Vol. 29, No. 1, 1969

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The power-commutator structure of finite p-groups

Deane Eugene Arganbright

Vol. 29 (1969), No. 1, 11–17

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 Gp1 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.

Mathematical Subject Classification
Primary: 20.40
Received: 9 November 1967
Revised: 15 July 1968
Published: 1 April 1969
Deane Eugene Arganbright