Krull-Remak-Schmidt theorem

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 $$k$$

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.

Weaker facts

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

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 $$r$$, we can find an abelian group $$G$$ such that $$G^m \cong G^n$$ if and only if $$m \equiv n \pmod r$$.
 * Eilenberg swindle: In its sophisticated form, this states that for any collection of groups $$G_i, i \in I$$, there exists a group $$G$$ such that $$G$$ is isomorphic to each of the groups $$G \times G_i$$.