Krull–Remak–Schmidt Theorem (Groups)

Under chain conditions, direct product decompositions into indecomposable normal factors are unique up to order

Krull–Remak–Schmidt Theorem (Groups)

Krull–Remak–Schmidt Theorem (Groups). Let $G$ be a group that satisfies both the ascending and descending chain conditions on normal subgroups (in particular, any finite group satisfies these conditions). Suppose

G \cong G_1 \times \cdots \times G_n

is a direct product decomposition in which each $G_i$ is nontrivial, normal in $G$ , and directly indecomposable (meaning $G_i$ is not isomorphic to $A\times B$ with $A,B$ both nontrivial). If also

G \cong H_1 \times \cdots \times H_m

is another such decomposition with directly indecomposable normal factors, then $n=m$ and, after permuting indices, there are isomorphisms $G_i \cong H_i$ for all $i$ .

Equivalently: the multiset of isomorphism types of directly indecomposable factors in an internal direct product decomposition is an invariant of $G$ (under the stated chain hypotheses). This is the group analogue of Krull–Schmidt–Azumaya for modules .

Proof sketch. One compares projections of one decomposition onto the factors of the other, showing that indecomposability forces certain endomorphisms to be invertible or nilpotent. Chain conditions ensure termination of the resulting refinement process, yielding a bijection between indecomposable factors up to isomorphism.