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 is prime and is any positive integer, then the additive group
is indecomposable. Hence, not every indecomposable group is simple.
- 3.
The additive groups and are indecomposable, but the additive group is decomposable.
- 4.
If and are relatively prime integers (and both greater than one), then the additive group 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.