A note on abelian groups
L. Carlitz · Proceedings of the American Mathematical Society · 1953
Vijayaraghavan and Chowla [2] have proved the following result. If n=2 or has no primitive root, then there exist suitable reduced residue systems rl, r2, , * * , rh and sl, S2 , . * Sh, where h= 4(n), such that risi, r2s2, * , rhsh is also a complete residue system (mod n). Since the numbers of a reduced residue system (mod n) form an abelian group with respect to multiplication, it seems natural to raise the following question. Let a,, a2, * * *, ah denote the elements of an abelian group A. For what groups A is it possible to find a permutation bi, b2, * , bh of the a's such that albi, a2b2, * , ahbh are distinct? For brevity let us call this property M. Clearly if A and B have property M then the same is true of the direct product A XB. We now prove the following