Translation triple systems (Q1262526)

The author proves the following characterization of the affine plane of order 3, AG(2,3). Let \(\pi\) be a finite projective plane of order n. Assume an Steiner triple system STS(v), S, is embedded in \(\pi\) as a maximal arc and \(\ell\) is an exterior line to S. Denote by \(T_{\ell}(S)\) the group of elations of \(\pi\) with axis \(\ell\) each of which fixes S (setwise). Suppose that G is a collineation group of \(\pi\) fixing both S and \(\ell\), and block-transitive on S. Then \(G\supseteq T_{\ell}(S)\), \(\pi\) is the projective plane of order 3 and \(S=AG(2,3)\).