If C is a smooth curve over an algebraically closed field k of characteristic exponent p, then the structure of the maximal prime-to-p quotient of the étale fundamental group is known by analytic methods. In this paper, we discuss the properties of the fundamental group that can be deduced by purely algebraic techniques. We describe a general reduction from an arbitrary curve to the projective ...

Polynomial-time algorithms are given to find a central decomposition of maximum size for a finite p-group of class 2 and for a nilpotent Lie ring of class 2. The algorithms use Las Vegas probabilistic routines to compute the structure of finite ∗-rings and also the Las Vegas C-MeatAxe. When p is small, the probabilistic methods can be replaced by deterministic polynomial-time algorithms. The me...

As the heat kernel plays an important role in many problems in harmonic analysis, an explicit usable expression is very much desirable. An explicit expression for the heat kernel for the Heisenberg group Hn = C ×R was obtained by Hulanicki [9] and by Gaveau [7]. Gaveau [7] also obtained the heat kernel for free nilpotent Lie groups of step two. Cygan [4] obtained the heat kernel for all nilpote...

Let G be a classical group and let g be its Lie algebra. For a nilpotent element X E g, the ring R(Ox) of regular functions on the nilpotent orbit Ox is a Gmodule. In this thesis, we will decompose it into irreducible representations of G for some spherical nilpotent orbits. Thesis Supervisor: David Alexander Vogan Title: Professor of Mathematics

For a finite group G and an arbitrary prime p, let S (G) denote the P intersection of all maximal subgroups M of G such that [G:M] is both composite and not divisible by p; if no such M exists we set S (G) G. Some properties of P G are considered involving S (G). In particular, we obtain a characterization of P G when each M in the definition of S (G) is nilpotent. P

Let G be a complex simple Lie group and let g be its Lie algebra. Then G has the adjoint action on g. The orbit Ox of a nilpotent element x ∈ g is called a nilpotent orbit. A nilpotent orbit Ox admits a non-degenerate closed 2-form ω called the Kostant-Kirillov symplectic form. The closure Ōx of Ox then becomes a symplectic singularity. In other words, the 2-form ω extends to a holomorphic 2-fo...

Let G be a complex simple Lie group and let g be its Lie algebra. Then G has the adjoint action on g. The orbit Ox of a nilpotent element x ∈ g is called a nilpotent orbit. A nilpotent orbit Ox admits a non-degenerate closed 2-form ω called the Kostant-Kirillov symplectic form. The closure Ōx of Ox then becomes a symplectic singularity. In other words, the 2-form ω extends to a holomorphic 2-fo...

Suppose GR is the real points of a complex connected reductive algebraic group G defined over R. (The class of such groups is exactly the class treated by the software package atlas.) Write g R for the dual of the Lie algebra of GR and N R for the nilpotent elements in gR. Then GR acts with finitely many orbits on N via the coadjoint action, and it is clearly very desirable to parametrize these...

We say that a finite group G is DRR-detecting if, for every subset S of G, either the Cayley digraph Cay(G,S) digraphical regular representation (that is, its automorphism acts regularly on vertex set) or there nontrivial φ such φ(S) = S. show nilpotent p-group, but wreath product Zp wr not DRR-detecting, odd prime p. also if G1 and G2 are groups admit gcd(|G1|, |G2|) 1, then direct x DRR-detec...