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|Title:||The Locally Free Cancellation Property of The Group Ring Zg (Eichler Condition, Quaternion Group, Kernel Group)|
|Department / Program:||Mathematics|
|Degree Granting Institution:||University of Illinois at Urbana-Champaign|
|Abstract:||It is of the utmost importance to know whether ZG has locally free cancellation in many applications where ZG is the integral group ring of a finite group G over Z. Jacobinski's cancellation theorem implies that cancellation holds unless some quotient G/N is a binary poly- hedral group. Swan's theorem which investigates the extent to which the converse of Jacobinski's cancellation theorem holds says that cancellation fails for ZG if G has a quotient which is a binary poly- hedral group but not one of the following seven groups Q(,8), Q(,12), Q(,16), Q(,20), (')T, (')O, and (')I. However, this does not yet characterize the groups for which cancellation fails.
Two problems discussed in this thesis are (1) If G has a unique binary polyhedral quotient which is one of the seven groups listed above, then cancellation holds. (2) If G has more than one binary polyhedral quotient which is one of the first four groups listed above, then cancellation fails. The problems arise naturally because "one binary polyhedral quotient is bad, two should be worse". In this thesis both problems are partially solved.
Our approach to both problems is to discuss the locally free cancellation for G (TURNEQ) C(,p) x H where C(,p) is a cyclic group of prime order p and H is one of the seven good groups listed above. Com- plete classification of such type of G such that cancellation fails for ZG is answered for p (NOT=) 2. When G = C(,2) x Q(,4n) for n = 2, 3, 4, 5, the classification is also obtained. The latter result is the essential step to solve the second problem.
Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1986.
|Date Available in IDEALS:||2014-12-16|