Glasnik Matematicki, Vol. 44, No.1 (2009), 167-175.

FINITE p-GROUPS IN WHICH SOME SUBGROUPS ARE GENERATED BY ELEMENTS OF ORDER p

Yakov Berkovich

Department of Mathematics, University of Haifa, Mount Carmel, Haifa 31905, Israel

Abstract.   We prove that if a p-group G of exponent pe > p has no subgroup H such that |Ω1(H)| = pp and H/Ω1(H) is cyclic of order pe-1 ≥ p and H is regular provided e = 2, then G is either absolutely regular or of maximal class. This result supplements the fundamental theorem of Blackburn on p-groups without normal subgroups of order pp and exponent p. For p > 2, we deduce even stronger result than (respective result for p = 2 is unknown) a theorem of Bozikov and Janko.

2000 Mathematics Subject Classification.   20D15.

Key words and phrases.   p-groups of maximal class, regular and absolutely regular p-groups, metacyclic p-groups, Lp-groups.

Full text (PDF) (free access)

DOI: 10.3336/gm.44.1.09

References:

  1. Y. Berkovich, On subgroups and epimorphic images of finite p-groups, J. Algebra 248 (2002), 472-553.
    MathSciNet     CrossRef

  2. Y. Berkovich, Groups of Prime Power Order, volume 1, Walter de Gruyter, 2008.
    MathSciNet

  3. Y. Berkovich, A generalization of theorems of Ph. Hall and N. Blackburn and an application to non-regular p-groups, Math. USSR Izv. 35 (1971), 815-844.
    MathSciNet     CrossRef

  4. Y. Berkovich and Z. Janko, Structure of finite $p$-groups with given subgroups, Contemp. Math. 402 (2006), 13-93.
    MathSciNet

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

  6. N. Blackburn, On a special class of p-groups, Acta Math. 100 (1958), 45-92.
    MathSciNet     CrossRef

  7. N. Blackburn, Note on a paper of Berkovich, J. Algebra 24 (1973), 323-334.
    MathSciNet     CrossRef

  8. Z. Bozikov and Z. Janko, Finite 2-groups all of whose nonmetacyclic subgroups are generated by involutions, Arch. Math. 90 (2008), 14-17.
    MathSciNet     CrossRef

  9. P. Hall, A contribution to the theory of groups of prime power order, Proc. London Math. Soc. 2, No. 36 (1933), 29-95.

  10. B. Huppert, Endliche Gruppen. I., Springer, Berlin, 1967.
    MathSciNet

Glasnik Matematicki Home Page