locally cyclic groupGeneralizedCyclicGroup2003-07-23 11:06:472007-06-13 14:36:33Definition\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\def\genby#1{{\left\langle #1\right\rangle}}
\newcommand{\R}[0]{\mathbb{R}}
\newcommand{\C}[0]{\mathbb{C}}
\newcommand{\Q}[0]{\mathbb{Q}}
\newcommand{\N}[0]{\mathbb{N}}
\newcommand{\Z}[0]{\mathbb{Z}}
\newtheorem{theorem}{Theorem}\PMlinkescapeword{clear}
\PMlinkescapeword{equivalent}
\PMlinkescapeword{generates}
\PMlinkescapeword{properties}
\PMlinkescapeword{quotient}
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\PMlinkescapeword{subgroup}
\PMlinkescapeword{subgroups}
\section*{Definition}
A {\em locally cyclic} group is a group in which every finite subset generates a cyclic subgroup.
\section*{Properties}
From the definition we see that every finitely generated locally cyclic group
(and, in particular, every finite locally cyclic group) is cyclic.
The following can all be shown to be equivalent for a group $G$:
\begin{itemize}
\item $G$ is locally cyclic.
\item For all $a,b\in G$, the \PMlinkname{subgroup}{Subgroup} $\genby{a,b}$ is cyclic.
\item $G$ is the union of a chain of cyclic subgroups.
\item The lattice of subgroups of $G$ is \PMlinkname{distributive}{DistributiveLattice}.
\item $G$ embeds in $\Q$ or $\Q/\Z$.
\item $G$ is isomorphic to a subgroup of a \PMlinkname{quotient}{QuotientGroup} of $\Q$.
\item $G$ is \PMlinkname{involved in}{SectionOfAGroup} $\Q$.
\end{itemize}
From the last of these equivalent properties it is clear that
every locally cyclic group is countable and abelian,
and that subgroups and quotients
of locally cyclic groups are locally cyclic.