On group ring automorphisms. (Q1878309)

Let \(G\) be a finite group and \(R\) be a complete discrete valuation ring of characteristic \(0\). The authors study the group of those automorphisms \(\text{Outcent}(RG)\) of the group ring \(RG\) which fix the center of \(RG\) pointwise. As a main result of the paper the authors show that if \(B\) is a block of the group ring of \(G\) over the \(p\)-adic integers, then \(\text{Outcent}(B)\) is the trivial group, provided \(B\) has cyclic defect. The authors apply this to the special case where \(G\) is the alternating group of degree \(6\). In [\textit{A. Zimmermann}, Algebr. Represent. Theory 7, No. 1, 19-34 (2004)], the reviewer obtained the same result for more general orders, including blocks of group rings \(RG\) with cyclic defect and without exceptional vertex. While the reviewer uses an approach by Roggenkamp to this class of orders, the authors of the paper under review use an alternative approach due to Plesken.