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Abstract

This. thesis studies subgroup theoretical properties and classes of groups defined in teriis of subgroup theoretical properties. After an introductory chapter, containing some definitions and some general results, two particular subgroup theoretical properties are examined. These two represent the concepts of (mathematical symbol and, read abstract from pdf file) centrality, for a class of groups (mathematical symbol) after Stanley and Arrell. They are generalizations of the more familiar subgroup theoretical property "is a central subgroup of".

Some connections are established between classes of groups defined in terms of (mathematical equation) centrality, notably that the classes of groups with (mathematical equation) central series coincide, and that when (mathematical symbol) is restricted to be a variety, similar results hold for ascending and descending series. Some results of Petty are used to prove certain closure properties for some of the classes under discussion. In particular, the local closure of the class of groups with a V-central series, for a variety V, is proved.

A generalization of the class of residually commutable groups is introduced. Some of Ayoub's results about residually commutable groups, including her local theorem, are generalized accordingly.

When (mathematical symbol) is the class of nilpotent groups, (mathematical symbol)- centrality is shown to yield a class of hypercentral groups. The relationships between this class and other classes of generalized nilpotent groups are studied. Also, when (mathematical symbol) is the class of soluble groups, (mathematical symbol)- centrality gives rise to several new classes of generalized soluble groups. Taking (mathematical symbol) to be the class of finite groups provides a characterization of the class of FC-groups. This characterization is used to investigate the effect of the condition of being an FC-group on generalized soluble and nilpotent groups. In particular, it is proved that residually central FC-groups are hypercentral, with hypercentral length not exceeding w.

Publication Type:	Thesis (Doctoral)
Subjects:	Q Science > QA Mathematics
Departments:	School of Science & Technology > Department of Mathematics School of Science & Technology > School of Science & Technology Doctoral Theses Doctoral Theses

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