The Hurwitz bound on the order of the $\mathbb K$-automorphism group ${\rm{Aut}}({\mathcal{X}})$ of an algebraic curve ${\mathcal{X}}$ of genus $g(\mathcal{X})\ge 2$ defined over a field $\mathbb K$ of zero characteristic states that $|{\rm{Aut}}({\mathcal{X}})|\le 84(g(\mathcal{X})-1)$. Improved bounds are available for the order of certain types of subgroups within automorphism groups. For instance, if a subgroup $H$ of ${\rm{Aut}}({\mathcal{X}})$ is dihedral, then in the complex case, $|H| \leq 4g(\mathcal{X}) + 4$. More recently it has been shown that a tighter bound holds for $H$ a generalized quasi-dihedral group. In this paper we explore the more general setting of a curve defined over a field of any characteristic, and $H$ a group admitting a cyclic subgroup of index two. We show that the same upper bound for the size of a dihedral group of automorphisms holds for curves defined over an algebraically closed field of characteristic $p\ne 2$. Then we provide some classification results about (non-dihedral) groups of size larger than $4g(\mathcal{X})+4$ admitting a cyclic subgroup of index $2$.
Given a finite transitive group $G\leq \operatorname{Sym}(\Omega)$, the derangement graph $\Gamma_G$ is the graph whose vertex set is $G$, and two vertices $g$ and $h$ are adjacent if the ratio $h^{-1}g$ is a fixed-point-free permutation. In this paper, we show that the automorphism group of the derangement graph of th...
For every group $\Gamma$, we define a homomorphism $d^\Gamma : \mathrm{Comm}(\Gamma) \to \mathbb{Z}^{(\mathcal{FS})}$ from the abstract commensurator $\mathrm{Comm}(\Gamma)$ to the free abelian group $\mathbb{Z}^{(\mathcal{FS})}$ with basis the collection $\mathcal{FS}$ of isomorphism classes of finite simple groups. W...
We show that the homeomorphism groups of the following spaces have precisely $2^{2^{\aleph_0}}$ maximal subgroups: the rational numbers $\mathbb{Q}$, the Baire space $\mathbb{N}^{\mathbb{N}}$, the space $\mathbb{N}\times 2^{\mathbb{N}}$ where $2^{\mathbb{N}}$ is the Cantor set, the ordinal $\omega^2$ under its order to...
Let $\delta\in\mathbb{F}_{2^n}$ satisfy $\operatorname{Tr}_{\mathbb{F}_{2^n}/\mathbb{F}_2}(\delta)=1$. We study the permutation behavior of $$ f(x) = \left(\frac{1}{x^2+x+\delta}\right)^{2^k}+x $$ over $\mathbb{F}_{2^n}$. Helleseth and Zinoviev proved that $f(x)$ is a permutation for $k=0,1$, and remarked that numerica...
Let $\mathcal{F}$ be a canonical foliation of general type on a smooth projective surface $X$, and write $\mathrm{vol}(\mathcal{F}):=\mathrm{vol}(K_{\mathcal{F}})$. Let $G\subseteq\operatorname{Bir}(X,\mathcal{F})$ be a finite subgroup, and let $\mathcal{G}:=\mathcal{F}/G$ be the quotient foliation in the birational se...
For a finite group $G$, define the field $\mathbb{Q}(G) := \mathbb{Q}(\{\chi(g) : \chi \in \operatorname{Irr}(G) , g \in G\})$. In this article we compute generating sets for the fields $\mathbb{Q}(G)$ when $G$ is a finite general or special linear group, a finite general or special unitary group, or one of the simple...
Eden Ketchum· 0 citations
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