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.

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DOI: 10.3336/gm.45.2.10

References:

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