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

摘要

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 p(k(p-1)+3), k > 2, and exponent > p(k) with Omega(k)(G) = G, then either G is of maximal class or G possesses a normal subgroup H of order p(p) and exponent p such that G/H is of maximal class. Counting theorems play important role in this note. (c) 2005 Elsevier Inc. All rights reserved.