Direct sums of local torsion-free Abelian groups (Q2781328)

Let \(G\) be a torsion-free Abelian group of finite rank. \(G\) is said to be \(p\)-local if there exists a fixed prime \(p\) such that \(G\) is divisible by all primes \(q\not=p\). It is well-known that the category \(TF\) of all torsion-free \(p\)-local Abelian groups of finite rank has the \(n\)-th root property (if \(G^n\cong H^n\), then \(G\cong H\)) and the cancellation property (if \(G\oplus N\cong G\oplus K\), then \(N\cong K\)). However, there are counterexamples showing that the Krull-Schmidt property fails for \(TF\). Recall that an Abelian group \(G\) is Krull-Schmidt if any two direct decompositions of \(G\) into indecomposable summands are equivalent up to isomorphism and order of summands. The author proves that \(10\) is the minimal rank of a counterexample to the Krull-Schmidt property in \(TF\) and therefore answers a question posed by M.~C.~R.~Butler in the 1960's.