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An isomorphism between two geometries is a 1-1 mapping from one geometry to another, sending points to points and lines to lines. That is, the point set of one geometry is mapped to the point set of another such that there also exists a 1-1 correspondence between the line sets of each geometry. In fact, the isomorphis is defined by how it acts on the point set. If an isomorphism exists between two geometries, we will say that these geometries are isomorphic.
We define any isomorphism of a geometry onto itself as an automorphism. Of course, any geometry is isomorphic to itself, since the identity automorphism (which sends each point to itself, and each line to itself) is an automorphism.
The following two figures are isomorphic:
We can define one isomorphism as follows:
where the point set P = {1, 2, 3, 4} and the point set P` = {a, b, c, d}, and thus the induced map on the line sets is
where the line set L = {{1,2}, {1,3}, {2,3}, {2,4}, {3,4}} and the line set L` = {{a,d}, {c,d}, {a,c}, {a,b}, {b,c}}.
In addition, we can draw the automorphisms of this figure. The identity automorphism, that is, the mapping of a figure to itself, is also included.
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PROBLEM 6(a): Are the following two figures isomorphic? If yes, give an explicit map. If no, then explain why not.
(b) Show that the Fano Plane has exactly 168 automorphisms (including the identity transformation).
(c) Given a geometry, show that its set of automorphisms forms a group.
Recall: To show that a given set G is a group under some operation * we must show:
- that the set is closed under a given operation (i.e., a*b is an element of G for every a, b in G),
- the set has an identity element (i.e., a*e = e*a = a for any a in G)
- each element of the set has an inverse element (i.e., for any a in G, there is a b in G such that a*b = b*a = e).
- the set is associative (i.e., (a*b)*c = a*(b*c) for every a,b,c in G)
In the case of automorphisms of a given geometry, our elements are the mappings and our operation is composition.
Solutions
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References: Batten, Bennett, Polster.
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