Difference Bases in Dihedral Groups
نویسنده
چکیده
A subset B of a group G is called a difference basis of G if each element g ∈ G can be written as the difference g = ab−1 of some elements a, b ∈ B. The smallest cardinality |B| of a difference basis B ⊂ G is called the difference size of G and is denoted by ∆[G]. The fraction ð[G] := ∆[G]/ √ |G| is called the difference characteristic of G. We prove that for every n ∈ N the dihedral group D2n of order 2n has the difference characteristic √ 2 ≤ ð[D2n] ≤ 48 586 ≈ 1.983. Moreover, if n ≥ 2 · 10 , then ð[D2n] < 4 6 ≈ 1.633. Also we calculate the difference sizes and characteristics of all dihedral groups of cardinality ≤ 80.
منابع مشابه
Calculations of Dihedral Groups Using Circular Indexation
In this work, a regular polygon with $n$ sides is described by a periodic (circular) sequence with period $n$. Each element of the sequence represents a vertex of the polygon. Each symmetry of the polygon is the rotation of the polygon around the center-point and/or flipping around a symmetry axis. Here each symmetry is considered as a system that takes an input circular sequence and g...
متن کاملOn the eigenvalues of Cayley graphs on generalized dihedral groups
Let $Gamma$ be a graph with adjacency eigenvalues $lambda_1leqlambda_2leqldotsleqlambda_n$. Then the energy of $Gamma$, a concept defined in 1978 by Gutman, is defined as $mathcal{E}(G)=sum_{i=1}^n|lambda_i|$. Also the Estrada index of $Gamma$, which is defined in 2000 by Ernesto Estrada, is defined as $EE(Gamma)=sum_{i=1}^ne^{lambda_i}$. In this paper, we compute the eigen...
متن کاملAsymptotic Nonexistence of Difference Sets in Dihedral Groups
Almost all known results on difference sets need severe restrictions on the parameters. The main purpose of this paper is to provide an asymptotic nonexistence result free from any assumptions on the parameters. The only assumption we make is that the underlying group is dihedral. Difference sets originally mainly were studied in cyclic groups where they exist in abundance. For example, for any...
متن کاملThe number of Fuzzy subgroups of some non-abelian groups
In this paper, we compute the number of fuzzy subgroups of some classes of non-abeilan groups. Explicit formulas are givenfor dihedral groups $D_{2n}$, quasi-dihedral groups $QD_{2^n}$, generalized quaternion groups $Q_{4n}$ and modular $p$-groups $M_{p^n}$.
متن کاملClassifying fuzzy normal subgroups of finite groups
In this paper a first step in classifying the fuzzy normalsubgroups of a finite group is made. Explicit formulas for thenumber of distinct fuzzy normal subgroups are obtained in theparticular cases of symmetric groups and dihedral groups.
متن کامل