Krull-Remak-Schmidt theorem

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Statement

Suppose G is a group of finite chief length, i.e., it satisfies the following two conditions:

  1. The ascending chain condition on normal subgroups: There is no strictly ascending infinite chain of normal subgroups of the group
  2. The descending chain condition on normal subgroups: There is no strictly descending infinite chain of normal subgroups of the group

A finite group has finite chief length, so the above always hold for it.

Then, the following three facts are true:

  1. G can be expressed as an internal direct product of finitely many directly indecomposable groups
  2. If we have two such expressions, say G \cong G_1 \times G_2 \times \dots G_m and G \cong H_1 \times H_2 \times \dots H_n, then n = m, and, after reordering, each G_i \cong H_i
  3. Further, after reordering, we have G \cong G_1 \times G_2 \times \dots G_k \times H_{k+1} \times \dots \times H_n for every k

Examples

Elementary abelian groups

Recall that an elementary abelian group is a vector space over a prime field, and the subgroups of an elementary abelian group are its vector subspaces. The only directly indecomposable subgroups here are the groups of prime order, or the one-dimensional subspaces. The Krull-Remak-Schmidt theorem thus tells us that:

  1. Any finite-dimensional vector space can be expressed as a direct sum of one-dimensional vector spaces (i.e., every vector space has a basis)
  2. The number of one-dimensional vector spaces used in the direct summation is independent on the choice of direct sum (i.e., the dimension is an invariant)
  3. Given two such decompositions, we can rearrange them so that the condition (3) is satisfied. This translates to checking an easy fact about bases and their rearrangement.

Finite abelian groups

For finite abelian groups, this result follows from the structure theorem for finitely generated abelian groups.

Related facts

Weaker facts

Opposite facts

External links