Krull-Schmidt theorem

In mathematics, the Krull-Schmidt theorem states that a group $G$, subjected to certain finiteness conditions of chains of subgroups, can be uniquely written as a finite direct product of indecomposable subgroups.

Definitions

We say that a group $G$ satisfies the ascending chain condition (ACC) on subgroups if every sequence of subgroups of $G$:

:$1\; =\; G\_0\; le\; G\_1\; le\; G\_2\; le\; dots$

is eventually constant, i.e., there exists $N$ such that $G\_N\; =\; G\_\{N+1\}\; =\; G\_\{N+2\}=\; dots$. We say that $G$ satisfies the ACC on normal subgroups if every such sequence of normal subgroups of $G$ eventually becomes constant.

Likewise, one can define the descending chain condition on (normal) subgroups, by looking at all decreasing sequences of (normal) subgroups:

:$G\; =\; G\_0\; ge\; G\_1\; ge\; G\_2\; ge\; ldots.$

Clearly, all finite groups satisfy both ACC and DCC on subgroups. The infinite cyclic group $mathbf\{Z\}$ satisfies ACC but not DCC, since $(2)\; >\; (2^2)\; >\; (2^3)\; >\; ldots$ is an infinite decreasing sequence of subgroups. On the other hand, the $p^infty$-torsion part of $mathbf\{Q\}/mathbf\{Z\}$ (the quasicyclic "p"-group) satisfies DCC but not ACC.

We say a group $G$ is indecomposable if it cannot be written as a direct product of non-trivial subgroups $G\; =\; H\; imes\; K$.

Krull-Schmidt theorem

The theorem says:

If $G$ is a group that satisfies ACC and DCC on normal subgroups, then there is a unique way of writing $G$ as a direct product $G\_1\; imes\; G\_2\; imesldots\; imes\; G\_k$ of finitely many indecomposable subgroups of $G$. Here, uniqueness means: suppose $G\; =\; H\_1\; imes\; H\_2\; imes\; ldots\; imes\; H\_l$ is another expression of $G$ as a product of indecomposable subgroups. Then $k=l$ and there is a reindexing of the $H\_i$'s satisfying

* $G\_i$ and $H\_i$ are isomorphic for each $i$;
* $G\; =\; G\_1\; imes\; ldots\; imes\; G\_r\; imes\; H\_\{r+1\}\; imesldots\; imes\; H\_l$ for each $r$.

Krull-Schmidt theorem for modules

If $E\; eq\; 0$ is a module that satisfies the ACC and DCC on submodules (that is, it is both Noetherian and Artinian), then $E$ is a direct sum of indecomposable modules. Up to a permutation, the indecomposable components in such a direct sum are uniquely determined up to isomorphism.

History

The present-day Krull-Schmidt theorem is the result of work by Robert Remak (1911), Wolfgang Krull (1925) and Otto Schmidt (1928) in a paper "Über unendliche Gruppen mit endlicher Kette".

Further reading

* Hungerford, Thomas W. "Algebra, Graduate Texts in Mathematics Volume 73". ISBN 0-387-90518-9

External links

* [http://planetmath.org/encyclopedia/KrullRemakSchmidtTheorem.html Page at PlanetMath]