The hypercenter of an algebraic group

TL;DR

This paper introduces the hypercenter of a connected algebraic group as the final term of its transfinitely extended upper central series, showing it leads to a quotient with trivial center.

Contribution

It constructs the hypercenter of an algebraic group and proves its properties, including an analogue of Fitting's theorem for algebraic groups.

Findings

Existence of a nilpotent hypercenter for any connected algebraic group.

The quotient of the group by its hypercenter has trivial center.

Extension of classical theorems to the setting of algebraic groups.

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.