On collination groups of finite planes

Points will always be denoted by small latin letters, lines by capitals (unlike Bonisoli’s notations). By (a, b), with a, b ∈ N, we denote the greatest common divisor of a and b. Suppose π is a projective plane of order n, and suppose (p, L) is a point-line pair. Then a collineation θ of π is a (p, L)-collineation if θ fixes any point on L and every line through p. If (p, L) is a flag, then θ is also called a (p, L)elation; if (p, L) is an anti-flag, then θ is said to be a (p, L)-homology. In both cases, L is called the axis and p the center. A collineation of a finite projective plane has a center if and only if it has an axis. Collineations with centers are often called central collineations or perspectivities. A projective plane π of order n is called (p, L)-transitive for a point-line pair (p, L) if the group of (p, L)-collineation is maximal (if (p, L) is a flag, the size