It is our purpose to study
abstractly and in general, the structure of the set of all double cosets of a group
with respect to a subgroup. In our flrst section we allow the groups to be
infinite, and focus on a ternary collineation relation. If X, Y , Z are double
cosets, we say they are collinear if there exists x ∈ X, y ∈ Y , z ∈ Z with
x ⋅ y ⋅ z = 1. This relation is abstracted and forms the basis of that section. In the
second section we essentially insist the groups are finite, and count the double
cosets which products from two cosets can appear in (i.e., count all Z with
x ⋅ y ∈ Z for x ∈ X, y ∈ Y ). We abstract the properties that these numbers
possess, and study their implications. The orbit space of conjugacy classes of a
finite group can be taken as a set of double cosets (in the holomorph of the
group). This set has a natural dual, and in certain instances other sets of
double cosets have one also. We study this situation in the third and last
section by use of a complex matrix which enjoys some of the properties of the
character table of a finite group; this may be thought of as another approach
(slightly different than Brauer’s pseudogroups) to character tables as a thing in
themselves.
