Linear representations of subgeometries

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© 2014, Springer Science+Business Media New York. The linear representation (Formula presented.) of a point set K in a hyperplane of (Formula presented.) is a point-line geometry embedded in this projective space. In this paper, we will determine the isomorphisms between two linear representations (Formula presented.) and (Formula presented.), under a few conditions on (Formula presented.). First, we prove that an isomorphism between (Formula presented.) and (Formula presented.) is induced by an isomorphism between the two linear representations (Formula presented.) and (Formula presented.) of their closures (Formula presented.) and (Formula presented.). This allows us to focus on the automorphism group of a linear representation (Formula presented.) of a subgeometry (Formula presented.) embedded in a hyperplane of the projective space (Formula presented.). To this end we introduce a geometry X(n,t,q) and determine its automorphism group. The geometry X(n,t,q) is a straightforward generalization of (Formula presented.) which is known to be isomorphic to the linear representation of a Baer subgeometry. By providing an elegant algebraic description of X(n,t,q) as a coset geometry we extend this result and prove that X(n,t,q) and (Formula presented.) are isomorphic. Finally, we compare the full automorphism group of (Formula presented.) with the “natural” group of automorphisms that is induced by the collineation group of its ambient space.

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Designs, Codes, and Cryptography