Quivers of groups with normal _p-subgroups

Abstract

Let K be a sufficiently large field of characteristic p. We determine the quiver of a semidirect product H = P ⋊ [Hbar] of a finite p-group [Hbar] as the join of the quiver of [Hbar] and the McKay graph D([Hbar], [Rbar]') of the conjugation representation [Rbar]' of [Hbar] on . More generally, whenever , we show that the quiver Q_KH of H is a subgraph of the above join, and give a necessary and sufficient condition on the radicals for the quiver of H to exactly equal the join . Finally, we identify the “transgressing” arrows of the McKay graph, those which do not appear in the quiver of H, with basis elements in the kernel of the transgression map.