Finite quotients of ultraproducts of finite perfect groups (Q6169106)

This paper is concerned with the problem: Given an ultraproduct of finite groups, what can be said about the possible quotients? For example, \textit{B. Zilber} [Lect. Notes Ser., Inst. Math. Sci., Natl. Univ. Singap. 25, 199--223 (2014; Zbl 1321.03052)] asks whether a compact simple Lie group can be a quotient of an ultraproduct of finite groups. The answer is ``no'' as shown by \textit{N. Nikolov} et al. [J. Éc. Polytech., Math. 5, 239--258 (2018; Zbl 1452.20025)]. Since an ultraproduct of groups is a quotient of the direct product of those same groups, the answer to the question above in many cases is clear. For example, the quotients of an ultraproduct of abelian groups must be abelian. In [\textit{L. Ribes} and \textit{P. Zalesskii}, Profinite groups. Berlin: Springer. 19--77 (2000; Zbl 0949.20017)], it is shown that a finite quotient of an ultraproduct of finite \(p\)-groups is a \(p\)-group and that a finite quotient of an ultraproduct of finite soluble groups is a soluble group. In this paper, the following main result is proved: For any finite group \(G\), \(G\) is an abstract quotient of an ultraproduct of finite perfect groups; in particular, \(G\) is an abstract quotient of a direct product of finite perfect groups. The proof is done via an explicit construction.