Influence of normality conditions on almost minimal subgroups of a finite group

Abstract

Information about the maximal subgroups of a finite group often gives a good insight into the structure of the group. Wielandt’s characterization of a finite nilpotent group as a finite group in which each maximal subgroup is normal is an example. Finite groups all of whose r-maximal subgroups are normal (Y = 2, 3, 4, 5) are considered in [6, 8, 91. Dualising the above result of Wielandt, Gaschiitz and Ito proved that a finite group all of whose minimal subgroups are normal is a solvable group of Fitting length at most 3 [5, Satz 5.7, p. 4361. In [l], Buckley considered p-groups, p an odd prime, all of whose minimal subgroups are normal and obtained results on some of their images. Let Y be a natural number. A subgroup H of a finite group G is called an r-minimal subgroup of G if there is a chain of subgroups 1 = HO C HI C ... C H, = H with each Hi maximal in Hi+, . Clearly a subgroup can sometimes be both an Yand an s-minimal subgroup for Y # s and a subgroup is an r-minimal subgroup for a unique r if and only if it is supersolvable. See [5, Satz 9.7, p. 7191. A subgroup of G which permutes with every subgroup of G is called a quasinormal subgroup of G. Extending the definition of a PN-group in [l], we say that a group G is a PN-Y group (SN-Y group, QN-Y group) if each r-minimal subgroup of G is normal (subnormal, quasinormal) in G. These properties are obviously inherited by subgroups of G. In Section 1, we show that PN-Y groups are solvable of Fitting length at most Y for Y = 2 and 3. Simple groups enter into the consideration of PN-Y groups for Y > 4. We obtain some partial results about PN-Y groups for r > 4 and analyze the PN-4 groups more closely. In Section 2, using a construction of Gaschiitz, we show that SN-r groups can be “quite arbitrary.” In fact, the construction of Gaschiitz shows that an SN-1 group which has a subnormal chain of length at most 2 through each of its minimal subgroups can also be “quite arbitrary.” Thus, the sub-