Some properties of dihedral group codes

Lie algebras with Frobenius dihedral groups of automorphisms

Suppose that a Lie algebra $L$ admits a finite Frobenius group of automorphisms $FH$ with cyclic kernel $F$ and complement $H$ of order 2, such that the fixed-point subalgebra of $F$ is trivial and the fixed-point subalgebra of $H$ is metabelian. Then the derived length of $L$ is bounded by a constant.

Fuchs' problem for dihedral groups

More than 50 years ago, Laszlo Fuchs asked which abelian groups can be the group of units of a ring. Though progress has been made, the question remains open. One could equally well pose the question for various classes of nonabelian groups. In this paper, we prove that D_2, D_4, D_6, D_8, and D_12 are the only dihedral groups that appear as the group of units of a ring of positive characteristic (or, equivalently, of a finite ring), and D_2 and D_4k, where k is odd, are the only dihedral groups that appear as the group of units of a ring of characteristic 0.

On Commutation Semigroups of Dihedral Groups

For G a group and g in G, we define mappings pg(G) and lg(G) from G into G by pg(x)=[x,g] and lg(x)=[g,x]. We let P(G) and L(G) denote the subsemigroups of the set of all mappings from G to G generated by {pg: g in G} and {lg: g in G}, respectively. P(G) and L(G) are called the right and left commutation semigroup of G, respectively. In this paper we will give explicit formulas for the orders of both P(G) and L(G) where G is a dihedral group.

关于四元数与二面体brace的数量及Hopf--Galois结构

On the number of quaternion and dihedral braces and Hopf--Galois structures

本研究证明了乘群为广义四元数群的给定阶braces数量的相关猜想，给出了二面体群情形的类似结果，并枚举了对应Galois扩张上的阿贝尔型Hopf-Galois结构。

Primitive decompositions of idempotents of the group algebras of dihedral groups and generalized quaternion groups

In this paper, we introduce a method computing the primitive decomposition of idempotents of any semisimple finite group algebra based on its matrix representations and Wedderburn decomposition. Particularly, we use this method to calculate the examples of the dihedral group algebras $\mathbb{C}[D_{2n}]$ and generalized quaternion group algebras $\mathbb{C}[Q_{4m}]$. Inspired by the orthogonality relations of the character tables of these two families of groups, we obtain two sets of trigonometric identities. Furthermore, a group algebra isomorphism between $\mathbb{C}[D_{8}]$ and $\mathbb{C}[…

Public key cryptography based on twisted dihedral group algebras

In this paper, we propose to use a twisted dihedral group algebra for public-key cryptography. For this, we introduce a new $2$-cocycle $α_λ$ to twist the dihedral group algebra. Using the ambient space $\mathbb{F}^{α_λ} D_{2n}$, we then introduce a key exchange protocol and present an analysis of its security. Moreover, we explore the properties of the resulting twisted algebra $\mathbb{F}^{α_λ}D_{2n}$, exploiting them to enhance our key exchange protocol. We also introduce a probabilistic public-key scheme derived from our key-exchange protocol and obtain a key encapsulation mechanism (KEM) …

具有二面体中心商和同构模群代数的非同构2-群的识别

Identification of non-isomorphic 2-groups with dihedral central quotient and isomorphic modular group algebras

本文将非同构有限p-群具有同构模群代数的否定解嵌入到具有二面体中心商的二生成有限2-群类中，解决了该类所有群的原始问题，发现了新否定解并揭示了其特殊性。

Public key cryptography based on skew dihedral group rings

In this paper, we propose to use a skew dihedral group ring given by the group $D_{2n}$ and the finite field $\mathbb{F}_{q^2}$ for public-key cryptography. Using the ambient space $\mathbb{F}_{q^{2}}^θ D_{2n}$ and a group homomorphism $θ: D_{2n} \rightarrow \mathrm{Aut}(\mathbb{F}_{q^2})$, we introduce a key exchange protocol and present an analysis of its security. Moreover, we explore the properties of the resulting skew group ring $\mathbb{F}_{q^{2}}^θ D_{2n}$, exploiting them to enhance our key exchange protocol. We also introduce a probabilistic public-key scheme derived from our key exc…

Automorphic Lie Algebras with dihedral symmetry

The concept of Automorphic Lie Algebras arises in the context of reduction groups introduced in the early 1980s in the field of integrable systems. Automorphic Lie Algebras are obtained by imposing a discrete group symmetry on a current algebra of Krichever-Novikov type. Past work shows remarkable uniformity between algebras associated to different reduction groups. For example, if the base Lie algebra is $\mathfrak{sl}_2(\mathbb{C})$ and the poles of the Automorphic Lie Algebra are restricted to an exceptional orbit of the symmetry group, changing the reduction group does not affect the Lie a…

二面体群的非交换不变量

Noncommutative invariants of dihedral groups

本文研究复数域上二元生成自由亚交换结合代数和李代数在有限阶二面体群作用下的不变量，给出了结合情形的有限生成元集、李情形的极小生成元系统，并计算了希尔伯特级数。