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indecomposable group (Definition)

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.




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See Also: Krull-Schmidt theorem

Other names:  indecomposable
Also defines:  decomposable, indecomposable module
Keywords:  indecomposable, decomposable
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Cross-references: isomorphism, abelian group, finitely generated, relatively prime, simple, additive group, integer, positive, prime, simple group, submodules, direct sum, theory, module, normal subgroups, direct product, group
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This is version 5 of indecomposable group, born on 2005-07-16, modified 2005-12-23.
Object id is 7232, canonical name is IndecomposableGroup.
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Classification:
AMS MSC20-00 (Group theory and generalizations :: General reference works )

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