6
votes
2
answers
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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.
- 4,715
4
votes
1
answer
430
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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 \...
- 32.5k