indecomposable groupIndecomposableGroup2005-07-16 15:23:412005-12-23 16:09:54Definitiondecomposableindecomposable moduleindecomposabledecomposable% this is the default PlanetMath preamble. as your knowledge
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% define commands hereBy definition, an \emph{indecomposable group} is a nontrivial group that cannot be expressed as the internal direct product of two proper normal subgroups. A group that is not indecomposable is called, predictably enough, \emph{decomposable}.
The analogous concept exists in module theory. An indecomposable module is a nonzero module that cannot be expressed as the direct sum of two nonzero submodules.
The following examples are left as exercises for the reader.
\begin{enumerate}
\item Every simple group is indecomposable.
\item If $p$ is prime and $n$ is any positive integer, then the additive group $\mathbb{Z}/p^n\mathbb{Z}$ is indecomposable. Hence, not every indecomposable group is simple.
\item The additive groups $\mathbb{Z}$ and $\mathbb{Q}$ are indecomposable, but the additive group $\mathbb{R}$ is decomposable.
\item If $m$ and $n$ are relatively prime integers (and both greater than one), then the additive group $\mathbb{Z}/mn\mathbb{Z}$ is decomposable.
\item Every finitely generated abelian group can be expressed as the direct sum of finitely many indecomposable groups. These summands are uniquely determined up to isomorphism.
\end{enumerate}
{\bf References}.
\begin{itemize}
\item Dummit, D. and R. Foote, \emph{Abstract Algebra}. (2d ed.), New York: John Wiley and Sons, Inc., 1999.
\item Goldhaber, J. and G. Ehrlich, \emph{Algebra}. London: The Macmillan Company, 1970.
\item Hungerford, T., \emph{Algebra}. New York: Springer, 1974.
\end{itemize}