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Krull-Schmidt theorem
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 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:
is eventually constant, i.e. there exists N such that 
. 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:
Clearly, all finite groups satisfy both ACC and DCC on subgroups. The infinite cyclic group 
satisfies ACC but not DCC, since 
is an infinite decreasing sequence of subgroups. On the other hand, the 
-torsion part of 
satisfies DCC but not ACC.
We say a group G is indecomposable if it cannot be written as a product of subgroups 
.
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 product 
of finitely many subgroups of G. Here, uniqueness means: suppose 
is another expression of G as a product of subgroups. Then k = l and there is a reindexing of the Hi's satisfying
- Gi and Hi are isomorphic for each i;
-
for each r.
External links
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