We compute the integral cohomology of the minimal non-trivial nilpotent orbit in a complex simple (or quasi-simple) Lie algebra. We find by a uniform approach that the middle cohomology group is isomorphic to the fundamental group of the sub-root system generated by the long simple roots. The modulo l reduction of the Springer correspondent representation involves the sign representation exactl...

Maximal and minimal conditions for ideals in associative rings have often been considered, but little seems to be known of these conditions in non-associative rings, or of chain conditions on the non-normal subgroups of a group. Moreover, it is usual to assume the condition for one-sided ideals in noncommutative rings, and the weaker condition for two-sided ideals rarely appears. In this note w...

Let G0 denote a compact semisimple Lie algebra and U a finite dimensional real G0 module. The vector space N0 = U ⊕ G0 admits a canonical 2-step nilpotent Lie algebra structure with [N0,N0] = G0 and an inner product 〈, 〉, unique up to scaling, for which the elements of G0 are skew symmetric derivations of N0. Let N0 denote the corresponding simply connected 2-step nilpotent Lie group with Lie a...

The notable exclusions from the family of automatic groups are those nilpotent groups which are not virtually abelian, and the fundamental groups of compact 3-manifolds based on the Nil or Sol geometries. Of these, the 3-manifold groups have been shown by Bridson and Gilman to lie in a family of groups defined by conditions slightly more general than those of automatic groups, that is, to have ...

Let [Formula: see text] be a finite group and fixed prime divisor of text]. We prove that if every maximal subgroup is nilpotent, or normal, has text]-order, then (1) solvable; (2) Sylow tower; (3) there exists at most one such neither text]-nilpotent nor text]-closed.

In this work we study some topics in logic and group theory, and naturally, their intersection. One important theme in this work is the notion of random groups, which is informally the study of “what properties do ‘most’ groups satisfy?” In Chapter 2, we are interested in the theory of a random group, i.e., the properties we are interested here are first-order properties. Knight conjectured tha...

Let G be a finite group and α be an automorphism of G of order p n for an odd prime p. Suppose that α acts fixed point freely on every α-invariant p-section of G, and acts trivially or exceptionally on every elementary abelian α-invariant p-section of G. It is proved that G is a solvable p-nilpotent group of nilpotent length at most n + 1, and this bound is best possible.

We describe a nilpotent quotient algorithm for a certain type of infinite presentations: the so-called finite L-presentations. We then exhibit finite L-presentations for various interesting groups and report on the application of our nilpotent quotient algorithm to them. As result, we obtain conjectural descriptions of the lower central series structure of various interesting groups including t...

A W-algebra (of finite type) W is a certain associative algebra associated with a semisimple Lie algebra, say g, and its nilpotent element, say e. The goal of this paper is to study the categoryO for W introduced by Brundan, Goodwin and Kleshchev. We establish an equivalence of this category with a certain category of g-modules. In the case when e is of principal Levi type (this is always so wh...

We study the Ramond twisted representations of the affine W algebra W(ḡ, f) in the case that f admits a good even grading. We establish the vanishing and the almost irreducibility of the corresponding BRST cohomology. This confirms some of the recent conjectures of Kac and Wakimoto [KW5]. In type A, our results give the characters of all irreducible ordinary Ramond twisted representations of W(...