A Note on Abelian Groups
نویسنده
چکیده
Vijayaraghavan and Chowla [2] have proved the following result. If n = 2 or has no primitive root, then there exist suitable reduced residue systems ft, r», • • • , f » and Si, s2, ■ ■ ■ , Sh, where h=in), such that riSi, r2s2, • • • , r^Sh 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 Oi, a2, • • • , a* denote the elements of an abelian group A. For what groups A is it possible to find a permutation 61, b2, • • • , bh of the a's such that aj>i, a2b2, • • ■ , anbh 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
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