Addendum to "Modular Representations of Metabelian Groups"

Abstract

In this note the principal indecomposable modules of QG are determined where G is a finite metabelian group and Q is an algebraically closed field with characteristic p dividing |G|. The notations are the same as of [1]. Let P be a p-Sylow subgroup of K(H). Since K(H)/K(H)' is abelian, there exist subgroups V1 2 K(H)' and V2 D K(H)' such that K(H)/K(H)'' V1/K(H)' x V2/K(H)', V1/K(H)' is a p-group and p{ IV2/K(H)'I. Let P1 be a p-sylow subgroup of V2, then P1 C K(H)' and thus P1 is normal in V2* Hence there exists a subgroup V of V2 such that V2 P1 o V, the semidirect product, and pt IVI. Clearly K(H)= (P, V), P n V = 1, and IVI = IK(H)I/IPI. For each K(H), A/H cyclic and pt IA/HI, fix a subgroup V with the above properties. Let r' be a linear representation of K(H) with ker T' n A = H such that 7K is conjugate to a where K= K(A). Then 'G is irreducible and r G e B(r, H). Let x c G and define