Questions tagged [cyclic-groups]

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

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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 & \...
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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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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 $...