Ehsaan Hossain, Department of Pure Mathematics, University of Waterloo
“Polycyclic groups and primitive group rings”
If g ∈ G has finitely many conjugates, then the sum c := h∈[g] h is central in K[G]. In fact these sums span the center. Since primitive K-algebras often have center K, it follows that if K[G] is primitive then G has no finite conjugacy classes, i.e. ∆(G) = 1. We’ll examine a theorem of Passman which gives the converse for polycyclic groups: if G is a polycyclic-by- finite group of rank r with ∆(G) = 1, and if K is a field of transcendence degree ≥ r, then K[G] is primitive. Roseblade showed that K[G] is never primitive if K is algebraic over a finite field, so the restrictions on K are somewhat necessary.
MC 5403