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'Indecomposable Group'
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| Title of object: |
Indecomposable Group |
| Canonical Name: |
IndecomposableGroup |
| Type: |
Definition |
| Created on: |
2005-07-16 15:23:41 |
| Modified on: |
2005-07-16 18:35:02 |
| Classification: |
msc:20-00 |
| Keywords: |
indecomposable, decomposable |
| Defines: |
indecomposable group, decomposable group, indecomposable module |
| Synonyms: |
Indecomposable Group=indecomposable |
Preamble:
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Content:
By 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 Every infinite cyclic group is indecomposable. Hence, the additive group of integers is indecomposable.
\item The additive group of rationals is indecomposable.
\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} |
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