Factoring Elementaryp-Groups

Letpbe an odd prime and letGbe an elementaryp-group. In other words letGbe a direct product of groups of orderp. Further letB, A1,…,Anbe subsets ofGsuch that B = p2, A1 = ••• = An = p, and each ofA1,…,Andiffers from a subgroup of orderpofGin at most one element. If the productBA1•••Anis direct and is equal toG, then at least one of the factorsB, A1,…,Anmust be periodic. A subset ofGis periodic if it is a direct product of a subset and a proper subgroup ofG.