Department of Mathematics
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A finite projective plane of order n can be viewed as a configuration K = (k, n+1) of k (= n^2+n+1) points with every line being incident with exactly n+1 points. A tight embedding of a configuration K is a group representation of K where the group is of cardinality k+1. Here we show that every finite desarguesian projective plane has a tight embedding in an abelian group. Also we prove that there are infinitely many other configurations of type (n,3) having similar tight embeddings. While no finite projective plane is geometrically realizable over any field of characteristic zero, we use these group embeddings to show that there are infinitely many (n, 3)'s which are realizable over the real projective plane.

Some of these are new results obtained in collaboration with Eric Ens.