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Fuchs' problem
for small groups
Joshua A. Carr, William J. Cook and Lindsey M. Wise |
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Vol. 17 (2024), No. 3, 441–463
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Abstract |
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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. |
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Keywords
Fuchs' problem, group of units, dicyclic groups
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Mathematical Subject Classification
Primary: 16U60
Secondary: 20C05
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Milestones
Received: 5 April 2022
Revised: 22 December 2022
Accepted: 26 April 2023
Published: 17 July 2024
Communicated by Kenneth S. Berenhaut
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Authors |
| Joshua A. Carr | |
| Wilkes Community College Wilkesboro, NC United States |
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| William J. Cook | |
| Appalachian State University Boone, NC United States |
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| Lindsey M. Wise | |
| University of Southern
California Los Angeles, CA United States |
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| © 2024 MSP (Mathematical Sciences Publishers). |
