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The probability that an ordered pair of elements is an Engel pair. (English) Zbl 1454.20130

Summary: Let \(G\) be a finite group. We denote by \(ep(G)\) the probability that \([x_{,n}y] = 1\) for two randomly chosen elements \(x\) and \(y\) of \(G\) and some positive integer \(n\). For \(x\in G\) we denote by \(E_G(x)\) the subset \(\{y\in G: [y_{,n}x] =1\text{ for some integer }n\}\). \(G\) is called an \(E\)-group if \(E_G(x)\) is a subgroup of \(G\) for all \(x\in G\). Among other results, we prove that if \(G\) is an non-abelian \(E\)-group with \(ep(G)>\frac16\), then \(G\) is not simple and minimal non-solvable.

MSC:

20P05 Probabilistic methods in group theory
20D60 Arithmetic and combinatorial problems involving abstract finite groups
20F45 Engel conditions
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