Fuchs’ problem for small groups

Carr, Joshua A.; Cook, William J.; Wise, Lindsey M.

doi:10.2140/involve.2024.17.441

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Fuchs' problem for small groups

Joshua A. Carr, William J. Cook and Lindsey M. Wise

Vol. 17 (2024), No. 3, 441–463

DOI: 10.2140/involve.2024.17.441

Abstract

Fuchs’ problem asks which groups are realizable as the unit group of a ring. We solve Fuchs’ problem for dicyclic groups realized by finite rings. We also survey known results and give a complete list of realizable groups of order at most 15. For these realizable groups, we provide a ring in every viable characteristic. Consequently, we show the dicyclic group of order 12 is the smallest realizable group that cannot be realized by a finite ring.

Keywords

Fuchs' problem, group of units, dicyclic groups

Mathematical Subject Classification

Primary: 16U60

Secondary: 20C05

Milestones

Received: 5 April 2022

Revised: 22 December 2022

Accepted: 26 April 2023

Published: 17 July 2024

Communicated by Kenneth S. Berenhaut

Authors

Joshua A. Carr
Wilkes Community College Wilkesboro, NC United States

William J. Cook
Appalachian State University Boone, NC United States

Lindsey M. Wise
University of Southern California Los Angeles, CA United States

Vol. 17, No. 3, 2024