For any (Hausdorff) compact group G, denote by $\mathrm{cp}(G)$ the probability that a randomly chosen pair of elements of G commute. We prove that there exists a finite group H such that $\mathrm{cp}(G)= {\mathrm{cp}(H)}/{|G:F|^2}$ , where F is the FC-centre of G and H is isoclinic to F with $\mathrm{cp}(F)=\mathrm{cp}(H)$ whenever $\mathrm{cp}(G)>0$ . In addition, we prove that a compact group G with $\mathrm{cp}(G)>\tfrac {3}{40}$ is either solvable or isomorphic to $A_5 \times Z(G)$ , where $A_5$ denotes the alternating group of degree five and the centre $Z(G)$ of G contains the identity component of G.

中文翻译：

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紧凑群体的通勤概率

对于任何 (Hausdorff) 紧群G，表示为$\mathrm{cp}(G)$随机选择的一对元素的概率G通勤。我们证明存在一个有限群H这样$\mathrm{cp}(G)= {\mathrm{cp}(H)}/{|G:F|^2}$， 在哪里F是 FC 的中心G和H等斜于F和$\mathrm{cp}(F)=\mathrm{cp}(H)$每当$\mathrm{cp}(G)>0$. 此外，我们证明了紧群G和$\mathrm{cp}(G)>\tfrac {3}{40}$是可解的或同构的$A_5 \times Z(G)$， 在哪里$A_5$表示五度和中心的交替群$Z(G)$的G包含身份组件G.