# Krull-Remak-Schmidt theorem

## Contents

## Statement

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

- The ascending chain condition on normal subgroups: There is no strictly ascending infinite chain of normal subgroups of the group
- 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:

- can be expressed as an internal direct product of finitely many directly indecomposable groups
- If we have two such expressions, say and , then , and, after reordering, each
- Further, after reordering, we have for every

## 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:

- 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)
- 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)
- 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

- Direct product is cancellative for finite groups: This states that, for finite groups , if , then .
- Corollary of Krull-Remak-Schmidt theorem for cancellation of factors in direct product: This states that if all satisfy the ascending and descending chain conditions on normal subgroups, then implies .
- Corollary of Krull-Remak-Schmidt theorem for cancellation of powers: This states that if are groups satisfying the ascending and descending chain conditions on normal subgroups, and if , then .

### Opposite facts

- The existence of a group isomorphic to its square
- There exist abelian groups whose isomorphism classes of direct powers have any given period (proved by Corner, answering a conjecture of Pierce). This states that for any , we can find an abelian group such that if and only if .
- Eilenberg swindle: In its sophisticated form, this states that for any collection of groups , there exists a group such that is isomorphic to each of the groups .