The fixity of permutation groups

The fixity of a finite permutation group G is the maximal number of fixed points of a non-trivial element of G. We analyze the structure of non-regular permutation groups G with given fixity f. We show that if G is transitive and nilpotent, then it has a subgroup whose index and nilpotency class are both f-bounded.We also show that if G is primitive, then either it has a solublesubgroup of f-bounded index and derived length at most 4, or F*(G) is PSL(2, q) or Sz(q) in the natural permutation representations of degree q + 1, q² + 1 respectively.