On finite p-groups with few normal subgroups

Abstract

A [Formula: see text]-group [Formula: see text] is called a [Formula: see text]-group if [Formula: see text] [Formula: see text] or [Formula: see text] for every normal subgroup [Formula: see text] in [Formula: see text]. In this paper, we first investigate the properties of [Formula: see text]-groups, in particular, we give the upper bounds of the exponent of [Formula: see text] with [Formula: see text] and the order of [Formula: see text] for a [Formula: see text]-group [Formula: see text]. Then, we try to describe the structure of [Formula: see text]-groups, and a necessary condition for a [Formula: see text]-group to be a [Formula: see text]-group is given. We also give some elementary properties of [Formula: see text]-groups with very small derived subgroups by using the properties of capable [Formula: see text]-groups, which maybe could apply some other research of [Formula: see text]-groups.