Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The hypercenter of an algebraic group

Damian Sercombe

Abstract

We show that any connected algebraic group $G$ over a field admits a nilpotent normal subgroup $Z_\infty(G)$ such that the quotient $G/Z_\infty(G)$ has trivial center. We construct $Z_\infty(G)$ as the final term of the transfinitely extended upper central series of $G$; accordingly, we call it the hypercenter of $G$. We establish several related results about the upper central series of $G$, along with an analogue for algebraic groups of a well-known theorem of Fitting's.

The hypercenter of an algebraic group

Abstract

We show that any connected algebraic group over a field admits a nilpotent normal subgroup such that the quotient has trivial center. We construct as the final term of the transfinitely extended upper central series of ; accordingly, we call it the hypercenter of . We establish several related results about the upper central series of , along with an analogue for algebraic groups of a well-known theorem of Fitting's.

Paper Structure

This paper contains 5 sections, 14 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Schematic unions
  4. The upper central series
  5. Proofs

Key Result

Theorem 1.1

Let $G$ be a connected algebraic $k$-group. There exists a nilpotent normal subgroup $Z_\infty(G)$ of $G$ such that the center of $G/Z_\infty(G)$ is trivial. The formation of $Z_\infty(G)$ commutes with base change by algebraic field extensions.

Theorems & Definitions (29)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • ...and 19 more