Related papers: Maximal subgroups of maximal rank in the classical…

We study relatively minimal subgroups in topological groups. We find, in particular, some natural relatively minimal subgroups in unipotent groups which are defined over "good" rings. By "good" rings we mean archimedean absolute valued (not…

We describe simply connected compact exceptional simple Lie groups in very elementary way. We first construct all simply connected compact exceptional Lie groups G concretely. Next, we find all involutive automorphisms of G, and determine…

Finding the number of maximal subgroups of infinite index of a finitely generated group is a natural problem that has been solved for several classes of `geometric' groups (linear groups, hyperbolic groups, mapping class groups, etc). Here…

Let $G$ be a classical algebraic group, $X$ a maximal rank reductive subgroup and $P$ a parabolic subgroup. This paper classifies when $X\G/P$ is finite. Finiteness is proven using geometric arguments about the action of $X$ on subspaces of…

Suppose $\mathbb{F}$ is a field of prime characteristic $p$ and $E$ is a finite subgroup of the additive group $(\mathbb{F},+)$. Then $E$ is an elementary abelian $p$-group. We consider two such subgroups, say $E$ and $E'$, to be equivalent…

The number of maximal abelian subgroups of a finite p-group is shown to be congruent to 1 modulo p.

An equation to compute the dp-rank of any abelian group is given. It is also shown that its dp-rank, or more generally that of any one-based group, agrees with its Vapnik-Chervonenkis density. Furthermore, strong abelian groups are…

Let $K$ be an algebraically closed field of characteristic zero, $A= K[x_1, \dots, x_n]$ the polynomial ring in $n$ variables, and let $W_n(K)$ be the Lie algebra of all $K$-derivations of $A.$ This Lie algebra also is the free $A$-module…

We study the maximal rank in affine subspaces of symmetric or alternating matrices, in terms of the matching numbers of certain associated graphs. Applications include simple proofs of upper bounds on the dimension of such subspaces in…

Let $W$ be a vector space over an algebraically closed field $k$. Let $H$ be a quasisimple group of Lie type of characteristic $p\ne {\rm char}(k)$ acting irreducibly on $W$. Suppose also that $G$ is a classical group with natural module…

We describe the set of maximal orders in a 2-by-2 matrix algebra over a non-commutative local division algebra B containing a given suborder, for certain important families of such suborders, including rings of integers of division…

For an algebraic number $\alpha$ we consider the orders of the reductions of $\alpha$ in finite fields. In the case where $\alpha$ is an integer, it is known by the work on Artin's primitive root conjecture that the order is "almost always…

We begin with a review of the structure of simple, simply-connected complex Lie groups and their Lie algebras, describe the Chevalley lattice and the associated split group over the integers. This gives us a hyperspecial maximal compact…

We describe a new approach for classifying conjugacy classes of elementary abelian subgroups in simple algebraic groups over an algebraically closed field, and understanding the normaliser and centraliser structure of these. For toral…

We compute conjugacy classes in maximal parabolic subgroups of the general linear group. This computation proceeds by reducing to a ``matrix problem''. Such problems involve finding normal forms for matrices under a specified set of row and…

Let $G$ be a finite group. A proper subgroup $H$ of $G$ is said to be large if the order of $H$ satisfies the bound $|H|^3 \ge |G|$. In this note we determine all the large maximal subgroups of finite simple groups, and we establish an…

The rank of a graph is defined to be the rank of its adjacency matrix. A graph is called reduced if it has no isolated vertices and no two vertices with the same set of neighbors. A reduced graph $G$ is said to be maximal if any reduced…

We study the collection of group structures that can be realized as a group of rational points on an elliptic curve over a finite field (such groups are well known to be of rank at most two). We also study various subsets of this collection…

A Kac-Moody algebra is called hyperbolic if it corresponds to a generalized Cartan matrix of hyperbolic type. We study root subsystems of root systems of hyperbolic algebras. In this paper, we classify maximal rank regular hyperbolic…

Let $k$ be a nonperfect field of characteristic $2$. Let $G$ be a $k$-split simple algebraic group of type $E_6$ (or $G_2$) defined over $k$. In this paper, we present the first examples of nonabelian non-$G$-completely reducible…