Soluble groups with the minimum condition for normal subgroups

Abstract

A soluble group satisfying the minimum condition for subgroups is an extension of a periodic radicable abelian group by a finite group, by a wellknown result of Cernikov [3]. Such a group is necessarily periodic and countable. Baer [-1] has shown that soluble groups satisfying the minimum condition for normal subgroups must be periodic, but apart from this result very little has been published concerning such groups. 0arin [-2] has constructed a metabelian group satisfying the minimum condition for normal subgroups which does not satisfy the minimum condition for subgroups. We will show that a metabelian group with the minimum condition for normal subgroups is an extension by a finite group of a group which is very much like Carin's group. More precisely, we show that a metabelian group with the minimum condition for normal subgroups is countable (Theorem 4.4), and that in every subgroup the Sylow re-subgroups are conjugate for all sets of primes n (Theorem 3.5). Let 3~ denote the class of metabelian groups satisfying the minimum condition for normal subgroups and having no proper subgroups of finite index. It is can be shown that an arbitrary metabelian group satisfying the minimum condition for normal subgroups is an extension of an 3~ group by a finite group (Lemma 4.2). The main results in this paper concern X groups. We show in Theorem 3.2 that the Sylow p-subgroups of an 9 group are abelian for all primes p. Using this result we are able to show that an 3~ group splits over its derived group and that the complements are conjugate (Theorem 5.6 and Corollary 5.7). These results are proved for a class of groups which is shown, by Example 2 of Section 6, to be a strictly larger class than t;. With these results on metabelian groups at our disposal we naturally turn to the case of a soluble group of arbitrary derived length satisfying the minimum condition for normal subgroups. Unfortunately the situation immediately becomes more complex. In Section 6 we construct a centre-by-metabelian group satisfying the minimum condition for normal subgroups (Example 3). This group has no proper subgroups of finite index, has a non-abelian Sylow p-subgroup, and does not split over any term of its derived series. Finally we establish in Example 4 the existence of a soluble group with the minimum condition for normal subgroups, without proper subgroups of finite index, and of arbitrary derived length.