Z. Sepehrizadeh - Department of Pure Mathematics, Shahrekord University , P.O.Box ۱۱۵, Shahrekord, Iran. Email: zohreh.sepehri@gmail.com
M. R. Rismanchian - Department of Pure Mathematics, Shahrekord University , P.O.Box ۱۱۵, Shahrekord, Iran. Email: rismanchian۱۳۳@gmail.com, rismanchian@sku.ac.ir

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.

Autocommutativity degree, Characteristic degree, p-group