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Title:  The Occurrence of Certain Types of Groups as Automorphism Groups 
Author(s):  Zimmerman, Jay James 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Mathematics 
Abstract:  This dissertation is a study of the relationship between the structure of a group and that of its automorphism group. We examine the characteristics of groups G whose automorphism group Aut G has specified properties. If H = Aut G for some group G, we call H an automorphism group. We will see that there are many classes of groups which are not automorphism groups, and many other classes which are automorphism groups of certain types of groups only. We begin with the construction, by homological techniques, of a type of central extension which is closely related to a stem extension. These extensions are used to describe all groups G such that Aut G is a finite semisimple group. Consequently, we are able to deduce necessary and sufficient conditions for a finite semisimple group to be an automorphism group. These conditions will imply that certain central extensions of a simple group by an abelian group are not automorphism groups. Similar arguments allow us to construct an infinite number of slightly more complicated semisimple groups which are the automorphism group of uncountably many infinite groups, but not of any finite group. Define a purely nonabelian group (p.n.a. group) to be a finite group whose composition factors are nonabelian. Purely nonabelian groups cannot be automorphism groups of infinite groups and we briefly consider the question of when such groups are automorphism groups. However, we are mainly concerned with characterizing all complete p.n.a. groups. This problem is intimately connected with the existence of selfnormalizing subgroups of symmetric groups. We characterize those abstract groups which can occur as selfnormalizing subgroups of a symmetric group and then construct examples of complete p.n.a. groups using such selfnormalizing subgroups. Finally, we consider infinite torsion groups as automorphism groups. Let (omega)(H) be the set of all primes which divide the order of some torsion element of H. We show that if H is a countable torsion automorphism group with finite conjugacy classes and (omega)(H) finite, then H has finite exponent. This furnishes a large class of countable torsion groups which are not automorphism groups. 
Issue Date:  1983 
Type:  Text 
Description:  117 p. Thesis (Ph.D.)University of Illinois at UrbanaChampaign, 1983. 
URI:  http://hdl.handle.net/2142/71210 
Other Identifier(s):  (UMI)AAI8324680 
Date Available in IDEALS:  20141216 
Date Deposited:  1983 
This item appears in the following Collection(s)

Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois