We show that there is a positive constant δ < 1 such that the probability of satisfying either the 2-Engel identity [X1, X2, X2] = 1 or the metabelian identity [[X1, X2], [X3, X4]] = 1 in a finite group is either 1 or at most δ
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The first and the fourth authors have been supported by the “National Group for Algebraic and Geometric Structures, and their Applications” (GNSAGA - INdAM).
The second author has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 748129. He has also been supported by the Spanish Government grant MTM2017-86802-P and by the Basque Government grant IT974-16.
The third author has been partially supported by the Slovenian Research Agency (research core funding No. P1-0222, and projects No. J1-8132, J1-7256 and N1-0061).
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Delizia, C., Jezernik, U., Moravec, P. et al. Gaps in probabilities of satisfying some commutator-like identities. Isr. J. Math. 237, 115–140 (2020). https://doi.org/10.1007/s11856-020-1999-7