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Minimum product sets sizes in nonabelian groups. (English) Zbl 1278.20028

Summary: Given a group \(G\) and integers \(r\) and \(s\), let \(\mu_G(r,s)\) be the minimum cardinality of the product set \(AB\), where \(A\) and \(B\) are subsets of \(G\) of cardinality \(r\) and \(s\), respectively. We compute \(\mu_G\) for all nonabelian groups of order \(pq\), where \(p\) and \(q\) are distinct odd primes, thus proving a conjecture of A. Deckelbaum [J. Number Theory 129, No. 6, 1234-1245 (2009; Zbl 1165.20019]. In addition, we apply a theorem of Eliahou and Kervaire to compute \(\mu_G\) for all finite nilpotent groups.

MSC:

20D60 Arithmetic and combinatorial problems involving abstract finite groups
11B75 Other combinatorial number theory
11B30 Arithmetic combinatorics; higher degree uniformity

Citations:

Zbl 1165.20019
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References:

[1] Cauchy, Augustin-Louis, Recherches sur les nombres, J. École Polytechnique, 9, 99-123 (1813)
[2] Davenport, Harold, On the addition of residue classes, J. London Math. Soc., 10, 30-32 (1935) · Zbl 0010.38905
[3] Deckelbaum, Alan, Minimum product set sizes in nonabelian groups of order pq, Journal of Number Theory, 129, 6, 1234-1245 (2009) · Zbl 1165.20019
[4] Eliahou, Shalom; Kervaire, Michel, The small sumsets property for solvable finite groups, European J. Combin., 27, 7, 1102-1110 (2006), MR MR2259941 (2008f:11116) · Zbl 1099.11008
[5] Eliahou, Shalom; Kervaire, Michel, Bounds on the minimal sumset size function in groups, Int. J. Number Theory, 3, 4, 503-511 (2007), MR MR2371772 (2008m:11022) · Zbl 1160.20018
[6] Eliahou, Shalom; Kervaire, Michel, Some extensions of the Cauchy-Davenport theorem, (6th Czech-Slovak International Symposium on Combinatorics, Graph Theory, Algorithms and Applications. 6th Czech-Slovak International Symposium on Combinatorics, Graph Theory, Algorithms and Applications, Electron. Notes Discrete Math., vol. 28 (2007), Elsevier: Elsevier Amsterdam), 557-564, MR MR2324064 · Zbl 1161.20020
[7] Eliahou, Shalom; Kervaire, Michel, Some results on minimal sumset sizes in finite non-abelian groups, J. Number Theory, 124, 1, 234-247 (2007), MR MR2321003 (2008d:11022) · Zbl 1130.20022
[8] Eliahou, Shalom; Kervaire, Michel, Minimal sumsets in finite solvable groups, Discrete Mathematics, 310, 3, 471-479 (2010) · Zbl 1196.20030
[9] Eliahou, Shalom; Kervaire, Michel; Plagne, Alain, Optimally small sumsets in finite abelian groups, J. Number Theory, 101, 2, 338-348 (2003), MR MR1989891 (2004i:11015c) · Zbl 1043.11021
[10] Kemperman, J. H.B., On complexes in a semigroup, Nederl. Akad. Wetensch. Proc. Ser. A., 59, 247-254 (1956), MR MR0085263 (19, 13h) · Zbl 0072.25605
[11] Mann, Henry B., Addition Theorems: The Addition Theorems of Group Theory and Number Theory (1965), Interscience Publishers John Wiley & Sons: Interscience Publishers John Wiley & Sons New York-London-Sydney, MR MR0181626 (31 #5854) · Zbl 0127.27203
[12] Vosper, A. G., The critical pairs of subsets of a group of prime order, J. London Math. Soc., 31, 200-205 (1956), MR MR0077555 (17, 1056c) · Zbl 0072.03402
[13] Zémor, Gilles, A generalisation to noncommutative groups of a theorem of Mann, Discrete Math., 126, 1-3, 365-372 (1994), MR MR1264502 (95a:20023) · Zbl 0791.05055
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