How many tuples of group elements have a given property?

TL;DR

This paper extends a theorem on the divisibility of solution counts for systems of equations in groups, including equations with coefficients and more complex formulas, revealing new divisibility properties.

Contribution

It generalizes Solomon's theorem to equations with coefficients and arbitrary first-order formulas in groups, broadening the scope of solution count divisibility results.

Findings

Number of solutions to certain equations is divisible by group order.

Divisibility holds for solutions with elements' squares in a subgroup.

Generalization includes equations with coefficients and complex formulas.

Abstract

Generalising Solomon's theorem, C. Gordon and F. Rodriguez-Villegas have proven recently that, in any group, the number of solutions to a system of coefficient-free equations is divisible by the order of this group whenever the rank of the matrix composed of the exponent sums of i-th unknown in j-th equation is less than the number of unknowns. We generalise this result in two directions: first, we consider equations with coefficients, and secondly, we consider not only systems of equations but also any first-order formulae in the group language (with constants). Our theorem implies some amusing facts; for example, the number of group elements whose squares lie in a given subgroup is divisible by the order this subgroup.