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.
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1.
Every simple group is indecomposable.
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2.
If is prime and is any positive integer, then the additive group is indecomposable. Hence, not every indecomposable group is simple.
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3.
The additive groups and are indecomposable, but the additive group is decomposable.
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4.
If and are relatively prime integers (and both greater than one), then the additive group is decomposable.
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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.
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Dummit, D. and R. Foote, Abstract Algebra. (2d ed.), New York: John Wiley and Sons, Inc., 1999.
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Goldhaber, J. and G. Ehrlich, Algebra. London: The Macmillan Company, 1970.
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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 |