Lower central words in finite p-groups.

  1. de las Heras, Iker
  2. Morigi, Marta
Zeitschrift:
Publicacions matematiques

ISSN: 0214-1493

Datum der Publikation: 2021

Ausgabe: 65

Nummer: 1

Seiten: 243-269

Art: Artikel

DOI: 10.5565/PUBLICACIONSMATEMATIQUES.V65I1.383697 DIALNET GOOGLE SCHOLAR lock_openDDD editor

Andere Publikationen in: Publicacions matematiques

Zusammenfassung

It is well known that the set of values of a lower central word in a group G need not be a subgroup. For a fixed lower central word γr and for p ≥ 5, Guralnick showed that if G is a finite p-group such that the verbal subgroup γr(G) is abelian and 2-generator, then γr(G) consists only of γr-values. In this paper we extend this result, showing that the assumption that γr(G) is abelian can be dropped. Moreover, we show that the result remains true even if p= 3. Finally, we prove that the analogous result for pro-p groups is true.

Bibliographische Referenzen

  • C. Acciarri and P. Shumyatsky, On profinite groups in which commutators are covered by finitely many subgroups, Math. Z. 274(1–2) (2013), 239–248. DOI: 10.1007/s00209-012-1067-z.
  • Y. Berkovich, “Groups of Prime Power Order”, Vol. 1, With a foreword by Zvonimir Janko, De Gruyter Expositions in Mathematics 46, Walter de Gruyter GmbH & Co. KG, Berlin, 2008. DOI: 10.1515/9783110208221.
  • N. Blackburn, On prime-power groups in which the derived group has two generators, Proc. Cambridge Philos. Soc. 53(1) (1957), 19–27. DOI: 10.1017/ s0305004100031959.
  • R. S. Dark and M. L. Newell, On conditions for commutators to form a subgroup, J. London Math. Soc. (2) 17(2) (1978), 251–262. DOI: 10.1112/jlms/ s2-17.2.251.
  • [I. de las Heras, Commutators in finite p-groups with 3-generator derived subgroup, J. Algebra 546 (2020), 201–217. DOI: 10.1016/j.jalgebra.2019.10.027.
  • J. D. Dixon, M. P. F. du Sautoy, A. Mann, and D. Segal, “Analytic Pro-p Groups”, Second edition, Cambridge Studies in Advanced Mathematics 61, Cambridge University Press, Cambridge, 1999. DOI: 10.1017/CBO9780511470882.
  • G. A. Fernandez-Alcober and I. de las Heras ´ , Commutators in finite p-groups with 2-generator derived subgroup, Israel J. Math. 232(1) (2019), 109–124. DOI: 10.1007/s11856-019-1864-8.
  • R. M. Guralnick, Commutators and commutator subgroups, Adv. in Math. 45(3) (1982), 319–330. DOI: 10.1016/S0001-8708(82)80008-X.
  • R. M. Guralnick, Generation of the lower central series, Glasgow Math. J. 23(1) (1982), 15–20. DOI: 10.1017/S0017089500004742.
  • R. M. Guralnick, Generation of the lower central series II, Glasgow Math. J. 25(2) (1984), 193–201. DOI: 10.1017/S0017089500005619.
  • L.-C. Kappe and R. F. Morse, On commutators in groups, in: “Groups St Andrews 2005”, Vol. 2, London Math. Soc. Lecture Note Ser. 340, Cambridge Univ. Press, Cambridge, 2007, pp. 531–558. DOI: 10.1017/CBO9780511721205.018.
  • E. I. Khukhro, “p-Automorphisms of Finite p-Groups”, London Mathematical Society Lecture Note Series 246, Cambridge University Press, Cambridge, 1998. DOI: 10.1017/CBO9780511526008.
  • M. W. Liebeck, E. A. O’Brien, A. Shalev, and P. H. Tiep, The Ore conjecture, J. Eur. Math. Soc. (JEMS) 12(4) (2010), 939–1008. DOI: 10.4171/JEMS/ 220.
  • I. D. Macdonald, On cyclic commutator subgroups, J. London Math. Soc. 38(1) (1963), 419–422. DOI: 10.1112/jlms/s1-38.1.419.
  • I. D. Macdonald, “The Theory of Groups”, Clarendon Press, Oxford, 1968.
  • D. J. S. Robinson, “A Course in the Theory of Groups”, Second edition. Graduate Texts in Mathematics 80, Springer-Verlag, New York, 1996. DOI: 10.1007/ 978-1-4419-8594-1. [17] D. M. Rodney, On cyclic derived subgroups, J. London Math. Soc. (2) 8(4) (1974), 642–646. DOI: 10.1112/jlms/s2-8.4.642.
  • D. M. Rodney, Commutators and abelian groups, J. Austral. Math. Soc. Ser. A 24(1) (1977), 79–91. DOI: 10.1017/s1446788700020073.
  • D. Segal, “Words: Notes on Verbal Width in Groups”, London Mathematical Society Lecture Note Series 361, Cambridge University Press, Cambridge, 2009. DOI: 10.1017/CBO9781139107082.
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