Projective Planes of Order 12 Do Not Have a Four Group as a Collineation Group

We have shown in [2] that the full collineation group of any projective plane of order 12 is a (2, 3) group. It is of interest to determine the structure of this (2,3} group. As a first step in that direction, we have shown in [3] that a non-Abelian group of order 6 cannot act as a collineation group on any projective plane of order 12. As a second step, we have shown in [4] that there is no projective plane of order 12 which possesses a collineation group of order 4 consisting of elations with a fixed point as a center and a fixed line as its axis. As a third and essential step, we prove the following result: