Abstract
Abstract Collineation groups of finite projective planes are studied which do not leave invariant any point, line or triangle and contain a non-trivial perspectivity. In many instances the collineation group G can be determined, and it can be proved that the underlying projective plane contains a desarguesian subplane, whose order is related to the order of G . This can be done, because one has rather strong results about the structure of G , in particular about the first terms of a chief series of G . Projective planes are taken here as a suitable testing ground for methods, which can be used for many classes of designs to quickly find new examples and obtain some classification.
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