Glasnik Matematicki, Vol. 43, No.1 (2008), 59-95.

ON THE NUMBER OF SUBGROUPS OF GIVEN TYPE IN A FINITE p-GROUP

Yakov Berkovich

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

Abstract.   We study the p-groups G containing exactly p+1 subgroups of order pp and exponent p. A number of counting theorems and results on subgroups of maximal class and p-groups with few subgroups of given type are also proved. Counting theorems play crucial role in the whole paper.

2000 Mathematics Subject Classification.   20D15.

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

Full text (PDF) (free access)

DOI: 10.3336/gm.43.1.06

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, Part I, in preparation.

  3. Y. Berkovich, On subgroups of finite p-groups, J. Algebra 240 (2000), 198-240.
    MathSciNet     CrossRef

  4. Y. Berkovich, On abelian subgroups of p-groups, J. Algebra 199 (1998), 262-280.
    MathSciNet     CrossRef

  5. Y. Berkovich, Alternate proofs of some basic theorems of finite group theory, Glas. Mat. 40 (2005), 207-233.
    MathSciNet     CrossRef

  6. Y. Berkovich, On the number of elements of given order in a finite p-group, Israel J. Math. 73 (1991), 107-112.
    MathSciNet     CrossRef

  7. Y. Berkovich and Z. Janko, Groups of Prime Power Order, Part II, in preparation.

  8. Y. Berkovich and Z. Janko, Structure of finite p-groups with given subgroups, Contemporary Mathematics 402 (2006), 13-93.
    MathSciNet

  9. N. Blackburn, On a special class of p-groups, Acta Math. 1 (1958), 45-92.
    MathSciNet     CrossRef

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

  11. P. Hall, A contribution to the theory of groups of prime power order, Proc. London Math. Soc. 36 (1933), 29-95.
    Jahrbuch

  12. P. Hall, On a theorem of Frobenius, Proc. London Math. Soc. 40 (1936), 468--501.
    Jahrbuch

  13. M. Hall, Jr. and J.K. Senior, The groups of order 2n (n ≤ 6), Macmillan, NY, 1964.
    MathSciNet

  14. R. James, 2-groups of almost maximal class, J. Austral. Math. Soc. (Ser A) 19 (1975), 343-357; corrigendum, ibid. 35 (1983), 307.
    MathSciNet

  15. Z. Janko, A classification of finite 2-groups with exactly three involutions, J. Algebra 291 (2005), 505-533.
    MathSciNet     CrossRef

  16. Z. Janko, Finite 2-groups with exactly four cyclic subgroups of order 2n, J. reine angew. Math. 566 (2004), 135-181.
    MathSciNet     CrossRef

  17. Z. Janko, Finite 2-groups G with |Ω2(G)| = 16, Glas. Mat. 40 (2005), 71-86.
    MathSciNet     CrossRef

  18. A. Mann, Conjugacy classes in finite groups, Israel J. Math. 31 (1978), 78-84.
    MathSciNet     CrossRef

  19. D. S. Passman, Nonnormal subgroups of p-groups, J. Algebra 15 (1970), 352-370.
    MathSciNet     CrossRef
Glasnik Matematicki Home Page