indecomposable group
By definition, an 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, 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.

1.
Every simple group^{} is indecomposable.

2.
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.

3.
The additive groups $\mathbb{Z}$ and $\mathbb{Q}$ are indecomposable, but the additive group $\mathbb{R}$ is decomposable.

4.
If $m$ and $n$ are relatively prime integers (and both greater than one), then the additive group $\mathbb{Z}/mn\mathbb{Z}$ is decomposable.

5.
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^{}.
References.

•
Dummit, D. and R. Foote, Abstract Algebra. (2d ed.), New York: John Wiley and Sons, Inc., 1999.

•
Goldhaber, J. and G. Ehrlich, Algebra^{}. London: The Macmillan Company, 1970.

•
Hungerford, T., Algebra. New York: Springer, 1974.
Title  indecomposable group 

Canonical name  IndecomposableGroup 
Date of creation  20130322 15:23:46 
Last modified on  20130322 15:23:46 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  8 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 2000 
Synonym  indecomposable 
Related topic  KrullSchmidtTheorem 
Defines  decomposable 
Defines  indecomposable module 