# Questions tagged [cyclic-groups]

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

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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{...
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### 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 ...
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### $P\in \mathbb{F}_{2}[x]$ 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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### 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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### 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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### 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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### 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 ...
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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 ...
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 \...