We introduced linear algebra lingo in Problem 7, and as you may recall, the real action in linear algebra takes place when we can "transform" a linear vector space in such a way that the linear structure is respected: this amounts to the study of linear transformations. We now want to investigate appropriate actions on near-linear and linear geometries. So what we are looking at then are linear maps T from one near-linear geometry S = (P, L) to another S' = (P', L' ) that respect the geometric structures of S and S'. This means that T must map points to points and lines to lines, but it must also preserve incidence relationships. More exactly,
Definition. A transformation  is said to be linear if T(l) is a line of S' whenever l is a line of S. So the order of T(l) can be no greater than that of l.
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Akin to the study of linear vector spaces, we will say that S and S' are isomorphic if there is a bijective linear transformation such that the inverse transformation  is also linear. In this case we will call T an isomorphism. In the case where S = S' we will call any isomorphism beween S and S' an automorphism. Since the identity transformation is linear and its own inverse, any near-linear geometry is isomorphic to itself.
Note that we will also reter to automormphisms from a near-linear space to itself by the term collineations.
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PROBLEM 13a: (a) Show that if is linear and bijective where both S and S' are linear spaces, then S and S' are isomorphic. That is, show that T -1 must be linear. Show that this need not be the case if S and S' are near-linear spaces. Solution
13a: (b) Show that the set of all automorphisms of any given linear geometry S forms a group, the group of automorphisms of S. Solution
13a: (c) Find the group of automorphisms of the Fano Plane. Solution
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References: Batten, Batten & Beutelspacher.
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