Alternating groups and flag-transitive triplanes

Let $${\mathcal{D}}$$ be a nontrivial triplane, and G be a subgroup of the full automorphism group of $${\mathcal{D}}$$. In this paper we prove that if $${\mathcal{D}}$$ is a triplane, $${G\leq Aut(\mathcal{D})}$$ is flag-transitive, point-primitive and Soc(G) is an alternating group, then $${\mathcal{D}}$$ is the projective space PG2(3, 2), and $${G\cong A_7}$$ with the point stabiliser $${G_x\cong PSL_3(2)}$$.

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