Krull-Schmidt theorem

A group $G$ is said to satisfy the ascending chain condition (or ACC) on normal subgroups if there is no infinite ascending proper chain$G_1⊊G_2⊊G_3\mathrm{\cdots }$ with each $G_i$ a normal subgroup of $G$.

Similarly, $G$ is said to satisfy the descending chain condition (or DCC) on normal subgroups if there is no infinite descending proper chain of normal subgroups of $G$.

One can show that if a nontrivial group satisfies either the ACC or the DCC on normal subgroups, then that group can be expressed as the internal direct product of finitely many indecomposable subgroups. If both the ACC and DCC are satisfied, the Krull-Schmidt theorem guarantees that this “decomposition into indecomposables” is essentially unique. (Note that every finite group satisfies both the ACC and DCC on normal subgroups.)

Krull-Schmidt theorem: Let $G$ be a nontrivial group satisfying both the ACC and DCC on its normal subgroups. Suppose $G=G_1\times \mathrm{\cdots }\times G_n$ and $G=H_1\times \mathrm{\cdots }\times H_m$ (internal direct products) where each $G_i$ and $H_i$ is indecomposable. Then $n=m$ and, after reindexing, $G_i\cong H_i$ for each $i$. Moreover, for all , $G=G_1\times \mathrm{\cdots }\times G_k\times H_{k+1}\times \mathrm{\cdots }\times H_n$.

For proof, see Hungerford’s Algebra.

Noetherian [resp. artinian] modules satisfy the ACC [resp. DCC] on submodules. Indeed the Krull-Schmidt theorem also appears in the context of module theory. (Sometimes, as in Lang’s Algebra, this result is called the Krull-Remak-Schmidt theorem.)

Krull-Schmidt theorem (for modules): A nonzero module that is both noetherian and artinian can be expressed as the direct sum of finitely many indecomposable modules. These indecomposable summands are uniquely determined up to isomorphism and permutation.

References.

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

• Lang, S., Algebra. (3d ed.), New York: Springer, 2002.

Title	Krull-Schmidt theorem
Canonical name	KrullSchmidtTheorem
Date of creation	2013-03-22 15:24:00
Last modified on	2013-03-22 15:24:00
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	24
Author	CWoo (3771)
Entry type	Theorem
Classification	msc 16P40
Classification	msc 16P20
Classification	msc 16D70
Classification	msc 20E34
Classification	msc 20-00
Synonym	Krull-Remak-Schmidt theorem
Related topic	IndecomposableGroup
Defines	ascending chain condition
Defines	descending chain condition