ABSTRACT
We determine all finitely generated centre-by-finite groups that can be embedded into finite-dimensional division algebras. The division algebras generated by such groups are called projective Schur division algebras (there are also more general projective Schur algebras) by Lorenz and Opolka, [6]. In [1], Aljadeff and Sonn prove that such division algebras have abelian splitting fields. For their purposes, the determination of the exact structure of these groups was not necessary. These groups, however, are a relatively useful ingredient in the study of projective representations of finite groups over arbitrary fields, so their structure is of some interest. In addition, such groups can also be regarded as central extensions of those
finite groups, which possess a faithful projective representation of reduced degree equal to 1. Here the term reduced degree is used in an obvious extension of the term as used for ordinary representations of finite groups (e.g., [5], Section 5): If G is a finite group, then the reduced ordinary degrees of the irreducible representations of G over a field k of characteristic zero are the matrix degrees of the simple components of the group ring kG (the ordinary degrees are then the product of the reduced degrees and the Schur indices). In the same way, the reduced projective degrees of a finite group G may be defined as the matrix degrees of finite-dimensional simple k-algebras R, which are spanned over k by a group X of units, with the property that
X is a central extension of G. Having reduced degree 1 then means that the algebra R is a division algebra. Unlike reduced ordinary degrees, nontrivial division algebras can occur
in positive characteristic. The results depend on whether the characteristic is zero or not, and are stated separately. Recall that an Amitsur group is a finite subgroup of a division ring of characteristic zero (for a list, see 2.1 below).
