The \(Z_ p^*\)-theorem and units in blocks
The paper under review points out a surprising and interesting connection between the \(Z^*_ p\)-theorem in local group theory and the isomorphism problem for group rings. Let \(p\) be a prime, let \(G\) be a finite group, let \(x\) be an element of order \(p\) in \(G\), and suppose that \(x^ g=x\) for any \(g\in G\) such that \(x^ g\) and \(x\) lie in a common Sylow \(p\)-subgroup. The \(Z^*_ p\)-conjecture says that \(G=O_{p'}(G)C_ G(x)\) in this case. It was proved for \(p=2\) by G. Glauberman and is known to be a consequence of the classification of finite simple groups for arbitrary \(p\). Without using the classification, the author constructs, for a minimal counterexample \(G\) and a prime \(p\geq 5\), a unit \(u\) of order \(p\) and augmentation 1 in the center of the principal block \(B_ 0(G)\) of \(G\) over \({\mathbb{Z}}_{(p)}\) such that \(u\) is not conjugate to \(e_ 0g\) for any \(g\in G\) (where \(e_ 0\) denotes the principal block idempotent). The existence of such a unit contradicts the following conjecture by K. Roggenkamp and L. Scott: Let \(G_ 1\), \(G_ 2\) be finite groups such that \(B_ 0(G_ 1)=B_ 0(G_ 2)\) as augmented algebras, and let \(D_ 1\), \(D_ 2\) be Sylow \(p\)-subgroups of \(G_ 1\), \(G_ 2\), respectively. Then \(e_ 0D_ 1\) and \(e_ 0D_ 2\) are conjugate via a unit in \(B_ 0(G_ 1)\).
scientific article; zbMATH DE number 4158933