Inverse Problems on Cirtical Number in Finite Groups

DOI: 10.4236/ojdm.2013.32018   PDF   HTML   XML   2,815 Downloads   4,883 Views  

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.

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Q. Wang and J. Zhuang, "Inverse Problems on Cirtical Number in Finite Groups," Open Journal of Discrete Mathematics, Vol. 3 No. 2, 2013, pp. 93-96. doi: 10.4236/ojdm.2013.32018.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] P. Erdõs and H. Heibronn, “On the Addition of Residue Classes Mod p,” Acta Arithmetica, Vol. 9, 1964, pp. 149- 159.
[2] 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
[3] 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
[4] 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
[5] W. Gao and Y. O. Hamidoune, “On Additive Bases,” Acta Arithmetica, Vol. 88, 1999, pp. 233-237.
[6] 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
[7] 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
[8] 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
[9] 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
[10] 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.
[11] 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
[12] 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
[13] 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.
[14] H. B. Mann, “Addition Theorems,” 2nd Edition, R. E. Krieger, New York, 1976.
[15] J. E. Olson, “Sum of Sets of Group Elements,” Acta Arithmetica, Vol. 28, No. 76, 1975, pp. 147-156.
[16] Y. O. Hamidoune, “Adding Distinct Congruence Classes,” Combinatorics, Probability and Computing, Vol. 7, No. 1, 1998, pp. 81-87. doi:10.1017/S0963548397003180

  
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