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Inverse Problems on Cirtical Number in Finite Groups
Colleage of Science, Tianjin University of Technology, Tianjin, China
Department of Mathematics, Dalian Maritime University, Dalian, China
- 1 Colleage of Science, Tianjin University of Technology, Tianjin, China
- 2 Department of Mathematics, Dalian Maritime University, Dalian, China
Open Journal of Discrete Mathematics·Volume 03 (2013)·Pages 93–96·Published 24 April 2013·DOI10.4236/ojdm.2013.32018
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Abstract
Let G be a finite nilpotent group of odd order and S be a subset of G\{0} . We say that S is complete if every element of G can be represented as a sum of different elements of S and incomplete otherwise. In this paper, we obtain the characterization of large incomplete sets.
KeywordsCritical NumberIncomplete SetFinite Nilpotent Group
- P. Erdõs and H. Heibronn, “On the Addition of Residue Classes Mod p,” Acta Arithmetica, Vol. 9, 1964, pp. 149- 159.
- J. A. Dias da Silva and Y. O. Hamidoune, “Cyclic Spaces for Grassmann Derivatives and Additive Theory,” Bulletin London Mathematical Society, Vol. 26, No. 2, 1994, pp. 140-146. doi:10.1112/blms/26.2.140
- G. T. Diderrich, “An Addition Theorem for Abelian Groups of Order pq,” Journal of Number Theory, Vol. 7, No. 1, 1975, pp. 33-48. doi:10.1016/0022-314X(75)90006-2
- J. E. Olson, “An Addition Theorem Mod p,” Journal of Combinatorial Theory, Vol. 5, No. 1, 1968, pp. 45-52. doi:10.1016/S0021-9800(68)80027-4
- W. Gao and Y. O. Hamidoune, “On Additive Bases,” Acta Arithmetica, Vol. 88, 1999, pp. 233-237.
- H. B. Mann and Y. F. Wou, “An Addition Theorem for the Elementary Abelian Group of Type (p,p),” Monatshefte für Mathematik, Vol. 102, No. 4, 1986, pp. 273-308. doi:10.1007/BF01304301
- M. Freeze, W. D. Gao and A. Geroldinger, “The Critical Number of Finite Abelian Groups,” Journal of Number Theory, Vol. 129, No. 11, 2009, pp. 2766-2777. doi:10.1016/j.jnt.2009.05.016
- Q. H. Wang and J. J. Zhuang, “On the Critical Number of Finite Groups of Order pq,” International Journal of Number Theory, Vol. 8, No. 5, 2012, pp. 1271-1280. doi:10.1142/S1793042112500741
- J. R. Griggs, “Spanning Subset Sums for Finite Abelian Groups,” Discrete Mathematics, Vol. 229, No. 1-3, 2001, pp. 89-99. doi:10.1016/S0012-365X(00)00203-X
- Q. H. Wang and Y. K. Qu, “On the Critical Number of Finite Groups (II),” Ars Combinatoria, Accepted for Publication in December 2009, to Appear.
- W. Gao, Y. O. Hamidoune, A. Llad and O. Serra, “Covering a Finite Abelian Group by Subset Sums,” Combinatorica, Vol. 23, No. 4, 2003, pp. 599-611. doi:10.1007/s00493-003-0036-x
- V. H. Vu, “Structure of Large Incomplete Sets in Finite Abelian Groups,” Combinatorica, Vol. 30, No. 2, 2010, pp. 225-237. doi:10.1007/s00493-010-2336-2
- D. Guo, Y. K. Qu, G. Q. Wang and Q. H. Wang, “Extremal Incomplete Sets in Finite Abelian Groups,” Ars Combinatoria, Accepted for Publication in December 2011, to Appear.
- H. B. Mann, “Addition Theorems,” 2nd Edition, R. E. Krieger, New York, 1976.
- J. E. Olson, “Sum of Sets of Group Elements,” Acta Arithmetica, Vol. 28, No. 76, 1975, pp. 147-156.