Compact groups with probabilistically central monothetic subgroups

If K is a closed subgroup of compact group G, the probability that randomly chosen pair elements from and G commute denoted by Pr(K, G). Say ≤ ϵ-central in if Pr(〈g〉, G) ≥ ϵ for any g K. Here 〈g〉 denotes monothetic generated ∈ G. Our main result then there an ϵ-bounded number e normal T such both index [G:T] order commutator [Ke, T] are finite ϵ-bounded. In particular, which > 0 [Ge,