On the fixing sets of dihedral groups

The fixing number of a graph $\Gamma$ is the minimum number of labeled vertices that, when fixed, remove all nontrivial automorphisms from the automorphism group of $\Gamma$. The fixing set of a finite group $G$ is the set of all fixing numbers of graphs whose automorphism groups are isomorphic to $G$. Previously, authors have studied the fixing sets of both abelian groups and symmetric groups. In this article, we determine the fixing set of the dihedral group.     

A result concerning nilpotent values with generalized skew derivations on Lie ideals

ABSTRACT

Let n≥1 be a fixed integer, R a prime ring with its right Martindale quotient ring Q, C the extended centroid, and L a non-central Lie ideal of R. If F is a generalized skew derivation of R such that (F(x)F(y)?yx)ⁿ = 0 for all x,y∈L, then char(R) = 2 and R?M₂(C), the ring of 2×2 matrices over C.     

On the arithmetically cohen-macaulay property for monomial curves

Let C be a monomial curve in three dimensional projective space over a field of characteristic zero . We give a necessary criterion for a monomial curve to be set theoretic complete intersection on bihomogeneous surfaces. Using this criterion we prove several results concerning the arithmetically Cohen-Macaulay property for monomial curves.     

On the Automorphism Group of the First Weyl Algebra

Let A ₁: = 𝕜[t, ?] be the first algebra over a field 𝕜 of characteristic zero. Let Aut_𝕜(A ₁) be the automorphism group of the ring A ₁. One can associate to each right ideal I of A ₁ a subgroup of Aut_𝕜(A ₁) called the isomorphism subgroup of I. In this article, we show that each such isomorphism subgroup is equal to its normalizer. For that, we study when the isomorphism subgroup of a right ideal of A ₁ contains a given isomorphism subgroup.     

On Rings with Few Orbits

In this article we investigate the structure of rings with some strong symmetry condition.