“School of Mathematics”
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| Paper IPM / M / 17867 | ||||||||||||||||||
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Let k be any positive integer and G a compact (Hausdorff) group. Let \mfnpk(G) denote the probability that k+1 randomly
chosen elements x1,…,xk+1 satisfy [x1,x2,…,xk+1]=1.
We study the following problem: If \mfnpk(G)>0 then, does there exist an open nilpotent subgroup of class at most k? The answer is positive for profinite groups and we give a new proof. We also prove that the connected component G0 of G is abelian and there exists a closed normal nilpotent subgroup N of class at most k such that G0N is open in G. In particular, a connected compact group G with \mfnpk(G)>0 is abelian.
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