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

TL;DR

This paper generalizes classical theorems on counting solutions to equations in groups by extending to arbitrary first-order formulas, unifying and broadening previous results like Frobenius and Solomon theorems.

Contribution

It introduces a general theorem that extends solution counting from equations to arbitrary first-order formulas in groups, encompassing several classical results.

Findings

Generalizes Frobenius theorem to first-order formulas

Extends Solomon theorem to broader formulas

Unifies solution counting in groups under a common framework

Abstract

Our result contains as special cases the Frobenius theorem (1895) on the~number of solutions to the equation $x^n=1$ 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.