Minimal orbit sizes in nilpotent group actions

Abstract

Let [Formula: see text] be a finite nilpotent group. We prove the following results. (1) If [Formula: see text] is of class 2 and acts faithfully and irreducibly on an elementary abelian group [Formula: see text], then all nontrivial orbits of [Formula: see text] on [Formula: see text] have sizes larger than [Formula: see text]. (2) If [Formula: see text] is cyclic, then every subgroup of [Formula: see text] intersecting trivially with the center of [Formula: see text] has order less than [Formula: see text]. We also show that a result like (2) cannot be obtained when the hypothesis that [Formula: see text] is cyclic is replaced by the hypothesis that the center of [Formula: see text] is cyclic.