Questions tagged [cyclic-groups]
Questions about the branch of algebra that deals with cyclic groups.
16
questions
3
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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 $X_G:=\cup_{H\le G} G/H$, ...
- 1,243
4
votes
0
answers
161
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} =
\...
- 1,243
2
votes
0
answers
53
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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\...
- 551
5
votes
2
answers
298
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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
368
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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
74
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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 & \...
- 2,598
0
votes
1
answer
179
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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
104
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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
192
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
1k
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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
257
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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
353
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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
990
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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
605
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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 ...
- 15.7k
6
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2
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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.
- 4,565
4
votes
1
answer
386
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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 \...
- 31.5k