DM-GAA

ISSN 1509-9415 (print version)

ISSN 2084-0373 (electronic version)

https://doi.org/10.7151/dmgaa

Discussiones Mathematicae - General Algebra and Applications

Cite Score: 0.4

SJR: 0.203

SNIP: 0.562

MCQ: 0.12

Index Copernicus: 121.02

Discussiones Mathematicae - General Algebra and Applications

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Discussiones Mathematicae General Algebra and Applications 20(1) (2000) 129-139
DOI: https://doi.org/10.7151/dmgaa.1011

EQUIVALENT CONDITIONS FOR P-NILPOTENCE

Keresztély Corrádi and Erzsébet Horváth

Eötvös Loránd University, Department of Computer Techn.
H-1088 Budapest, Múzeum krt. 6-8
Technical University of Budapest, Department of Algebra
H-1521 Budapest, Muegyetem rkp. 3-9

e-mail: he@math.bme.hu

Abstract

In the first part of this paper we prove without using the transfer or characters the equivalence of some conditions, each of which would imply p-nilpotence of a finite group G. The implication of p-nilpotence also can be deduced without the transfer or characters if the group is p-constrained. For p-constrained groups we also prove an equivalent condition so that Oq′(G)P should be p-nilpotent. We show an example that this result is not true for some non-p-constrained groups.

In the second part of the paper we prove a generalization of a theorem of Itô with the help of the knowledge of the irreducible characters of the minimal non-nilpotent groups.

Keywords: p-nilpotent group, p-constrained group, character of a group, Schmidt group, Thompson-ordering, Sylow p-group.

1991 Mathematics Subject Classification: 20D15 and 20C15.

References

[1] J.L. Alperin, Centralizers of abelian normal subgroups of p-groups, J. Algebra 1 (1964), 110-113.
[2] K. Corrádi, On certain properties of centralizers hereditary to the factor group, Publ. Math. (Debrecen) 37 (1990), 203-206.
[3] K. Corrádi and E. Horváth, Steps towards an elementary proof of Frobenius' theorem, Comm. Algebra 24 (1996), 2285-2292.
[4] K. Corrádi and E. Horváth, Normal π-complement theorems, Arch. Math. (Basel) 71 (1998), 262-269.
[5] D. Gorenstein, ``Finite groups", Chelsea Publ. Comp., New York 1980.
[6] B. Huppert, ``Endliche Gruppen", Springer-Verlag, Berlin 1967.
[7] I.M. Isaacs, ``Character theory of finite groups", Dover Publ., Inc., New York 1994.
[8] N. Itô, On a theorem of H.F. Blichfeldt, Nagoya Math. J. 5 (1953), 75-77.

Received 25 January 1999


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