Induction from elementary abelian subgroups
Let \(G\) be a finite group, \(R\) a ring with unit element, \(R*G\) a crossed product of \(G\) over \(R\), and let \(M\) be an \(R*G\)-module. The main result of this paper tells that \(M\) is weakly projective (that is, relatively 1-projective), respectively projective iff \(_{R*H}M\) is weakly projective, respectively projective for every elementary abelian subgroup \(H\) of \(G\). This result generalises a theorem of \textit{L. G. Chouinard} [J. Pure Appl. Algebra 7, 287-302 (1976; Zbl 0327.20020)], which is also used in the proof. The final part of the paper deals with finding an expression for \(x_G\) with \(\text{tr}(x_G)=1_R\) in terms of elements \(y_H\) with \(\text{tr}_H(y_H)=1_R\) for every \(H\leq G\) elementary abelian.
scientific article; zbMATH DE number 843881