We take the base field to be the field of complex numbers in these lectures. The varieties are, by definition, quasi-projective, reduced (but not necessarily irreducible) schemes. Let G be a semisimple, simply-connected, complex algebraic group with a fixed Borel subgroup B, a maximal torus H ⊂ B, and associated Weyl group W . (Recall that a Borel subgroup is any maximal connected, solvable sub...

Consider a smooth connected solvable group G over a field k. If k is algebraically closed then G = T nRu(G) for any maximal torus T of G. Over more general k, an analogous such semi-direct product structure can fail to exist. For example, consider an imperfect field k of characteristic p > 0 and a ∈ k−kp, so k′ := k(a1/p) is a degree-p purely inseparable extension of k. Note that k′ s := k ′ ⊗k...

in this paper, we classify the indecomposable non-nilpotent solvable lie algebras with $n(r_n,m,r)$ nilradical,by using the derivation algebra and the automorphism group of $n(r_n,m,r)$.we also prove that these solvable lie algebras are complete and unique, up to isomorphism.

applications of hypergroups have mainly appeared in special subclasses. one of the important subclasses is the class of polygroups. in this paper, we study the notions of nilpotent and solvable polygroups by using the notion of heart of polygroups. in particular, we give a necessary and sufficient condition between nilpotent (solvable) polygroups and fundamental groups.

In this paper we introduce the concept of α-commutator which its definition is based on generalized conjugate classes. With this notion, α-nilpotent groups, α-solvable groups, nilpotency and solvability of groups related to the automorphism are defined. N(G) and S(G) are the set of all nilpotency classes and the set of all solvability classes for the group G with respect to different automorphi...

We introduce finite spaces and explain their relations to sets with reflexive and transitive relations and to simplicial complexes. We introduce Quillen’s conjecture: If Ap(G) is weakly contractible then G has a nontrivial normal p-subgroup. We use Quillen’s method to show this is true for solvable groups.

We classify measures on a homogeneous space which are invariant under a certain solvable subgroup and ergodic under its unipotent radical. Our treatment is independent of characteristic. As a result we get the first measure classification for the action of semisimple subgroups without any characteristic restriction.

LetN be a normal subgroup of a groupG and let p be a prime. We prove that if the p-part of jx j is a constant for every prime-power order element x 2 N n Z.N /, then N is solvable and has normal p-complement.

In this note we want to study compact homogeneous spaces G/Γ, where G is a connected and simply-connected Lie group and Γ a discrete subgroup in G. It is well known that the existence of such a Γ implies the unimodularity of the Lie group G. Recall that a Lie group G is called unimodular if for all X ∈ g holds tr adX = 0, where g denotes the Lie algebra of G. If we further demand G/Γ to be symp...

I would like to discuss two results about finite groups: (0) All finite groups of odd order are solvable. (F) A finite group is nilpotent if it admits a fixed point free automorphism of prime order. Walter Feit and I proved (0) after a prolonged joint effort [5]. A critical special case of (0) was proved by Walter Feit, Marshall Hall, Jr. and me [4]. A slightly stronger result than (F) is in th...