Jim Propp asks about the uniqueness of the projective plane of order 2.
LEMMA: Up to isomorphism, there is just one projective plane of order 2.
(Def. A proj. plane of order n is a set S of n^2+n+1 points having a
collection (denoted Lns) of n^2+n+1 specified subsets of size n+1 called
"lines", such that a) any two points lie in a unique line, and b) any two
lines intersect in a unique point.)
Given a proj. plane (S;Lns) of order 2, then #(S) = 7. Pick any point p, and
note that there must be exactly 3 lines containing p. WLOG we denote these
lines as {p,a2,a3}, {p,b2,b3}, {p,c2,c3}, where these 6 new elements must all
be distinct. WLOG we may assume the line determined by {a2,b2} is
{a2,b2,c2}. The remaining 3 lines are now completely determined as
{a2,b3,c3},{a3,b2,c3} and {a3,b3,c2}.
--Dan
