A note on an “Anzahl” theorem of P. Hall

Abstract

Let [Formula: see text] be a prime and [Formula: see text] a finite [Formula: see text]-group. Hall proved [Formula: see text] [Formula: see text] for [Formula: see text], where [Formula: see text] is the number of subgroups of index [Formula: see text] which do not contain the Frattini subgroup of [Formula: see text]. We determine the minimal values of [Formula: see text] for [Formula: see text] and [Formula: see text].