Descriptive Complexity of Finite Abelian Groups

We investigate the descriptive complexity of finite abelian groups. Using Ehrenfeucht-Fräıssé 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, and let m be a number that divides one of the two groups’ orders. Then the following hold: (1) there exists a first-order sentence φ that distinguishes G1 and G2 such that φ is existential, has quantifier depth O(logm), 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(logm). 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of a single element on the class of finite groups (their undefinability were shown by A. Koponen and K. Luosto in an unpublished paper).