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Collineation Groups Strongly Irreducible on an Oval

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DOI: 10.1016/s0304-0208(08)73127-x

Abstract

Publisher Summary There is a well-developed theory of strongly irreducible collineation groups containing perspectivities, which has significant applications. To investigate local versions of the concept of irreducibility, a finite projective plane π of even order with a collineation group Γ and a Γ-invariant oval Ω are considered in the chapter, such that Γ does not leave invariant any point, chord, or suboval of Ω. A suboval of Ω is a subset of points of R that is an oval in a proper subplane of π. Gamma is strongly irreducible on the oval omega. Γ is not strongly irreducible on π as it fixes the knot K of Ω. The main result in the chapter states that if Γ has even order then Γ contains some involutorial perspectivities—that is, elations. The subgroup <Δ> generated by all involutorial elations is essentially determined. If Γ has a fixed line, then <Δ> is the semidirect product of 0(<Δ>) with a subgroup of order two generated by an elation. If Γ has no fixed line, then Γ acts as a bewegend group on the dual affine plane of π with respect to the line at infinity K.

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