We prove that the order of the abelian automorphism group G of a minimal complex surface S of general type is bounded from above by 12.5K +100 provided the geometric genus of the surface is at least 7.The upper bound is reached for infinitely many families of surfaces whose geometric genus can be arbitrarily large.We will present also an example to show that the lower bound 7 on the geometric genus can not be replaced by 3.This is a joint work with Xin Lv.