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Extra info for Algorithmic Geometry [auth. unkn.]

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Furthermore, let G be an automorphism group of & such that (R, cp) is G-admissible and let G{x) be the action which G(x) induces on resg(x), then (R\(x),cpx) is G(x)-admissible. 7 c-extensions 35 Proof. For y € FIX the order of '3) = 1) which implies (R3). D The above result possesses the following reformulation in terms of the collinearity graph T of'S. 3 Let (R,(p) be a representation of^ which is G-admissible for an automorphism group G of (S, let Y be the collinearity graph of

R j= 1) then cp(u) is abelian of order 21 whenever u is an element of type i in ^. 7 c-extensions Let'S be a geometry of rank n > 2 with its diagram of the form X 2 2 2 2 q (in particular, ^ can be a P - or T-geometry), and let G be a flag-transitive automorphism group of 'S. Let {R, maps the point-set of \$ into the set of 36 General features involutions in R.

In particular if u is a plane of IS then (cp*(u), cpu) is a representation of the projective plane pg(2,2) of order 2 formed by the points and lines of ^ incident to u, in particular cp'(u) is abelian of order at most 2 3 . Let x be a point in <§ and ^ = (I1X,LX) be the point-line system of res#(x), which means that ITX and Lx are the lines and planes in ^ incident to x. 2 In the above terms let (R, cp) be a representation of <§, x be a point ofy, R\(x) be the subgroup in R generated by the elements cp(y) taken for all points y collinear to x, R\(x) = R\{x)/cp{x).