Commuting Probability of Compact Groups

2021/03/21 by Abdollahi, Alireza, Malekan, Meisam soleimani
#20E18 #20P05 #FOS: Mathematics #Group Theory (math.GR) #Probability (math.PR)

paper · doi:10.48550/arxiv.2103.11336

Abstract

For any (Hausdorff) compact group G with the normalized Haar measure \mathbf m_G, denote by \rm cp(G) the probability \mathbf m_{G× G}(\(x,y)∈ G× G | xy=yx\) of commuting a randomly chosen pair of elements of G. Here we prove that if \rm cp(G)>0, then there exists a finite group H such that \rm cp(G)= \frac\rm cp(H)|G:F|², where F is the FC-center of G i.e. the set of all elements of G whose conjugacy classes are finite and H is isoclinic to F with \rm cp(F)=\rm cp(H). The latter equality enables one to transfer many existing results concerning commuting probability of finite groups to one of compact groups. For example, here for a compact group G we prove that if \rm cp(G)>(3)/(40) then either G is solvable or, else G≅ A₅ × T for some abelian group T, in which case \rm cp(G)=(1)/(12); where A₅ denotes the alternating group of degree 5.

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