A new class of generalized nilpotent groups

Abstract

In 1967, A. Rae introduced the concept of a locally subnormal subgroup of a group. (If A is a group and H is a subgroup of A such that H is subnormal in every subgroup of A that can be generated by H and finitely many elements of A, then H is called a locally subnormal subgroup of A.) In Chapter II of this dissertation, some basic properties of locally subnormal subgroups are developed. An example is presented to show that the property of local subnormality is not necessarily a transitive property. Also, an example is presented to show that the join of two locally subnormal subgroups of a group need not be locally subnormal in the group. In Chapter III, we examine the class L of groups in which every subgroup is locally subnormal. We investigate necessary and sufficient conditions on groups A and B in order that the wreath product of A and B is an L-group. It is proved that if A wr B is an L-group, then A and B are p-groups (for the same prime p), A is hypercentral of length w, every subgroup of A is subnormal in A (A in G), and B is cyclic of prime order p. Also, it is proved that if A is a nilpotent p-group and B is cyclic of prime power order p (for the same prime p), then A wr B is an L-group. These results indicate that the class of L-groups properly contains the class of G-groups, and is properly contained in the class that consists of all groups in which every proper subgroup is distinct from its normalizer. Also, in Chapter III, examples are given which show that the class of groups that are both L-groups and Baer-groups is properly contained in the class of Baer-groups, and properly contains the class of G-groups.