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Equivalent conditions for p-nilpotence

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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 $O^{q'}(G)P$ 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.
Twórcy
  • 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
  • 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
Bibliografia
  • [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.
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Bibliografia
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bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1011
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