Questions tagged [cyclic-groups]

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

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3 votes
1 answer
243 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 $X_G:=\cup_{H\le G} G/H$, ...
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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} = \...
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2 votes
0 answers
53 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\...
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5 votes
2 answers
298 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 ...
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8 votes
1 answer
368 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{...
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4 votes
0 answers
74 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 & \...
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  • 2,598
0 votes
1 answer
179 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 $\...
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1 vote
0 answers
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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 $...
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  • 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 ...
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  • 39
8 votes
1 answer
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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?
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6 votes
1 answer
257 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) ...
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  • 509
2 votes
0 answers
353 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, ...
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6 votes
2 answers
990 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
605 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 ...
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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.
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  • 4,565
4 votes
1 answer
386 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 \...
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