Graduation Term

Spring 2026

Degree Name

Master of Science (MS)

Department

Department of Mathematics

Committee Chair

Sunil Chebolu

Committee Member

Charlotte Ure

Abstract

Fuchs’ problem asks which groups can arise as the group of units of a ring. Although the finite cyclic case has been completely classified, much less is known in the infinite setting. This thesis contributes to this problem by investigating quasi-cyclic. (Pr¨ufer) groups and their finite direct products. We show that for every odd prime p, there is no commutative ring R such that R×∼= Cp∞. This obstruction arises from characteristic restrictions and the algebraic structure of finite fields. More generally, we prove that any group in which every element has order a power of an odd prime p and which contains an element of order p2 cannot be realized as the group of units of a commutative ring. As a consequence, finite direct products of Cp∞ for a fixed odd prime p are also not realizable.

Access Type

Thesis-Open Access

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Included in

Algebra Commons

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