DESCRIPTIVE COMPLEXITY OF FINITE ABELIAN GROUPS

Abstract

We investigate the descriptive complexity of finite abelian groups. Using Ehrenfeucht–Fraïssé games we find upper and lower bounds on quantifier depth, quantifier alternations, and number of variables of a first-order sentence that distinguishes two finite abelian groups. Our main results are the following. Let G1 and G2 be a pair of non-isomorphic finite abelian groups. Then there exists a positive integer m that divides one of the two groups' orders such that the following holds: (1) there exists a first-order sentence φ that distinguishes G1 and G2 such that φ is existential, has quantifier depth O(log m), and has at most 5 variables and (2) if φ is a sentence that distinguishes G1 and G2 then φ must have quantifier depth Ω( log m). These results are applied to (1) get bounds on the first-order distinguishability of dihedral groups, (2) to prove that on the class of finite groups both cyclicity and the closure of a single element are not first-order definable, and (3) give a different proof for the first-order undefinability of simplicity, nilpotency, and the normal closure of a single element on the class of finite groups (their undefinability were shown by Koponen and Luosto in an unpublished paper).