Algorithms to tile the infinite grid with finite clusters

We say that a subset T of Z², the two dimensional infinite grid, tiles Z² if we can cover Z² with non-overlapping translates of T. No algorithm is known to decide whether a finite T⊆Z² tiles Z². Here we present two algorithms, one for the case when |T| is prime, and another for the case when |T|=4. Both algorithms generalize to the case, where we replace Z ² with all arbitrary finitely generated Abelian group. As a by-product of our results we partially settle the Periodic Tiling Conjecture raised by J. Lagarias and Y. Wang (1997), and we also get the following generalization of a theorem of L.Redei (1965): Let G be a (finite or infinite) Abelian group G with a generator set T of prime cardinality such, that 0∈T, and there is a set T´⊆G with the property that for every g∈G there are unique t∈T, t´∈T´ such that g=t+t´. Then T´ can be replaced with a subgroup of G, that also has the above property