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The subdivision graph S(Σ) of a graph Σ is obtained from Σ by 'adding a vertex' in the middle of every edge of Σ. Various symmetry properties of S(Σ) are studied. We prove that, for a connected graph Σ, S(Σ) is locally s-arc transitive if and only if Σ is ⌈ s+1 2 ⌉-arc transitive. The diameter of S(Σ) is 2d + δ, where Σ has diameter d and 0 δ 2, and local s-distance transitivity of S(Σ) is defined for 1 s 2d + δ. In the general case where s 2d − 1 we prove that S(Σ) is locally s-distance transitive if and only if Σ is ⌈ s+1 2 ⌉-arc transitive. For the remaining values of s, namely 2d s 2d + δ, we classify the graphs Σ for which S(Σ) is locally s-distance transitive in the cases, s 5 and s 15 + δ. The cases max{2d, 6} s min{2d + δ, 14 + δ} remain open.

In this article, we study symmetric designs with λ prime admitting a flag-transitive and point-primitive automorphism group G of almost simple type with socle X. We prove that either D is one of the six well-known examples of biplanes and triplanes, or D is the pointhyperplane design of PG(n−1, q) with λ = (q n−2 −1)/(q −1) prime and X = PSL n (q).