Krull-Schmidt theorem

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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 product of indecomposable subgroups.

1 Definitions
2 Krull-Schmidt theorem
3 Krull-Schmidt theorem for modules
4 History
5 Further reading
6 External link

[edit] 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:

$G = G_0 \ge G_1 \ge G_2 \ge \dots$

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 $\mathbf{Q}/\mathbf{Z}$ satisfies DCC but not ACC.

We say a group $G$ is indecomposable if it cannot be written as a product of subgroups $G = H \times K$ .

[edit] 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 $G_1 \times G_2 \times\ldots \times G_k$ of finitely many indecomposable subgroups of $G$ . Here, uniqueness means: suppose $G = H_1 \times H_2 \times \ldots \times H_l$ is another expression of $G$ as a product of subgroups. Then $k = l$ and there is a reindexing of the $H i$ 's satisfying

$G i$ and $H i$ are isomorphic for each $i$ ;
$G = G_1 \times \ldots \times G_r \times H_{r+1} \times\ldots\times H_l$ for each $r$ .

[edit] Krull-Schmidt theorem for modules

If $E \neq 0$ is a module that satisfies the ACC and DCC on submodules (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.

[edit] 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.