Vertex-minimal graphs with dihedral symmetry I

Let $D_{2n}$ denote the dihedral group of order $2n$, where $n\ge 3$. In this article, we build upon the findings of Haggard and McCarthy who, for certain values of $n$, produced a vertex-minimal graph with dihedral symmetry. Specifically, Haggard considered the situation when $\frac{n}{2}$ or $n$ is a prime power, and McCarthy investigated the case when $n$ is not divisible by $2$, $3$ or $5$. In this article, we assume $n$ is not divisible by $4$ and construct a vertex-minimal graph whose automorphism group is isomorphic to $D_{2n}$.     

What is the probability an automorphism fixes a group element?

Extending the notion of probability to the automorphisms of a group, we find the probability of an arbitrarily chosen automorphism of a group fixing an arbitrary element of the group.     

Number of solutions for cubic Thue equations with automorphisms

Let be an irreducible cubic form with positive discriminant, and with non-trivial automorphisms. We show that the Thue equation F(x,y) = 1 has at most three integer solutions except for a few known cases. For the proof, we use an explicitly expressed cubic form which is equivalent to F. To obtain an upper bound for the size of solutions, we use the Padé approximation method developed in our former work. To obtain a lower bound for the size of solutions, we use a result of R. Okazaki on gaps between solutions, which is obtained by geometric consideration. 2000 Mathematics Subject Classification Primary—11D25, 11D59     

一类无限Cernikov p-群的自同构群北大核心CSCD

设G是无限Cernikov p-群,且G的每个真商群是Abel群,但G不是Abel群,本文确定了G的自同构群.     

A Galois correspondence for compact group actions on C-algebras

In this paper, we prove a Galois correspondence for compact group actions on C^?-algebras in the presence of a commuting minimal action. Namely, we show that there is a one-to-one correspondence between the C^?-subalgebras that are globally invariant under the compact action and the commuting minimal action, that in addition contain the fixed point algebra of the compact action and the closed, normal subgroups of the compact group.     

On the Automorphism Group of a Semifield of Order 54

The purpose of this article is to determine Aut(A) where A is a semifield of order 5⁴ admitting an automorphism group E ? Z ₂ × Z ₂ acting freely on A.     

Matrix Classes Realized by Automorphisms of Compact Riemann Surfaces of Genus Two

A complete list of realizable conjugacy classes of torsion in the 4 × 4 integral symplectic group by analytic automorphisms of compact connected Riemann surfaces of genus two is given.     

Ore Extensions of Ore Domains

Let R be a right Ore domain and φ a derivation or an automorphism of R. We determine the right Martindale quotient ring of the Ore extension R[t; φ] (Theorem 1.1). As an attempt to generalize both the Weyl algebra and the quantum plane, we apply this to rings R such that k[x] ? R ? k(x), where k is a field and x is a commuting variable. The Martindale Quotient quotient ring of R[t; φ] and its automorphisms are computed. In this way, we obtain a family of non-isomorphic infinite dimensional simple domains with all their automorphisms explicitly described.