Finite Groups Whose «-maximal Subgroups Are Subnormal

Introduction. Dedekind has determined all groups whose subgroups are all normal (see, e.g., [5, Theorem 12.5.4]). Partially generalizing this, Wielandt showed that a finite group is nilpotent, if and only if all its subgroups are subnormal, and also if and only if all maximal subgroups are normal [5, Corollary 10.3.1, 10.3.4]. Huppert [7, Sätze 23, 24] has shown that if all 2nd-maximal subgroups of a finite group are normal, the group is supersolvable, while if all 3rd-maximal subgroups are normal, the group is solvable of rank at most 2 (see below for the definitions of the terms employed). Continuing this, Janko [8] has determined all nonsolvable finite groups, all of whose 4-maximal subgroups are normal, and also proved some results on the structure of solvable groups with the same property. Later, he has also determined all finite simple groups, all of whose 5-maximal subgroups are normal [9]. The present paper deals with similar problems. Our main concern is with the class of groups defined in the title. In §2, we derive some simple properties of these groups. For «^5, we extend the aforementioned results of Huppert and Janko to cover also our case (i.e., replacing normality by subnormality), and prove some results under slightly more general assumptions. In §3 we study solvable groups with subnormal «-maximal subgroups. If the order of the group in question has enough distinct prime divisors, the structure of the group is very limited. Thus, if the number of these primes exceeds «, the group is nilpotent, while if it is equal to «, we can still completely determine our groups. If the number is « — 1, the group has a Sylow tower. Then, imposing the stronger requirement that «-maximal subgroups are quasi-normal, we find that the rank of the group cannot exceed «— 1. In the last section we allow again our groups to be nonsolvable. The main result here states, that if each «-maximal subgroup is subnormal, and if the order of the groups has a large number of distinct prime divisors then the group is, in fact, solvable. Here, "large" is not only with respect to «, but also with respect to the number of prime divisors of the indices of maximal subgroups of subgroups. Most of the contents of this paper formed a part of the author's Ph.D. thesis prepared in the Hebrew University, Jerusalem, under the supervision of Professor S. A. Amitsur. I would like to take this opportunity to thank Professor Amitsur for his interest and encouragement, as well as for a number of valuable suggestions.