On the lattice of normal subgroups in ultraproducts of compact simple groups

TL;DR

This paper investigates the structure of normal subgroups in ultraproducts of compact simple groups, proving distributivity in general and linear ordering in specific cases, advancing understanding of their algebraic properties.

Contribution

It establishes that the lattice of normal subgroups in ultraproducts of compact simple groups is distributive and identifies conditions for linear ordering.

Findings

Lattice of normal subgroups is distributive in ultraproducts.

Normal subgroups are linearly ordered in ultraproducts of finite simple or compact Lie groups of bounded rank.

Provides new insights into the algebraic structure of ultraproducts of simple groups.

Abstract

We prove that the lattice of normal subgroups of ultraproducts of compact simple non-abelian groups is distributive. In the case of ultraproducts of finite simple groups or compact connected simple Lie groups of bounded rank the set of normal subgroups is shown to be linearly ordered by inclusion.