A note on the normal subgroup lattice of ultraproducts of finite quasisimple groups

TL;DR

This paper revises a previous claim about the linearity of the normal subgroup lattice in ultraproducts of finite simple groups, providing corrected results and insights into the structure for classical quasisimple groups.

Contribution

It corrects a false generalization about the normal subgroup lattice being linearly ordered and introduces relative bounded generation results for classical quasisimple groups.

Findings

The lattice of normal subgroups is not always linearly ordered in ultraproducts.

Relative bounded generation results influence the subgroup lattice structure.

Implications for the classification of normal subgroups in ultraproducts.

Abstract

In [A. Stolz and A. Thom, On the lattice of normal subgroups in ultraproducts of compact simple groups, PLMS 108(1), 2014] it was stated that the lattice of normal subgroups of an ultraproduct of finite simple groups is always linearly ordered. This is false in this form in most cases for classical groups of Lie type. We correct the statement and point out a version of 'relative' bounded generation results for classical quasisimple groups and its implications on the structure of the lattice of normal subgroups of an ultraproduct of quasisimple groups.