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
Department of Mathematics,
University of Haifa,
Mount Carmel, Haifa 31905,
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.
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)
- Y. Berkovich, Alternate proofs
of some basic theorems of finite group theory, Glas. Mat. Ser. III 40(60) (2005), 207-233.
- Y. Berkovich, Groups of Prime Power Order, Volume 1, Walter de Gruyter, Berlin, 2008.
- Y. Berkovich and Z. Janko, Groups of Prime Power Order, Volume 2, Walter de Gruyter, Berlin, 2008.
- Y. G. Berkovich and E. M. Zhmud, Characters of Finite Groups. Part 1, Translations of Mathematical Monographs, Volume 172, American Mathematical Society, Providence, 1998.
- M. Hall, Jr., The Theory of Groups, Macmillan, New York, 1959.
- Philip Hall, A note on solvable groups, J. London Math. Soc. 3 (1928), 98-105.
- N. Iivory and H. Yamaki, On a conjecture of Frobenius, Bull. Amer. Math. Soc. (N.S.) 25 (1991), 413-416.
Glasnik Matematicki Home Page