The Power Mapping as Endomorphism of a Group

Antonio Tortora

Bollettino dell'Unione Matematica Italiana (2013)

  • Volume: 6, Issue: 2, page 379-387
  • ISSN: 0392-4041

Abstract

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Let n ≠ 0 , 1 be an integer. A group G is said to be n -abelian if the mapping f n : x → x n is an endomorphism of G . Then ( x ⁢ y ) n = x n ⁢ y n for all x , y ∈ G , from which it follows [ x n , y ] = [ x , y ] n = [ x ; y n ] . In this paper we investigate groups G such that f n is a monomorphism or an epimorphism of G . We also deal with the connections between n -abelian groups and groups satisfying the identity [ x n , y ] = [ x , y ] n = [ x ; y n ] . Finally, we provide an arithmetic description of the set of all integers n such that f n is an automorphism of a given group G .

How to cite

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Tortora, Antonio. "The Power Mapping as Endomorphism of a Group." Bollettino dell'Unione Matematica Italiana 6.2 (2013): 379-387. <http://eudml.org/doc/294049>.

@article{Tortora2013,
abstract = {Let $n \neq 0$, $1$ be an integer. A group $G$ is said to be $n$-abelian if the mapping $f_\{n\} \colon x \to x^\{n\}$ is an endomorphism of $G$. Then $(xy)^\{n\} = x^\{n\}y^\{n\}$ for all $x$, $y \in G$, from which it follows $[x^\{n\}, y] = [x, y]^\{n\} = [x; y^\{n\}]$. In this paper we investigate groups $G$ such that $f_\{n\}$ is a monomorphism or an epimorphism of $G$. We also deal with the connections between $n$-abelian groups and groups satisfying the identity $[x^\{n\}, y] = [x, y]^\{n\} = [x; y^\{n\}]$. Finally, we provide an arithmetic description of the set of all integers $n$ such that $f_\{n\}$ is an automorphism of a given group $G$.},
author = {Tortora, Antonio},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {379-387},
publisher = {Unione Matematica Italiana},
title = {The Power Mapping as Endomorphism of a Group},
url = {http://eudml.org/doc/294049},
volume = {6},
year = {2013},
}

TY - JOUR
AU - Tortora, Antonio
TI - The Power Mapping as Endomorphism of a Group
JO - Bollettino dell'Unione Matematica Italiana
DA - 2013/6//
PB - Unione Matematica Italiana
VL - 6
IS - 2
SP - 379
EP - 387
AB - Let $n \neq 0$, $1$ be an integer. A group $G$ is said to be $n$-abelian if the mapping $f_{n} \colon x \to x^{n}$ is an endomorphism of $G$. Then $(xy)^{n} = x^{n}y^{n}$ for all $x$, $y \in G$, from which it follows $[x^{n}, y] = [x, y]^{n} = [x; y^{n}]$. In this paper we investigate groups $G$ such that $f_{n}$ is a monomorphism or an epimorphism of $G$. We also deal with the connections between $n$-abelian groups and groups satisfying the identity $[x^{n}, y] = [x, y]^{n} = [x; y^{n}]$. Finally, we provide an arithmetic description of the set of all integers $n$ such that $f_{n}$ is an automorphism of a given group $G$.
LA - eng
UR - http://eudml.org/doc/294049
ER -

References

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  1. ALPERIN, J. L., A classification of n-abelian groups, Canad. J. Math.21 (1969), 1238-1244. Zbl0213.29901MR248204DOI10.4153/CJM-1969-136-1
  2. BAER, R., Factorization of n-soluble and n-nilpotent groups, Proc. Amer. Math. Soc.4 (1953), 15-26. Zbl0050.02201MR53109DOI10.2307/2032195
  3. BRANDL, R., Infinite soluble groups with the Bell property: a finiteness condition, Monatsh. Math.104 (1987), 191-197. Zbl0626.20026MR918472DOI10.1007/BF01547952
  4. BRANDL, R. - KAPPE, L.-C., On n-Bell groups, Comm. Algebra, 17 (1989), 787-807. Zbl0672.20019MR990978DOI10.1080/00927878908823759
  5. DELIZIA, C. - MOGHADDAM, M. R. R. - RHEMTULLA, A., The structure of Bell groups, J. Group Theory, 9 (2006), 117-125. Zbl1108.20026MR2195841DOI10.1515/JGT.2006.007
  6. DELIZIA, C. - MORAVEC, P. - NICOTERA, C., Locally graded Bell groups, Publ. Math. Debrecen, 71 (2007), 1-9. Zbl1135.20028MR2340029
  7. DELIZIA, C. - TORTORA, A., On n-abelian groups and their generalizations, Groups St Andrews 2009 in Bath, Volume I, London Math. Soc. Lecture Note Ser.387 (Cambridge University Press, Cambridge, 2011), 244-255. Zbl1239.20049MR2858862
  8. DELIZIA, C. - TORTORA, A., Some special classes of n-abelian groups, International J. Group Theory, 1 no. 2 (2012), 19-24. Zbl1262.20034MR2925514
  9. KAPPE, L.-C., On n-Levi groups, Arch. Math.47 (1986), 198-210. Zbl0605.20033MR861866DOI10.1007/BF01191994
  10. KAPPE, L.-C. - MORSE, R. F., Groups with 3-abelian normal closure, Arch. Math.51 (1988), 104-110. Zbl0653.20037MR959384DOI10.1007/BF01206466
  11. KAPPE, L.-C. - MORSE, R. F., Levi-properties in metabelian groups, Contemp. Math.109 (1990), 59-72. Zbl0712.20020MR1076377DOI10.1090/conm/109/1076377
  12. KAPPE, W. P., Die A-Norm einer Gruppe, Illinois J. Math.5 (1961), 187-197. MR121399
  13. LEVI, F. W., Notes on Group Theory I, II, J. Indian Math. Soc.8 (1944), 1-9. MR10590
  14. LEVI, F. W., Notes on group theory VII, J. Indian Math. Soc.9 (1945), 37-42. Zbl0061.02611MR10590
  15. MACHALE, D., Power mappings and group morphisms, Proc. Roy. Irish. Acad. Sect. A74 (1974), 91-93. Zbl0274.20052MR357633
  16. SCHENKMAN, E. - WADE, L. I., The mapping which takes each element of a group onto its nth power, Amer. Math. Monthly, 65 (1958), 33-34. Zbl0079.02901MR103225DOI10.2307/2310303
  17. TORTORA, A., Some properties of Bell groups, Comm. Algebra, 37 (2009), 431-438. Zbl1167.20021MR2493711DOI10.1080/00927870802248613
  18. TROTTER, H. F., Groups in which raising to a power is an automorphism, Canad. Math. Bull.8 (1965), 825-827. Zbl0135.05101MR191975DOI10.4153/CMB-1965-063-4

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