Permutation groups with restricted stabilizers

Fix a positive integer d and let Γ be the class of finite groups without sections isomorphic to alternating group A . The in were studied by Babai, Cameron Pálfy 1980s they determined bounds on order primitive permutation with this property, which have found wide range applications. Subsequently, results base sizes such also obtained. In paper we replace structural conditions restrictions its point stabilizers, obtain similar, sometimes stronger conclusions. For example, prove that there is linear function f size any stabilizers at most ( ) This generalizes recent result first author solvable stabilizers. non-affine results, assuming only c points lie We show if G degree n whose -point , then | ⩽ 1 + o / e − asymptotically extends improves upper bound obtained ∈