Some remarks on subgroups of maximal class in a finite p-group

Abstract

The following characterizations of p-groups of maximal class are proved: (a) If a p-group of order > pp+2 contains a subgroup of maximal class and index p, then G possesses at most one normal subgroup of order pp and exponent p. (b) If the center of any nonabelian epimorphic image of a nonabelian two-generator p-group G is cyclic, then either G ≅ M pn or G is of maximal class. (c) An 𝒜n-group G, n > 1, is of maximal class ⇔ all its 𝒜2-subgroups of minimal order are of maximal class. (iv) If all factors of the lower central series of a nonabelian two-generator p-group are cyclic, then it is either of maximal class or ≅ M pn. (v) If a nonabelian p-group G is such that any s pairwise non-commuting elements generate a group of maximal class, where s is the fixed member of the set {3, …, p + 1} and p > 2 if s ≠ p + 1, then G is also of maximal class. We also study the noncyclic p-groups containing only one normal subgroup of a given order.