Minimum product set sizes in nonabelian groups of order pq
Let G be a nonabelian group of order pq, where p and q are distinct odd primes. We analyze the minimum product set cardinality μG(r,s)=min|AB|μG(r,s)=min|AB|, where A and B range over all subsets of G of cardinalities r and s , respectively. In this paper, we completely determine μG(r,s)μG(r,s) i...
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| Format: | Article |
| Language: | en_US |
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Elsevier
2015
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| Online Access: | http://hdl.handle.net/1721.1/96190 |
| _version_ | 1811075780956913664 |
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| author | Deckelbaum, Alan T. |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Deckelbaum, Alan T. |
| author_sort | Deckelbaum, Alan T. |
| collection | MIT |
| description | Let G be a nonabelian group of order pq, where p and q are distinct odd primes. We analyze the minimum product set cardinality μG(r,s)=min|AB|μG(r,s)=min|AB|, where A and B range over all subsets of G of cardinalities r and s , respectively. In this paper, we completely determine μG(r,s)μG(r,s) in the case where G has order 3p and conjecture that this result can be extended to all nonabelian groups of order pq. We also prove that for every nonabelian group of order pq there exist 1⩽r,s⩽pq1⩽r,s⩽pq such that μG(r,s)>μZ/pqZ(r,s)μG(r,s)>μ[subscript Z over pqZ(r,s)]. |
| first_indexed | 2024-09-23T10:11:46Z |
| format | Article |
| id | mit-1721.1/96190 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T10:11:46Z |
| publishDate | 2015 |
| publisher | Elsevier |
| record_format | dspace |
| spelling | mit-1721.1/961902022-09-26T16:19:58Z Minimum product set sizes in nonabelian groups of order pq Deckelbaum, Alan T. Massachusetts Institute of Technology. Department of Mathematics Deckelbaum, Alan T. Let G be a nonabelian group of order pq, where p and q are distinct odd primes. We analyze the minimum product set cardinality μG(r,s)=min|AB|μG(r,s)=min|AB|, where A and B range over all subsets of G of cardinalities r and s , respectively. In this paper, we completely determine μG(r,s)μG(r,s) in the case where G has order 3p and conjecture that this result can be extended to all nonabelian groups of order pq. We also prove that for every nonabelian group of order pq there exist 1⩽r,s⩽pq1⩽r,s⩽pq such that μG(r,s)>μZ/pqZ(r,s)μG(r,s)>μ[subscript Z over pqZ(r,s)]. National Science Foundation (U.S.) (Grant DMS-0447070-001) United States. National Security Agency (Grant H98230-06-1-0013) 2015-03-25T18:23:16Z 2015-03-25T18:23:16Z 2009-03 2009-02 Article http://purl.org/eprint/type/JournalArticle 0022314X 1096-1658 http://hdl.handle.net/1721.1/96190 Deckelbaum, Alan. “Minimum Product Set Sizes in Nonabelian Groups of Order Pq.” Journal of Number Theory 129, no. 6 (June 2009): 1234–1245. © 2009 Elsevier Inc. en_US http://dx.doi.org/10.1016/j.jnt.2009.02.006 Journal of Number Theory Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. application/pdf Elsevier Elsevier |
| spellingShingle | Deckelbaum, Alan T. Minimum product set sizes in nonabelian groups of order pq |
| title | Minimum product set sizes in nonabelian groups of order pq |
| title_full | Minimum product set sizes in nonabelian groups of order pq |
| title_fullStr | Minimum product set sizes in nonabelian groups of order pq |
| title_full_unstemmed | Minimum product set sizes in nonabelian groups of order pq |
| title_short | Minimum product set sizes in nonabelian groups of order pq |
| title_sort | minimum product set sizes in nonabelian groups of order pq |
| url | http://hdl.handle.net/1721.1/96190 |
| work_keys_str_mv | AT deckelbaumalant minimumproductsetsizesinnonabeliangroupsoforderpq |