Abstract
Let G be a finite p-group whose derived subgroup G′ can be generated by 2 elements. If G′ is abelian, Guralnick proved that every element of G′ is a commutator. In this paper, we prove that the condition that G′ should be abelian is not needed. Even more, we prove that every element of G′ is a commutator of the form [x, g] for afixed x ∊ G.
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References
Y. Berkovich, Groups of Prime Power Order. Vol. 1, de Gruyter Expositions in Mathematics, Vol. 46, Walter de Gruyter, Berlin, 2008.
N. Blackburn, On prime-power groups in which the derived group has two generators, Proceedings of the Cambridge Philosophical Society 53 (1957), 19–27.
J. D. Dixon, M. P. F. du Sautoy, A. Mann and D. Segal, Analytic Pro-p Groups, Cambridge Studies in Advanced Mathematics, Vol. 61, Cambridge University Press, Cambridge, s1999.
R. M. Guralnick, Commutators and commutator subgroups, Advances in Mathematics 45 (1982), 319–330.
L.-C. Kappe and R. F. Morse, On commutators in p-groups, Journal of Group Theory 8 (2005), 415–429.
L.-C. Kappe and R. F. Morse, On commutators in groups, in Groups St. Andrews 2005. Vol. 2, London Mathematical Society Lecture Note Series, Vol. 340, Cambridge University Press, Cambridge, 2007, pp. 531–558.
E. I. Khukhro, p-Automorphisms of Finite p-groups, London Mathematical Society Lecture Note Series, Vol. 246, Cambridge University Press, Cambridge, 1998.
M. W. Liebeck, E. A. O’Brien, A. Shalev and P. H. Tiep, The Ore conjecture, Journal of the European Mathematical Society 12 (2010), 939–1008.
I. D. Macdonald, On cyclic commutator subgroups, Journal of the London Mathematical Society 38 (1963), 419–422.
I. D. Macdonald, The Theory of Groups, Clarendon Press, Oxford, 1968.
D. M. Rodney, On cyclic derived subgroups, Journal of the London Mathematical Society 8 (1974), 642–646.
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Both authors are supported by the Spanish Government grant MTM2017-86802-P and by the Basque Government grant IT974-16. The first author is also supported by a predoctoral grant of the University of the Basque Country, and the second author, by the Spanish Government grant MTM2014-53810-C2-2-P. Grants of the Spanish Government are partially funded from FEDER funds.
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Fernández-Alcober, G.A., de las Heras, I. Commutators in finite p-groups with 2-generator derived subgroup. Isr. J. Math. 232, 109–124 (2019). https://doi.org/10.1007/s11856-019-1864-8
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DOI: https://doi.org/10.1007/s11856-019-1864-8