On a subalgebra of the centre of a group ring. II. (Q2480699)

Let \(FG\) denote the group algebra of a finite group \(G\) over a field \(F\) of characteristic \(p>0\). The author considers the question when \(Z'\), the \(F\)-subspace of the centre \(Z=Z(FG)\) of \(FG\) spanned by all \(p\)-regular class sums of \(G\), is in fact a subalgebra. His main result shows that this is the case whenever \(C_G(x)\) is \(p\)-nilpotent for every element of order \(p\) in \(G\). In particular, the question has a positive answer whenever \(p=2\) and the Sylow 2-subgroups of \(G\) are dihedral. For part I, see J. Algebra 295, No. 1, 293-302 (2006; Zbl 1110.20006).