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 following examples are left as exercises for the reader.
Every simple group is indecomposable.
If is prime and is any positive integer, then the additive group is indecomposable. Hence, not every indecomposable group is simple.
The additive groups and are indecomposable, but the additive group is decomposable.
If and are relatively prime integers (and both greater than one), then the additive group is decomposable.
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.
|Date of creation||2013-03-22 15:23:46|
|Last modified on||2013-03-22 15:23:46|
|Last modified by||CWoo (3771)|