Symmetries of biplanes

In this paper, we first study biplanes D with parameters (v, k, 2), where the block size k∈ { 13 , 16 }. These are the smallest parameter values for which a classification is not available. We show that if k= 13 , then either D is the Aschbacher biplane or its dual, or Aut(D) is a subgroup of the cyclic group of order 3. In the case where k= 16 , we prove that | Aut(D) | divides 2 ⁷· 3 ²· 5 · 7 · 11 · 13. We also provide an example of a biplane with parameters (16, 6, 2) with a flag-transitive and point-primitive subgroup of automorphisms preserving a homogeneous cartesian decomposition. This motivated us to study biplanes with point-primitive automorphism groups preserving a cartesian decomposition. We prove that such an automorphism group is either of affine type (as in the example), or twisted wreath type.