
3.4. FINDING GEOMETRIC AUTOMORPHISMS 97
First, suppose that G has a fixed point free automorphism β. It is clear that one can
extend β to G
′
to give an automorphism that satisfies part (a) of Theorem 3.2, and so G
′
has an axial geometric automorphism group.
Now suppose that G
′
has an axial geometric automorphism γ.
We claim that γ cannot map a vertex w of H to a vertex v of G, or to one of the new
vertices w
v
i
. This is bec aus e every vertex of G i s adjacent to a clique of size n + 1, while
no vertex of H has this property. Further, γ cannot map a vertex of G to one of the new
vertices w
v
i
, because each w
v
i
is in a clique of size n + 1, and none of the ...