Compact groups with high commuting probability of monothetic subgroups (Q2694731)

Let \(G\) be a compact group and let \(K\) be a subgroup of \(G\). Consider the set \(C=\{(x,y)\in K\times G\mid xy=yx\}\). Notice that \(C\) is closed in the Cartesian product \(K\times G\), since it is the preimage of \(1\) under the continuous map \(f:(x,y)\in K\times G\mapsto [x,y]\in G\). Denote now the normalized Haar measures of \(K\) and \(G\) by \(\nu\) and \(\mu\), respectively. Then, the probability that a random element from \(K\) commutes with a random element from \(G\) is defined as \(Pr(K,G):=(\nu\times\mu)(C)\). The main theorem of the paper states that if \(K\) is a subgroup of a compact group \(G\), then \(Pr(\langle x\rangle,G)>0\) for any \(x\in K\) if and only if \(G\) has an open normal subgroup \(T\) such that \(K/C_K (T)\) is torsion. Remark here that by \(\langle x\rangle\) we mean the subgroup of \(G\) topologically generated by \(\{x\}\). Moreover, some corollaries about the \(p\)-structure of a profinite group or about pro-\(p'\) group admitting an elementary abelian \(p\)-group of automorphisms are derived from this result.