Central units in blocks and the odd \(Z_p^*\)-theorem. (Q1012562)

Let \(p\) be an odd prime, \(G\) a finite group, \(x\) an element of order \(p\) in a \(p\)-Sylow subgroup \(P\). Glauberman's \(Z^*\) theorem states that if the element \(x\) commutes with none of its other conjugates, then \([x,G]\leq O^*_{p'}(G)\) holds. An analogue of this theorem for odd primes is that the subgroup \(O_{p'}(G)C_G(x)\) equals \(G\) if \(x\) is an isolated element in the subgroup \(P\), which follows from the classification of finite simple groups. In the present paper an independent proof is sought by means of representation theory. In the paper [J. Algebra 134, No. 2, 353-355 (1990; Zbl 0706.20012)] of the author it was shown that the property has a close relationship with units of prime order in the group ring. This idea is utilized also here: in the course of the investigations units of order \(p\) in the \(p\)-adic group ring are constructed valuable in proving the analogue of Glauberman's theorem and also of some independent interest.