Glasnik matematički, Vol. 45 No. 2, 2010.
Izvorni znanstveni članak
https://doi.org/10.3336/gm.45.2.10
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
Sažetak
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.
Ključne riječi
Solvable groups; Philip Hall's theorem on solvable groups; irregular p-groups; p-groups of maximal class
Hrčak ID:
62698
URI
Datum izdavanja:
24.12.2010.
Posjeta: 1.072 *