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

https://hrcak.srce.hr/62698

Datum izdavanja:

24.12.2010.

Posjeta: 1.027 *