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

Abstract Our result contains as special cases the Frobenius theorem (1895) on the number of solutions to the equation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>x</m:mi> <m:mi>n</m:mi> </m:msup> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> 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.