For a finite p-group G, Gn is
the n-th element in the descending central series of G; P(G) is the subgroup of G
generated by the set of all xp for x belonging to G; and Φ(G) is the Frattini subgroup
of G.
Hobby has characterized finite p-groups G (for p > 2) in which P(G) = Φ(G).
Since Φ(G) = G2P(G), the condition P(G) = Φ(G) is clearly equivalent to
G2⊆ P(G). In this paper we examine the class of finite p-groups G which have the
property that Gn⊆ P(Gm) for 1 < n∕m < p. In §2 we consider consequences of this
property in the case m = 1. For example, if Gp−1⊆ P(G), then the product of p-th
powers of elements of G is the p-th power of an element of G (Theorem 2). In §3 we
examine some connections between the property Gn⊆ P(Gm) and regularity, and
obtain a characterization of regular 3-groups (Theorem 4). In §4 we obtain bounds
on the number of generators of various commutator subgroups of G in the case
G3⊆ P(G), p > 3.
