206 CHAPTER 4 Abstract Algebra
Form an equivalence relationRonS^2 by declaring any point on the
sphere to be equivalent with its “antipode.” (Thus on the earth,
the north and south poles would be equivalent.) The quotient set
S^2 /Ris often called thereal projective planeand denotedRP^2.
(a) Give at least a heuristic argument that the constructions of
RP^2 given in this and Exercise 10 are equivalent.
(b) Show that onRP^2 that any pair of distinct points determine
a unique line and that any pair of distinct lines intersect in a
unique point.^74.2 Basics of Group Theory
4.2.1 Motivation—graph automorphisms
We shall start this discussion with one of my favorite questions, namely
which of the following graphs is more “symmetrical?”
While this question might not quite make sense at the outset, it is
my intention to have the reader rely mostly on intuition. Incidently,
(^7) This says that the real projective plane has a “point-line duality” not enjoyed by the usual
Euclidean plane.