摘要：Let f : G → H be a function, where ( G , ⋅ ) is a group and ( H , + ) is an abelian group. In this paper, the following third order Cauchy difference of f : C ( 3 ) f ( x 1 , x 2 , x 3 , x 4 ) = f ( x 1 x 2 x 3 x 4 ) − f ( x 1 x 2 x 3 ) − f ( x 1 x 2 x 4 ) − f ( x 1 x 3 x 4 ) − f ( x 2 x 3 x 4 ) + f ( x 1 x 2 ) + f ( x 1 x 3 ) + f ( x 1 x 4 ) + f ( x 2 x 3 ) + f ( x 2 x 4 ) + f ( x 3 x 4 ) − f ( x 1 ) − f ( x 2 ) − f ( x 3 ) − f ( x 4 ) ( ∀ x 1 , x 2 , x 3 , x 4 ∈ G ), is studied. We first give some special solutions of C ( 3 ) f = 0 on free groups. Then sufficient and necessary conditions on finite cyclic groups and symmetric groups are also obtained. MSC:39B52, 39A70.
关键词：Cauchy difference ; free group ; symmetric group ; cyclic group
