On flag collineations of finite projective planes
A flag collineation is a collineation which leaves some incident point- line pair invariant. The main result is the following: Let \(\Pi\) be a finite projective plane of order n and let G be a collineation group of \(\Pi\) which contains a non-trivial perspectivity. Suppose that G has a subgroup H of order n such that for some incident point-line pair P, \(\ell\) every element of H is a flag collineation leaving P, \(\ell\) invariant and that H has order n. Then one of the following holds: (1) G has an invariant point-line pair. (2) G has an invariant line but no invariant points and is a translation plane. (3) The dual of (2). (4) The plane \(\Pi\) is Desarguesian. A group G of collineations of a projective plane is said to be strongly irreducible if there is no invariant point, line, triangle, or proper subplane. The arguments depend on known results about strongly irreducible groups which contain non-trivial perspectivities (central collineations).
flag collineations of finite projective planes
scientific article; zbMATH DE number 3992299