Blocks of defect zero and products of elements of order \(p\) (Q1291071)

Let \(F\) be an algebraically closed field of characteristic \(p>0\), let \(G\) be a finite group, and let \(e_0\) be the sum of all block idempotents of defect zero in the group algebra \(FG\). Moreover, let \(\Omega^+\) be the sum (in \(FG\)) of all elements \(g\in G\) such that \(g^p=1\). One of the main results of this interesting paper shows that \(e_0=(\Omega^+)^2\) when \(p>2\), and that \(e_0=(\Omega^+)^3\) when \(p=2\). In the latter case, \((\Omega^+)^2=R^+\), where \(R\) is the sum of all real elements of 2-defect zero in \(G\). These results have applications to the existence and number of blocks of defect zero.