Glasnik Matematicki, Vol. 45, No.2 (2010), 431-439.

ALTERNATE PROOFS OF TWO CLASSICAL THEOREMS ON FINITE SOLVABLE GROUPS AND SOME RELATED RESULTS FOR P-GROUPS

Yakov Berkovich

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


Abstract.   We offer a new proof of the classical theorem asserting that if a positive integer n divides the order of a solvable group G and the set Ln of solutions of the equation xn=1 in G has cardinality n, then Ln is a subgroup of G. The second proof of that theorem is also presented. Next we offer an easy proof of Philip Hall's theorem on solvable groups independent of Schur-Zassenhaus' theorem. In conclusion, we consider some related questions for p-groups. For example, we study the irregular p-groups G satisfying |Lpk|≤ pk+p-1 for k>1.

2000 Mathematics Subject Classification.   20D15.

Key words and phrases.   Solvable groups, Philip Hall's theorem on solvable groups, irregular p-groups, p-groups of maximal class.


Full text (PDF) (free access)

DOI: 10.3336/gm.45.2.10


References:

  1. Y. Berkovich, Alternate proofs of some basic theorems of finite group theory, Glas. Mat. Ser. III 40(60) (2005), 207-233.
    MathSciNet     CrossRef

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

  3. Y. Berkovich and Z. Janko, Groups of Prime Power Order, Volume 2, Walter de Gruyter, Berlin, 2008.
    MathSciNet    

  4. Y. G. Berkovich and E. M. Zhmud, Characters of Finite Groups. Part 1, Translations of Mathematical Monographs, Volume 172, American Mathematical Society, Providence, 1998.
    MathSciNet    

  5. M. Hall, Jr., The Theory of Groups, Macmillan, New York, 1959. %nije referirano
    MathSciNet    

  6. Philip Hall, A note on solvable groups, J. London Math. Soc. 3 (1928), 98-105.

  7. N. Iivory and H. Yamaki, On a conjecture of Frobenius, Bull. Amer. Math. Soc. (N.S.) 25 (1991), 413-416.
    MathSciNet     CrossRef

Glasnik Matematicki Home Page