Relative commutativity degree for some topological groups

Assume that G is a compact Hausdorff topological group and H is a closed subgroup of G. In this paper, we will define the notion of relative commutativity degree for the subgroup H, which will be a generalization of the concept of relative commutativity degree in the case of finite groups. Then we will prove some properties of relative commutativity degree, that hold for finite groups, for these groups. In particular, we will derive an upper bound for the relative commutativity degree and state and prove a structural theorem for groups that have this upper bound. Additionally, we will provide some examples of compact infinite groups for which the mentioned bounds hold.