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Kernel of the second order Cauchy difference on groups

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Abstract

Let (G, ·) be a group, (H, +) be an abelian group, and \({f:G\rightarrow H}\). The second order Cauchy difference of f is

$$C^{(2)}f(x,y,z)=f(xyz)-f(xy)-f(yz)-f(xz)+f(x)+f(y)+f(z).$$

The functional equation

$$C^{(2)}f(x,y,z)=0$$

is studied. We present its general solution on free groups. Solutions on other selected groups are also given.

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Authors and Affiliations

  1. Department of Pure Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    Che Tat Ng

  2. School of Mathematics, Chongqing Normal University, Chongqing, 401331, People’s Republic of China

    Hou Yu Zhao

Authors
  1. Che Tat Ng
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  2. Hou Yu Zhao
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Correspondence to Hou Yu Zhao.

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Ng, C.T., Zhao, H.Y. Kernel of the second order Cauchy difference on groups. Aequat. Math. 86, 155–170 (2013). https://doi.org/10.1007/s00010-012-0174-6

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  • DOI: https://doi.org/10.1007/s00010-012-0174-6

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