ABSTRACT
In this chapter, we consider the possible groups of L(2; K2) that are Pappian parallelism inducing, where K2 denotes the quadratic …eld extension of K coordinatizing . We begin with the linear groups in GL(2; K2). In this setting, we need to …nd subgroups that satisfy the conditions: Sharp, Baer, and central …x, and we begin with this last condition. Let K2 f0g = Px Py be a group direct product decomposition
of the multiplicative group of K2 that also decomposes K f0g as a group. Assume also that the decomposition is invariant under GalKK2
(automatic if …nite). Furthermore, we choose the notation that K f0g = Px Py for Px Px and Py Py. We coordinatize a Pappian a¢ ne plane using K2 and consider the following homology groups:
Hx :
h =
1 0 0 h
;h 2 Px
and
Hy :
k =
k 0 0 1
; k 2 Py
.
