Alternate proofs of two theorems of Philip Hall on finite p-groups, and some related results

New very detailed proofs of Theorems 2.5 and 2.64 from the seminal paper of Philip Hall [P. Hall, On a theorem of Frobenius, Proc. London Math. Soc. Ser. (2) 40 (1936) 468-501] are given. A number of generalizations of these theorems are proved. For example, we show that if G is a p-group of order pk(p−1)+3, k>2, and exponent >pk with Ωk(G)=G, then either G is of maximal class or G possesses a normal subgroup H of order pp and exponent p such that G/H is of maximal class. Counting theorems play important role in this note.