We classify up an isomorphism all rings having expressed their elements by at most three commuting idempotents. Our main result considerably extends certain important achievements established by Hirano-Tominaga [3], Ying et al. [6] and Tang et al. [5] as well as it somewhat strengthens recent results proved by the author in [1] and [2].

We show that the $abc$ Conjecture implies the Weak Diversity Conjecture of Bilu and Luca. In addition, we unconditionally reduce the Weak Diversity Conjecture to the case of cyclic covers of prime order.

Around 1973, Macbeath provided a formula for the number of fixed points of an element in a group $G$ of conformal automorphisms of a closed Riemann surface $S$ of genus at least two. Such a formula was initially used to obtain the character of the representation associated to the induced action of $G$ on the first homology group of $S$, and later turned out to be extremely useful in many other contexts. By using a simple counting procedure, we provide a similar formula for the number of connected components of an element in a finite group of isometries of a hyperbolic manifold.

We give infinitely many examples to show that, even for simple groups $G$, it is possible for the lattice of overgroups of a subgroup $H$ to be the Boolean lattice of rank $2$, in such a way that the two maximal overgroups of $H$ are conjugate in $G$. This answers negatively a question posed by Aschbacher.

Reduction theory of binary forms has been studied by Julia in [23] and more recently by several other authors. In this paper we introduce the absolute reduction and give an algorithm to compute the absolutely reduced form of any binary form. Such method can be applied to determine the minimal Weierstrass equation of a superelliptic curve over an integral ring.

Let $𝒜$ be a finite-dimensional split basic algebra over a finite field $\mathbb{k}$ with odd characteristic, and assume that $𝒜$ is endowed with an involution $\sigma :𝒜\to 𝒜$. We determine the degrees of the irreducible characters of the group $C_G\left(\sigma \right)=\left\{x\in G:\sigma \left(x^{-1}\right)=x\right\}$ where $G={𝒜}^{\times }$ is the unit group of $𝒜$, and prove that every irreducible character of $C_G\left(\sigma \right)$ is induced by a linear character of some subgroup. As a particular case, our results hold for the Sylow $p$-subgroups of the finite classical groups of Lie type, and extend (in a uniform way) the results obtained by B. Szegedy in [11].

The problem of finding full automorphism groups of compact Riemann surfaces is classical, though complete results are only known for a few families. One tool used in some classification schemes is strong branching; a condition derived by Accola in [1]. In the following, we survey the main ideas behind strong branching including a general survey of current results. We also provide new results for families for which we can find the full automorphism group using strong branching and an inductive version of strong branching.

Schottky space ${\mathrm{𝒮}}_g$ is the space that parametrizes ${\mathrm{PSL}}_2\left(\mathrm{ℂ}\right)$-conjugacy classes of Schottky groups of rank $g\ge 2$. The branch locus ${ℬ}_g$ consists of the conjugacy classes of those Schottky groups which are a finite index proper subgroup of some Kleinian group. In a previous paper we observed that ${ℬ}_g$ was connected for $g\ge 3$ odd and that it has at most two components for $g\ge 4$ even. In this short note, we observe that ${ℬ}_g$ is always connected.

We study finite permutation groups with special properties, motivated from the theory of Riemann surfaces. In this article we focus on groups acting with fixity $4$ and analyse their Sylow structure and possible orbit lengths of Sylow subgroups.