on the characteristic degree of finite groups

In this article we introduce and study the concept of characteristic degree of a subgroup in a finite group. we define the characteristic degree of a subgroup h in a finite group g as the ratio of the number of all pairs (h,α) ∈ h×aut(g) such that h^α∈h, by the order of h × aut(g), where aut(g) is the automorphisms group of g. this quantity measures the probability that h can be characteristic in g. we determine the upper and lower bounds for this probability. we also obtain a special lower bound, when h is a cyclic p-subgroup of g.