Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

FINITE GROUPS WITH A CYCLIC NORM QUOTIENT

  • Wang, Junxin (Department of Applied Mathematics, Shanxi University of Finance and Economics)
  • Received : 2015.02.22
  • Published : 2016.03.31
Download PDF
⟨ Previous Next ⟩

Abstract

The norm N(G) of a group G is the intersection of the normalizers of all the subgroups of G. In this paper, the structure of finite groups with a cyclic norm quotient is determined. As an application of the result, an interesting characteristic of cyclic groups is given, which asserts that a finite group G is cyclic if and only if Aut(G)/P(G) is cyclic, where P(G) is the power automorphism group of G.

Keywords

References

  1. R. Baer, Der Kern, eine charakteristische Untergruppe, Compositio Math. 1 (1934), 254-283.
  2. R. Baer, Gruppen mit Hamilton Schem Kern, Compositio Math. 2 (1935), 241-246.
  3. R. Baer, Zentrum und Kern von Gruppen mit Elementen unendlicher Ordnung, Com-positio Math. 2 (1935), 247-249.
  4. R. Baer, Norm and hypernorm, Publ. Math. Debrecen 4 (1956), 347-350.
  5. C. D. H. Cooper, Power automorphisms of a group, Math. Z. 107 (1968), 335-356. https://doi.org/10.1007/BF01110066
  6. B. Huppert, Endliche Gruppen I, Springer-Verlag, New York, 1967.
  7. H. Kurzweil, Endliche Gruppen, Springer-Verlag, Berlin and New York, 1977.
  8. D. J. S. Robinson, A Course in the Theory of Groups, Springer-Verlag, New York, 1982.
  9. E. Schenkman, On the norm of a group, Illinois J. Math. 4 (1960), 150-152.
  10. J. X. Wang and X. Y. Guo, On the norm of finite groups, Algebra Colloq. 14 (2007), no. 4, 605-612. https://doi.org/10.1142/S1005386707000557
  11. J. X. Wang and X. Y. Guo, Finite groups with its power automorphism groups having small indices, Acta Math. Sin. (Engl. Ser.) 25 (2009), no. 7, 1097-1108. https://doi.org/10.1007/s10114-009-8060-4
  12. J. X. Wang and X. Y. Guo, Finite groups with a pre-fixed-point-free power automorphism, Comm. Algebra 41 (2013), no. 9, 3241-3251. https://doi.org/10.1080/00927872.2012.682673