 
Summary: ISRAEL JOURNAL OF MATHEMATICS121 (2001), 173198
DIVISION ALGEBRAS WITH A PROJECTIVE BASIS
BY
ELI ALJADEFF
Department of Mathematics, Technion  Israel Institute of Technology
Haifa 32000, Israel
email: aljade~@~echunix.technion.ac.il
AND
DARRELL HALLE
Department of Mathematics, Indiana University,
Bloomington, IN ~7405, USA
email: haile@indiana.edu
ABSTRACT
Let k be any field and G a finite group. Given a cohomology class a 6
H 2(G, k*), where G acts trivially on k*, one constructs the twisted group
algebra kaG. Unlike the group algebra kG, the twisted group algebra may
be a division algebra (e.g. symbol algebras, where G ~ Z, x Z,). This
paper has two main results: First we prove that if D = kaG is a division
algebra central over k (equivalently, D has a projective kbasis) then G is
nilpotent and G', the commutator subgroup of G, is cyclic. Next we show
