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 .