Krull–Schmidt theorem

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

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 G_N = G_N+1 = G_N+2 = ... . 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 ${\displaystyle \mathbf {Z} }$ satisfies ACC but not DCC, since (2) > (2)² > (2)³ > ... is an infinite decreasing sequence of subgroups. On the other hand, the ${\displaystyle p^{\infty }}$-torsion part of ${\displaystyle \mathbf {Q} /\mathbf {Z} }$ (the quasicyclic p-group) satisfies DCC but not ACC.

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