As G. M. Conwell pointed out in a 1910 paper, the group of all
40,320 permutations of an 8-element set is the same, in an
abstract sense, as the group of all collineations and dualities
of PG(3,2), the projective 3-space over the 2-element field.
This suggests we study the geometry related to the above group's
actions on the 105 partitions of an 8-set into four separate 2-sets.
Note that 105 equals 15×7 and also 35×3.
In such a study, the 15 points of PG(3,2) might correspond (somehow)
to 15 pairwise-disjoint seven-element subsets of the set of 105 partitions,
and the 35 lines of PG(3,2) might correspond (somehow) to 35 pairwise-
disjoint three-element subsets of the set of 105 partitions.
Exercise: Is this a mere pipe dream?
A search for such a study yields some useful background . . .
.
Taylor's Index of Names includes neither Conwell nor the
more recent, highly relevant, names Curtis and Conway .
