Second cohomology groups and finite covers

TL;DR

This paper classifies certain subgroups of a permutation group related to finite covers, connecting algebraic group theory with model theory problems involving infinite sets and their finite covers.

Contribution

It provides a classification of closed subgroups of a specific permutation group that project onto the symmetric group, linking algebraic structures to model-theoretic finite cover problems.

Findings

Classified closed subgroups projecting onto ext{Sym}(D)

Connected algebraic group theory with model theory

Solved a problem about finite covers in algebraic terms

Abstract

For D an infinite set, k>1 and W the set of k-sets from D, there is a natural closed permutation group G_k which is a non-split extension of \mathbb{Z}_2^W by \Sym(D). We classify the closed subgroups of G_k which project onto \Sym(D)$. The question arises in model theory as a problem about finite covers, but here we formulate and solve it in algebraic terms.