  This report is an account of freely representable groups, which are finite
groups admitting linear representations whose only fixed point for a
nonidentity element is the zero vector. The standard reference for such groups
is Wolf (1967) where such groups are used to classify spaces of constant
positive curvature. Such groups also arise in the theory of norm relations in
algebraic number theory, as demonstrated recently by Biasse, Fieker, Hofmann,
and Page (2020). This report aims to synthesize the information and results
from these and other sources to give a continuous, self-contained development
of the subject. I introduce new points of view, terminology, results, and
proofs in an effort to give a coherent, detailed, self-contained, and
accessible narrative.
