indecomposable group

By definition, an indecomposable group is a nontrivial group that cannot be expressed as the internal direct productMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of two proper normal subgroupsMathworldPlanetmath. A group that is not indecomposableMathworldPlanetmath is called, predictably enough, decomposableMathworldPlanetmathPlanetmath.

The analogous concept exists in module theory. An indecomposable module is a nonzero module that cannot be expressed as the direct sumMathworldPlanetmathPlanetmath of two nonzero submodulesMathworldPlanetmath.

The following examples are left as exercises for the reader.

  1. 1.

    Every simple groupMathworldPlanetmathPlanetmath is indecomposable.

  2. 2.

    If p is prime and n is any positive integer, then the additive groupMathworldPlanetmath /pn is indecomposable. Hence, not every indecomposable group is simple.

  3. 3.

    The additive groups and are indecomposable, but the additive group is decomposable.

  4. 4.

    If m and n are relatively prime integers (and both greater than one), then the additive group /mn is decomposable.

  5. 5.

    Every finitely generatedMathworldPlanetmathPlanetmathPlanetmath abelian groupMathworldPlanetmath can be expressed as the direct sum of finitely many indecomposable groups. These summands are uniquely determined up to isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath.


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

  • Goldhaber, J. and G. Ehrlich, AlgebraMathworldPlanetmathPlanetmathPlanetmath. London: The Macmillan Company, 1970.

  • Hungerford, T., Algebra. New York: Springer, 1974.

Title indecomposable group
Canonical name IndecomposableGroup
Date of creation 2013-03-22 15:23:46
Last modified on 2013-03-22 15:23:46
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 8
Author CWoo (3771)
Entry type Definition
Classification msc 20-00
Synonym indecomposable
Related topic KrullSchmidtTheorem
Defines decomposable
Defines indecomposable module