Some remarks on groups in which elements with the same p -power commute

Patrizia Longobardi; Mercede Maj

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1999)

  • Volume: 10, Issue: 1, page 11-15
  • ISSN: 1120-6330

Abstract

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In this paper we characterize certain classes of groups G in which, from x p = y p ( x , y ∈ G , p a fixed prime), it follows that x ⁢ y = y ⁢ x . Our results extend results previously obtained by other authors, in the finite case.

How to cite

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Longobardi, Patrizia, and Maj, Mercede. "Some remarks on groups in which elements with the same \( p \)-power commute." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 10.1 (1999): 11-15. <http://eudml.org/doc/252337>.

@article{Longobardi1999,
abstract = {In this paper we characterize certain classes of groups \( G \) in which, from \( x^\{p\} = y^\{p\} \) (\( x, y \in G \), \( p \) a fixed prime), it follows that \( xy = yx \). Our results extend results previously obtained by other authors, in the finite case.},
author = {Longobardi, Patrizia, Maj, Mercede},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {p-powers; p-elements; Locally nilpotent groups; -powers; -elements; locally nilpotent groups; regular -groups; hypercentral groups; central series; normal subgroups; finitely generated subgroups},
language = {eng},
month = {3},
number = {1},
pages = {11-15},
publisher = {Accademia Nazionale dei Lincei},
title = {Some remarks on groups in which elements with the same \( p \)-power commute},
url = {http://eudml.org/doc/252337},
volume = {10},
year = {1999},
}

TY - JOUR
AU - Longobardi, Patrizia
AU - Maj, Mercede
TI - Some remarks on groups in which elements with the same \( p \)-power commute
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1999/3//
PB - Accademia Nazionale dei Lincei
VL - 10
IS - 1
SP - 11
EP - 15
AB - In this paper we characterize certain classes of groups \( G \) in which, from \( x^{p} = y^{p} \) (\( x, y \in G \), \( p \) a fixed prime), it follows that \( xy = yx \). Our results extend results previously obtained by other authors, in the finite case.
LA - eng
KW - p-powers; p-elements; Locally nilpotent groups; -powers; -elements; locally nilpotent groups; regular -groups; hypercentral groups; central series; normal subgroups; finitely generated subgroups
UR - http://eudml.org/doc/252337
ER -

References

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  1. Bianchi, M. - Gillio Berta Mauri, A. - Verardi, L., Groups in which elements with the same p -power commute. Le Matematiche, Supplemento vol. LI, 1996, 53-62. Zbl0902.20008MR1485697
  2. Blackburn, N., Generalizations of certain elementary theorems on p -groups. Proc. London Math. Soc., 11, 1961, 1-22. Zbl0102.01903MR122876
  3. Brailovsky, L. - Freiman, G. A., On two-element subsets in groups. Ann. of the New York Academy of Sciences, 373, 1981, 183-190. Zbl0579.20017MR719039
  4. Brailovsky, L. - Herzog, M., Counting squares of two-subsets in finite groups. Ars Combinatoria, to appear. Zbl0861.20024MR1386941
  5. Bubboloni, D. - Corsi, G., Finite p -groups in which every element of order p is central. To appear. 
  6. Huppert, B., Endliche Gruppen I. Springer-Verlag, Berlin1967. Zbl0412.20002MR224703
  7. Laffey, T. J., A lemma on finite p -groups and some consequences. Proc. Camb. Phil. Soc., 75, 1974, 133-137. Zbl0277.20022MR332961
  8. Laffey, T. J., Centralizers of Elementary Abelian Subgroups in Finite p -groups. J. Algebra, 51, 1978, 88-96. Zbl0374.20024MR472997
  9. Mingyao Xu, , The power structure of finite p -groups. Bull. Austral. Math. Soc., 36, 1987, 1-10. Zbl0607.20010MR897416DOI10.1017/S0004972700026241
  10. Ol'shanskii, A. Yu., Geometry of Defining Relations in Groups. Nauka, Moscow1989 (English translation: Kluwer Academic Publisher, Dordrecht1991). Zbl0732.20019MR1024791
  11. Robinson, D. J. S., Finiteness conditions and generalized soluble groups. Springer-Verlag, Berlin1972. Zbl0243.20033

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