Factoring elementary \(p\)-groups (Q1270404)

Let \(G\) be an elementary \(p\)-group of order \(p^{n+2}\). Let \(A_0,\ldots,A_n\) be subsets of \(G\) verifying the following conditions: (i) \(| A_0|=p^2\) and \(| A_i|=p\), for all \(i\geq 1\). (ii) \(G=A_0+A_1+\cdots+A_n\). (iii) For all \(i\geq 1\), there are \(h,d\in G\) such that \(A_i=\{0,h,\ldots,(p-2)h,(p-1)h+d\}\). The author shows that for some \(i\geq 0\), \(A_i\) is periodic.