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

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.