Large p-groups of automorphisms of algebraic curves in characteristic p

Let S be a p-subgroup of the K-automorphism group Aut(X) of an algebraic curve X of genus g≥2 and p-rank γ defined over an algebraically closed field K of characteristic p≥3. Nakajima [27] proved that if γ≥2 then |S|≤pp−2(g−1). If equality holds, X is a Nakajima extremal curve. We prove that if|S|>p2p2−p−1(g−1) then one of the following cases occurs.