Bounds on the Action Degree of Groups
The commutativity degree is the probability that a pair of elements chosen randomly from a group commute. The concept of commutativity degree has been widely discussed by several authors in many directions. One of the important generalizations of commutativity degree is the probability that a random element from a finite group G fixes a random element from a non-empty set S that we call the action degree of groups. In this research, the concept of action degree is further studied where some inequalities and bounds on the action degree of finite groups are determined. Moreover, a general relation between the action degree of a finite group G and a subgroup H is provided. Next, the action degree for the direct product of two finite groups is determined. Previously, the action degree was only deﬁned for ﬁnite groups, the action degree for ﬁnitely generated groups will be deﬁned in this research and some bounds on them are going to be determined.