Geodesic
The Power Mapping as Endomorphism of a Group
Bollettino della Unione matematica italiana, Série 9, Tome 6 (2013) no. 2, pp. 379-387.

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

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 $G$. 
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Tortora, Antonio. The Power Mapping as Endomorphism of a Group. Bollettino della Unione matematica italiana, Série 9, Tome 6 (2013) no. 2, pp. 379-387. http://geodesic.mathdoc.fr/item/BUMI_2013_9_6_2_a6/

[1] J. L. Alperin, A classification of n-abelian groups, Canad. J. Math. 21 (1969), 1238-1244. | DOI | MR | Zbl

[2] R. Baer, Factorization of n-soluble and n-nilpotent groups, Proc. Amer. Math. Soc. 4 (1953), 15-26. | DOI | MR | Zbl

[3] R. Brandl, Infinite soluble groups with the Bell property: a finiteness condition, Monatsh. Math. 104 (1987), 191-197. | fulltext EuDML | DOI | MR | Zbl

[4] R. Brandl - L.-C. Kappe, On n-Bell groups, Comm. Algebra, 17 (1989), 787-807. | DOI | MR | Zbl

[5] C. Delizia - M. R. R. Moghaddam - A. Rhemtulla, The structure of Bell groups, J. Group Theory, 9 (2006), 117-125. | DOI | MR | Zbl

[6] C. Delizia - P. Moravec - C. Nicotera, Locally graded Bell groups, Publ. Math. Debrecen, 71 (2007), 1-9. | MR | Zbl

[7] C. Delizia - A. Tortora, 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. | MR | Zbl

[8] C. Delizia - A. Tortora, Some special classes of n-abelian groups, International J. Group Theory, 1 no. 2 (2012), 19-24. | MR | Zbl

[9] L.-C. Kappe, On n-Levi groups, Arch. Math. 47 (1986), 198-210. | DOI | MR | Zbl

[10] L.-C. Kappe - R. F. Morse, Groups with 3-abelian normal closure, Arch. Math. 51 (1988), 104-110. | DOI | MR | Zbl

[11] L.-C. Kappe - R. F. Morse, Levi-properties in metabelian groups, Contemp. Math. 109 (1990), 59-72. | DOI | MR | Zbl

[12] W. P. Kappe, Die A-Norm einer Gruppe, Illinois J. Math. 5 (1961), 187-197. | MR

[13] F. W. Levi, Notes on Group Theory I, II, J. Indian Math. Soc. 8 (1944), 1-9. | MR

[14] F. W. Levi, Notes on group theory VII, J. Indian Math. Soc. 9 (1945), 37-42. | MR | Zbl

[15] D. Machale, Power mappings and group morphisms, Proc. Roy. Irish. Acad. Sect. A 74 (1974), 91-93. | MR | Zbl

[16] E. Schenkman - L. I. Wade, The mapping which takes each element of a group onto its nth power, Amer. Math. Monthly, 65 (1958), 33-34. | DOI | MR | Zbl

[17] A. Tortora, Some properties of Bell groups, Comm. Algebra, 37 (2009), 431-438. | DOI | MR | Zbl

[18] H. F. Trotter, Groups in which raising to a power is an automorphism, Canad. Math. Bull. 8 (1965), 825-827. | DOI | MR | Zbl