On the computational complexity of the theory of Abelian groups

We develop a series of Ehrenfeucht games and prove the following results:

• 1.(i) The first order theory of the divisible and indecomposable p-group, the first order theory of the group of rational numbers with denominators prime to p and the first order theory of a cyclic group of prime power order can be decided in 2^{2cn log n} Turing time.
• 2.(ii) The first order theory of the direct sum of countably many infinite cyclic groups, the first order theory of finite Abelian groups and the first order theory of all Abelian groups can be decided in 2^2dn Turing space.