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| \section*{Definition} |
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| A {\em locally cyclic} group is a group in which every finite subset generates a \PMlinkname{cyclic}{CyclicGroup} \PMlinkname{subgroup}{Subgroup}. |
A {\em locally cyclic} group is a group in which every finite subset generates a \PMlinkname{cyclic}{CyclicGroup} \PMlinkname{subgroup}{Subgroup}. |
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| \section*{Properties} |
Some facts about locally cyclic groups: |
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A group $G$ is locally cyclic if and only if every pair of elements of $G$ generates a cyclic subgroup. |
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Every locally cyclic group is abelian. |
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Every finitely generated locally cyclic group is cyclic. |
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| From the definition we see that every finitely generated locally cyclic group |
A group is locally cyclic if and only if it is isomorphic to a subgroup of $\Q$ or $\Q/\Z$. |
| (and, in particular, every finite locally cyclic group) is cyclic. |
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| The following can all be shown to be equivalent for a group $G$: |
Subgroups and \PMlinkname{quotients}{QuotientGroup} of locally cyclic groups are also locally cyclic. |
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| \begin{itemize} |
A group is locally cyclic if and only if its lattice of subgroups is \PMlinkname{distributive}{DistributiveLattice}. |
| \item $G$ is locally cyclic. |
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| \item For all $a,b\in G$, the subgroup $\genby{a,b}$ is cyclic. |
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| \item $G$ is the union of a chain of cyclic subgroups. |
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| \item $G$ embeds in $\Q$ or $\Q/\Z$. |
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| \item $G$ is isomorphic to a subgroup of a quotient of $\Q$. |
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| \item $G$ is involved in $\Q$. |
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| \item The subgroup lattice of $G$ is \PMlinkname{distributive}{DistributiveLattice}. |
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| \end{itemize} |
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| From some of these equivalent properties it is clear that |
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| every locally cyclic group is countable and abelian, |
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| and that subgroups and \PMlinkname{quotients}{QuotientGroup} |
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| of locally cyclic groups are locally cyclic. |
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