Algorithms to Tile the Infinite Grid with Finite Clusters

We say that a subset T of Z 2 , the two dimensional innnite grid, tiles Z 2 if we can cover Z 2 with non-overlapping translates of T. No algorithm is known to decide whether a nite T Z 2 tiles Z 2. Here we present two algorithms, one for the case when jTj is prime, and another for the case when jTj = 4. Both algorithms generalize to the case, where we replace Z 2 with an arbitrary nitely generated Abelian group. As a by-product of our result we partially settle the Periodic Tiling Conjecture raised by J. Lagarias and Y. Wang 14], and we also get the following generalization of a theorem of R edei 16]: Let G be a ((nite or innnite) Abelian group G with a generator set T of prime cardinality such, that 0 2 T, and there is a set T 0 G with the property that for every g 2 G there are unique t 2 T, t 0 2 T 0 such that g = t + t 0. Then T 0 can be replaced with a subgroup of G, that also has the above property.