Some closure properties of classes of groups

Abstract

The study of weak homomorphic image closure properties of classes of groups defined by means of series is the primary purpose of this work. Of particular interest are classes of groups each of which has a series with factors determined by a variety of groups. Well-known groups of this kind are the Z-groups, SI-groups, SN-groups. Also of interest are classes of groups defined by means of normal factor coverings with factors determined by a variety of groups, such as the residually central groups. Classes of groups defined by means of composition series or chief series are also studied; e.g., Z-bar-groups, SI-bar-groups, SN-bar-groups. In Chapter II, results concerning closure with respect to reduced direct products are given. An example is given to illustrate what kind of behavior is possible for reduced direct products of Z-groups. This example is neither a Z-bar-group nor a ZD-group. Closure results regarding homomorphic images such that the kernel of the homomorphism is contained in the hyper V-marginal subgroup, where V is a variety of groups, are given in Chapter III. An example is included which is the homomorphic image of a Z-group. Again, it is demonstrated that this example is neither a Z-bar-group nor a ZD-group. In Chapter IV, characterizations of certain classes of groups defined by means of composition series or chief series are established. Typical of these characterizations is that a group G is a Z-bar-group if and only if each homomorphic image of G is a residually central group.