In this paper the question of
determining the absolutely irreducible constituents of an irreducible projective
representation of a finite group is considered from the viewpoint of the theory of
algebras. New proofs are given for several of the main results of the theory of
representations of finite dimensional associative algebras. This theory is applied to
determine the center of a simple component of a twisted group algebra modulo its
radical and sufficient conditions are given to insure that this center is a normal
extention of the base field. The Schur index of an absolutely irreducible projective
representation of a finite group is defined as in the theory of linear representations of
finite groups. It is shown that every irreducible complex projective representation of a
finite group is projectively equivalent to a representation whose associated factor
system has values which are roots of unity but whose Schur index over the rationals
is 1.
