A Sylowlike theorem for integral group rings of finite solvable groups (Q1325089)

Let \(G\) and \(H\) be finite solvable groups with \(\mathbb{Z} G = \mathbb{Z} H\) as augmented algebras. If \(S\) is a Sylow \(p\)-subgroup of \(H\), then there exists a unit \(a \in \mathbb{Q} G\) such that \(aSa^{-1}\) is a Sylow \(p\)- subgroup of \(G\).