On the number of tuples of group elements satisfying a first-order formula

2023/06/28 by Elena K. Brusyanskaya, Brusyanskaya, Elena K. · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Differential Equations and Dynamical Systems #FOS: Mathematics #Graph theory and applications #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.2306.16498

openalex publication_date 2023/06/28 · openalex created_date 2023/07/01 · openalex updated_date 2026/07/28

Abstract

Our result contains as special cases the Frobenius theorem (1895) on the~number of solutions to the equation xⁿ=1 in a finite group and the Solomon theorem (1969) on the number of solutions in a group to systems of equations with fewer equations than unknowns. Instead of systems of equations, we consider arbitrary first-order formulae in the group language with constants. Our result substantially generalizes the Klyachko--Mkrtchyan theorem (2014) on this topic.

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