Glasnik Matematicki, Vol. 43, No.2 (2008), 321-335.

FINITE NONNILPOTENT GROUPS WITH FEW HEIGHTS OF NONNORMAL SUBGROUPS

Yakov Berkovich

Department of Mathematics, University of Haifa, Mount Carmel, Haifa 31905, Israel
e-mail: berkov@math.haifa.ac.il


Abstract.   The number of prime factors of the order of a group G (multiplicities counted) is said to be the height of G and denoted by nλ(G). We classify the nonnilpotent groups G with nλ(G) = 2 and nonsolvable groups G with nλ(G) in {3,4}.

2000 Mathematics Subject Classification.   20D25.

Key words and phrases.   Height, solvable group, Carter subgroup.


Full text (PDF) (free access)

DOI: 10.3336/gm.43.2.07


References:

  1. Y. Berkovich, Nonnormal and minimal nonabelian subgroups of a finite group, submitted.

  2. Y. Berkovich, Some criteria for the solvability of finite groups, Sibirsk. Mat. Z. 4 (1963), 723-728.
    MathSciNet

  3. Ya. G. Berkovich and E. M. Zhmud, Characters of Finite Groups. Part 1, Translations of Mathematical Monographs 172, American Mathematical Society, Providence, RI, 1998.
    MathSciNet

  4. N. Blackburn, Generalizations of certain elementary theorems on p-groups, Proc. London Math. Soc. (3) 11 (1961), 1-22.
    MathSciNet     CrossRef

  5. L. E. Dickson, Linear groups with an exposition of the Galois field theory, Dover Publications, Inc., New York, 1958.
    MathSciNet

  6. B. Huppert, Endliche Gruppen. I, Springer-Verlag, Berlin-New York, 1967.
    MathSciNet

  7. Yu. A. Gol'fand, On groups all of whose subgroups are special, Doklady Akad. Nauk SSSR 60 (1948), 1313-1315.
    MathSciNet

  8. D. S. Passman, Nonnormal subgroups of p-groups, J. Algebra 15 (1970), 352-370.
    MathSciNet     CrossRef

  9. L. Redei, Die endlichen einstufig nichtnilpotenten Gruppen, Publ. Math. Debrecen 4 (1956), 303-324.
    MathSciNet

  10. O. Schmidt, Groups all of whose subgroups are nilpotent, Mat. Sb. 31 (1924), 366-372.
    Jahrbuch

  11. O. Schmidt, Groups having only one class of nonnormal subgroups, Mat. Sb. 33 (1926), 161-172.
    Jahrbuch

  12. O. Schmidt, Groups with two classes of nonnormal subgroups, Proc. Seminar on group theory (1938), 7-26.

  13. I. Schur, Über die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen, J. Reine Angew. Math. 127 (1904), 20-50.
    Jahrbuch

  14. J. G. Thompson, Nonsolvable finite groups all of whose local subgroups are solvable, Bull. Amer. Math. Soc. 74 (1968), 383-437.
    MathSciNet     CrossRef

  15. J. H. Walter, The characterization of finite groups with abelian Sylow 2-subgroup, Ann. Math. (2) 89 (1969), 405-514.
    MathSciNet     CrossRef

  16. G. Zappa, Finite groups in which all nonnormal subgroups have the same order, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 13 (2002), 5-16; II, ibid 14 (2003), 13-21.
    MathSciNet     MathSciNet


Glasnik Matematicki Home Page