How many tuples of group elements have a given property?

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.

With an appendix by D. V. Trushin, 9 pages. A Russian version of this paper is at http://mech.math.msu.su/department/algebra/staff/klyachko/papers.htm V3: a reference to Mathoverflow is added. V4: Corollary 5 is strengthened; the construction of the matrix $A(\varphi )$ is clarified slightly. V5: The main theorem is strengthened; a corollary is added. V6: misprint correction. V7: A reference added

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