A subset B of a group G is called a difference basis of G if each element g 2 G can be written as the difference g = ab 􀀀 1 of some elements a; b 2 B. The smallest cardinality jBj of a difference basis B  G is called the difference size of G and is denoted by Δ [G]. The fraction g[G]: = Δ [G]= √ jGj is called the difference characteristic of G. We prove that for every n 2 N the dihedral group D2n of order 2n has the difference characteristic p 2  g[D2n]  p48 586  1: 983. Moreover, if n  2
1015, then g[D2n] < p4 6  1: 633. Also we calculate the difference sizes and characteristics of all dihedral groups of cardinality  80.

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 generates a processed circular output sequence. The system can be described by a permutation function. Permutation functions can be written in a simple form using circular indexation. The operation between the symmetries of the polygon is reduced to the composition of permutation functions, which in turn is easily implemented using periodic sequences. It is also shown that each symmetry is effectively a pure rotation or a pure flip. It is also explained how to synthesize each symmetry using two generating symmetries: time-reversal (flipping around a fixed symmetry axis) and unit-delay (rotation around the center-point by 2 /n radians clockwise).

Diaconis and Isaacs de, ned the supercharacter theory for , nite groups as a natural generalization of the classical ordinary character theory of , nite groups. Supercharacter theory of many , nite groups such as the cyclic groups, the Frobenius groups, etc. are well studied and well-known. In this paper we , nd the normal and automorphic supercharacter theories of the dihedral groups in special cases.

‎The algebraic value of a chemical composition plays a crucial role in determining its physical properties‎, ‎chemical reactivity‎, ‎and biological activity‎. Algebraic graph theory investigates the connection between abstract algebra and graph theory‎. Focusing on the commuting graphs of semi-dihedral groups‎, ‎we examine their various topological properties‎. ‎Furthermore‎, ‎we provide a detailed exploration of key topological graph indices‎, ‎including the general Randi\'{c} index‎, ‎the Wiener index‎, ‎the atom-bond connectivity index (and its fourth variation)‎, ‎the Schultz molecular topological index‎, ‎the geometric-arithmetic index‎, ‎the harmonic index‎, ‎and the Harary index‎. This research reveals the structural as well as mathematical features of these graphs‎, ‎providing valuable insights into their possible applications‎. ‎

Let 􀀀 be a graph with adjacency eigenvalues  1   2  : : :   n. Then the energy of 􀀀 , a concept de ned in 1978 by Gutman, is de ned as E(G) = Σ n i=1 j ij. Also the Estrada index of 􀀀 , which is de ned in 2000 by Ernesto Estrada, is de ned as EE(􀀀 ) = Σ n i=1 e i. In this paper, we compute the eigenvalues, energy and Estrada index of Cayley graphs on generalized dihedral groups. As an application, we compute these items for honeycomb toroidal graphs and Cayley graphs on dihedral groups.

In this paper, we obtain the dimensions of symmetry classes of polynomials associated with the irreducible characters of the dihedral group as a subgroup of the full symmetric group. Then we discuss the existence of o-basis of these classes.

One of the significant aspects of fuzzy group theory is the classification of the fuzzy subgroups of finite groups. The main goal of this paper is to give an explicit formula for the number of fuzzy subgroups of the semi-dihedral group of order 8n in particular cases n=p^m and n=pq; by using the natural equivalence relation. The corresponding equivalence classes of fuzzy subgroups of the semi-dihedral group are closely connected to the chains of subgroups of the semi-dihedral group. ‎In this regard‎; Inclusion-Exclusion principle and structure of the maximal subgroups play an‎ ‎essential role and in some situations give relations which lead to explicit formulas. Finally; by the obtained results; we determine the fuzzy subgroups of all semi-dihedral groups of order less than 100.

In this paper, we introduce the concept of n-A-con-cos groups, n ≥ 2, mention some properties of them and determine all finite abelian groups with at most two direct factors. As a consequence, it is proved that dihedral groups D2m in which m has at most two prime factors are n-A-con-cos.

A , nite group G, in which two randomly chosen subgroups H and K commute, has been classi, ed by Iwasawa in 1941. It is possible to de, ne a probabilistic notion, which \measures the distance" of G from the groups of Iwasawa. Here we introduce the generalized subgroup commutativity degree gsd(G) for two arbitrary sublattices S(G) and T(G) of the lattice of subgroups L(G) of G. Upper and lower bounds for gsd(G) are shown and we study the behaviour of gsd(G) with respect to subgroups and quotients, showing new numerical restrictions.