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

作者：Berkovich Yakov^*

来源：Glasnik Matematicki, 2010, 45(2): 431-439.

DOI：10.3336/gm.45.2.10

• Solvable groups
• Philip Hall's theorem on solvable groups
• irregular p-groups
• p-groups of maximal class

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 L(n) of solutions of the equation x(n) = 1 in G has cardinality n, then L(n) 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 vertical bar L(pk)vertical bar <= p(k+p-1) for k > 1.

• 出版日期2010