The Occurrence of Certain Types of Groups as Automorphism Groups
Zimmerman, Jay James
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https://hdl.handle.net/2142/71210
Description
Title
The Occurrence of Certain Types of Groups as Automorphism Groups
Author(s)
Zimmerman, Jay James
Issue Date
1983
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Date of Ingest
2014-12-16T06:18:05Z
Keyword(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 non-abelian group (p.n.a. group) to be a finite group whose composition factors are non-abelian. Purely non-abelian 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 self-normalizing subgroups of symmetric groups. We characterize those abstract groups which can occur as self-normalizing subgroups of a symmetric group and then construct examples of complete p.n.a. groups using such self-normalizing 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.
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