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First, I construct an isomorphism between the categories of (topological) groups of nilpotency class 2 with 2-divisible center and (topological) Lie rings of nilpotency class 2 with 2-divisible center. That isomorphism allows us to…

We study the groups $G$ with the curious property that there exists an element $k\in G$ and a function $f\colon G\to G$ such that $f(xk)=xf(x)$ holds for all $x\in G$. This property arose from the study of near-rings and input-output…

An ordinary differential field $(F,d)$ of characteristic zero, a subgroup $H$ of affine group $ GL(n,C)\propto C^n$ with respect to its identical representation in $F^n$ and the following two fields of differential rational functions in…

We give the first examples of derived equivalences between varieties defined over non-closed fields where one has a rational point and the other does not. We begin with torsors over Jacobians of curves over Q and F_q(t), and conclude with a…

Let $k$ be an algebraically closed field of characteristic $p>0$ and let $\mathbb{F}$ be an algebraically closed field of characteristic $0$. Recently, together with Bouc, we introduced the notion of functorial equivalences between blocks…

In this article, we study the derivations of group algebras of some important groups, namely, dihedral ($D_{2n}$), Dicyclic ($T_{4n}$) and Semi-dihedral ($SD_{8n}$). First, we explicitly classify all inner derivations of a group algebra…

We classify elementary abelian 2 subgroups of compact simple Lie groups of adjoint type. This finishes the classification of elementary abelian $p$ subgroups of compact (or linear algebraic) simple groups of adjoint type.

Simple algebraic groups of type $F_4$ defined over a field $k$ are the full automorphism groups of Albert algebras over $k$. Let $A$ be an Albert algebra over a field $k$ of arbitrary characteristic. We prove that there is an isotope…

We prove the existence of certain rationally rigid triples in F_4(p) for good primes p (i.e., p>3), thereby showing that these groups occur as regular Galois groups over Q(t) and so also over Q. We show that these triples give rise to rigid…

We show that under the assumption of Artin's Primitive Root Conjecture, for all primes p there exist ordinary elliptic curves over $\bar F_p(x)$ with arbitrary high rank and constant j-invariant. For odd primes p, this result follows from a…

We study the possible structure of the groups of rational points on elliptic curves of the form y^2=(ax+1)(bx+1)(cx+1), where a,b,c are non-zero rationals such that the product of any two of them is one less than a square.

Every Ree group $R(q)$, with $q\neq 3$ an odd power of 3, is the automorphism group of an abstract regular polytope, and any such polytope is necessarily a regular polyhedron (a map on a surface). However, an almost simple group $G$ with…

We construct nontrivial homomorphisms from the quasi group of some cubic surfaces over $\bbF_{\!p}$ into a group. We show experimentally that the homomorphisms constructed are the only possible ones and that there are no nontrivial…

In this paper we study a general class of conics starting from a quotient field. We give a group structure over these conics generalizing the construction of a group over the Pell hyperbola. Furthermore, we generalize the definition of…

A simple undirected graph is said to be {\em semisymmetric} if it is regular and edge-transitive but not vertex-transitive. Every semisymmetric graph is a bipartite graph with two parts of equal size. It was proved in [{\em J. Combin.…

Let R be a semi-local regular ring containing an infinite perfect field, and let K be the field of fractions of R. Let H be a simple algebraic group of type F_4 over R such that H_K is the automorphism group of a 27-dimensional Jordan…

Let X be an affine irreducible variety over an algebraically closed field k of characteristic zero. Given an automorphism F, we denote by k(X)^F its field of invariants, i.e. the set of rational functions f on X such that f(F)=f. Let n(F)…

We consider the class $\mathfrak M$ of $\bf R$--modules where $\bf R$ is an associative ring. Let $A$ be a module over a group ring $\bf R$$G$ where $G$ is a group and let $\mathfrak L(G)$ be a set of all proper subgroups of $G$ such that…

In many situations, the monodromy group of enumerative problems will be the full symmetric group. In this paper, we study a similar phenomenon on the rational curves in $|\mathcal{O}(1)|$ on a generic K3 surface of fixed genus over…

Let $F$ be any field of characteristic $p$. It is well-known that there are exactly $p$ inequivalent indecomposable representations $V_1,V_2,...,V_p$ of $C_p$ defined over $F$. Thus if $V$ is any finite dimensional $C_p$-representation…