2021/09/30 by Azevedo, João, Shumyatsky, Pavel · 1 citation
#FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2110.00049
If K is a closed subgroup of a compact group G, the probability that randomly chosen pair of elements from K and G commute is denoted by Pr(K,G). Say that a subgroup K≤ G is ε-central in G if Pr(⟨ g ⟩,G)≥ ε for any g in K. Here ⟨ g ⟩ denotes the monothetic subgroup generated by g∈ G. Our main result is that if K is ε-central in G, then there is an ε-bounded number e and a normal subgroup T≤ G such that the index [G:T] and the order of the commutator subgroup [Ke,T] both are finite and ε-bounded. In particular, if G is a compact group for which there is ε>0 such that Pr(⟨ g ⟩,G)≥ ε for any g ∈ G, then there is an ε-bounded number e and a normal subgroup T such that the index [G:T] and the order of [Ge,T] both are finite and ε-bounded.