On a class of inseparable finite groups

Abstract

A finite group is called inseparable if the only normal subgroups over which it splits are the identity subgroup and the group itself. An approach to the study of inseparable finite groups is given by Bechtell [I]. It is related with one of the structural components within a group G that permit splitting over a normal subgroup, that is the E-residual GE for the formation of groups having all Sylow subgroups elementary abelian. Among others, Bechtell studies the class 01~ of all inseparable nonnilpotent soluble finite groups G with GE a p-group and proves that GE 01~ implies GE 2 Q(G) and [GE : Q(G)] > p2. In [2] he proves that a group G E 01~ with GE a metacyclic group is the inseparable extension of the quaternion group by the symmetric group of degree three. We will call this group the “Bechtell example.” The present paper is devoted to the development of some of the properties of groups in the class 01~ . Theorem 2.2 gives detailed information about a group G E (Ye , while Theorem 2.3 gives a remarkable improvement of Bechtell bound, that is, [GE : D(G)] >, pp. The remainder will examine the case that GE has a normal chain with cyclic factors and of length p, the minimal consistent with the given bound. We will call this class p, . We get again that ,L12 includes Bechtell example only and we prove that & is empty. The notation is standard (see [5]).