Projective Limits and Ultraproducts of Nonabelian Finite Groups

TL;DR

This paper investigates the structure of quotients of ultraproducts of finite groups, revealing conditions under which various classes of groups preserve their properties and constructing examples where any finite group can be a quotient.

Contribution

It characterizes finite quotients of ultraproducts of non-solvable finite groups and provides explicit constructions for arbitrary finite quotients of ultraproducts of perfect groups.

Findings

Finite quotients of pro- groups remain in  for certain classes.

Finite perfect groups with unbounded commutator width can produce any finite group as a quotient.

Explicit constructions of finite quotients of ultraproducts are provided.

Abstract

Groups that can be approximated by finite groups have been the center of much research. This has led to the investigations of the subgroups of metric ultraproducts of finite groups. This paper attempts to study the dual problem: what are the quotients of ultraproducts of finite groups? Since an ultraproduct is an abstract quotient of the direct product, this also led to a more general question: what are the abstract quotients of profinite groups? Certain cases were already well-studied. For example, if we have a pro-solvable group, then it is well-known that any finite abstract quotients are still solvable. This paper studies the case of profinite groups from a surjective inverse system of non-solvable finite groups. We shall show that, for various classes of groups $\mathcal{C}$, any finite abstract quotient of a pro-$\mathcal{C}$ group is still in $\mathcal{C}$. Here $\mathcal{C}$ can be the class of finite semisimple groups, or the class of central products of quasisimple groups, or the class of products of finite almost simple groups and finite solvable groups, or the class of finite perfect groups with bounded commutator width. Furthermore, in the case of finite perfect groups with unbounded commutator width, we show that this is NOT the case. In fact, any finite group could be a quotient of an ultraproduct of finite perfect groups. We provide an explicit construction on how to achieve this for each finite group.