Abstract
Let G be a finite group. The nonsoluble length \(\lambda (G)\) of G is the number of nonsoluble factors in a shortest normal series of G, each of whose factors either is soluble or is a direct product of nonabelian simple groups. In the present paper, we are concerned with bounding \(\lambda (G)\) in terms of coprime commutators, that is, commutators [a, b] with \((|a|,|b|)=1\). Let e be a positive integer and \(2^f\) the maximal 2-power dividing e. We show that if \(x^e=1\) whenever x is a coprime commutator in G, then \(\lambda (G)\le f\).
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Acknowledgements
This work was done during a visit of the first and the second authors at the Department of Mathematics of the University of Salerno. They thank the department for hospitality. The first author acknowledges financial support from FAPDF and GNSAGA - INdAM.
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Shumyatsky, P., Sica, C. & Tota, M. Coprime commutators and the nonsoluble length of a finite group. Arch. Math. 120, 3–8 (2023). https://doi.org/10.1007/s00013-022-01810-5
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DOI: https://doi.org/10.1007/s00013-022-01810-5