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 direct product of indecomposable subgroups.

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[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} (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 \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 direct 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 indecomposable subgroups. Then k = l and there is a reindexing of the Hi's satisfying

  • Gi and Hi 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 (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.

[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.

[edit] Further reading

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

[edit] External links