پیوندهای مفید
اخبار و اعلانات
آمار
| تعداد نشریات | 44 |
| تعداد شمارهها | 1,877 |
| تعداد مقالات | 15,278 |
| تعداد مشاهده مقاله | 43,731,372 |
| تعداد دریافت فایل اصل مقاله | 17,563,522 |
Difference bases in dihedral groups | ||
| International Journal of Group Theory | ||
| مقاله 5، دوره 8، شماره 1، خرداد 2019، صفحه 43-50 اصل مقاله (205.16 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/ijgt.2017.21612 | ||
| نویسندگان | ||
| Taras O. Banakh1؛ Volodymyr Gavrylkiv* 2 | ||
| 1Ivan Franko National University of Lviv (Ukraine), and Institute of Mathematics, Jan Kochanowski University in Kielce (Poland) | ||
| 2Vasyl Stefanyk Precarpathian National University, Ivano-Frankivsk, Ukraine | ||
| چکیده | ||
| A subset $B$ of a group $G$ is called a {\em difference basis} of $G$ if each element $g\in G$ can be written as the difference $g=ab^{-1}$ of some elements $a,b\in B$. The smallest cardinality $|B|$ of a difference basis $B\subset G$ is called the {\em difference size} of $G$ and is denoted by $\Delta[G]$. The fraction $\eth[G]:=\Delta[G]/{\sqrt{|G|}}$ is called the {\em difference characteristic} of $G$. We prove that for every $n\in N$ the dihedral group $D_{2n}$ of order $2n$ has the difference characteristic $\sqrt{2}\le\eth[D_{2n}]\leq\frac{48}{\sqrt{586}}\approx1.983$. Moreover, if $n\ge 2\cdot 10^{15}$, then $\eth[D_{2n}]<\frac{4}{\sqrt{6}}\approx1.633$. Also we calculate the difference sizes and characteristics of all dihedral groups of cardinality $\le80$. | ||
| کلیدواژهها | ||
| dihedral group؛ difference basis؛ difference characteristic | ||
| مراجع | ||
|
[1] T. Banakh and V. Gavrylkiv, Algebra in the sup erextensions of twinic groups, Dissertationes Math. 473 (2010) pp. 74. [2] T. Banakh and V. Gavrylkiv, Difference bases in cyclic groups, em J. Algebra Appl. bf 18 (2019) pp. 18. [3] T. Banakh, V. Gavrylkiv and O. Nykyforchyn, Algebra in sup erextension of groups, I: zeros and commutativity, Algebra Discrete Math. , 3 (2008) 1{29. [4] R. C. Bose, An affine analogue of Singer's theorem, J. Indian Math. Soc. , 6 (1942) 1{15. [5] R. C. Bose and S. Chowla, Theorems in the additive theory of numb ers, Comment. Math. Helv. , 37 (1962-63) 141{147. [6] P. Dusart, Autour de la fonction qui compte le nombre de nombres premiers , Ph.D. Thesis, Univ. de Limoges, 1998. [7] V. Gavrylkiv, Bases in dihedral and Bo olean groups, J. Integer Seq. , 20 (2017). [8] M. Golay, Notes on the representation of 1 ; 2 ; : : : ; n by differences, J. Lond. Math. Soc. , 4 (1972) 729{734. [9] J. Leech, On the representation of 1 ; 2 ; : : : ; n by differences, J. Lond. Math. Soc. , 31 (1956) 160{169. [10] G. Kozma and A. Lev, Bases and decomp osition numb ers of nite groups, Arch. Math. (Basel) , 58 (1992) 417{424. [11] L. Redei and A. Renyi, On the representation of the numb ers 1 ; 2 ; : : : ; N by means of differences, Math. Sbornik N.S. , 24 (1949) 385{389. [12] I. Z. Ruzsa, Solving a linear equation in a set of integers I, Acta Arith. , 3 (1993) 259{282. [13] J. Singer, A theorem in nite pro jective geometry and some applications to numb er theory, Trans. Amer. Math. Soc. , 43 (1938) 377{385. | ||
|
آمار تعداد مشاهده مقاله: 1,083 تعداد دریافت فایل اصل مقاله: 783 |
||