This paper is concerned with
the “ordinary” (over the complex numbers) representation theory of finite groups and
in particular with matrix groups of the first and second kinds (that is, matrix groups
which are similar to real groups or, alternatively, have real character but are not
similar to real groups. In the event that the character is non-real, we speak of the
third kind.)
The purpose of this paper is associate groups with exactly one involution with
representations of the second kind, and this we do in two ways: First, by showing
that any group possessing an irreducible representation of the second kind involves a
non-trivial group with only one involution. Second, by showing that a group with
only one involution cannot have a faithful irreducible representation of the first
kind. It is well and long known that groups of odd order possess nontrivial
irreducible representations of the third kind only, so that evenness of order
is a necessity if matrix groups of the first or second kind are to be dealt
with.
