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On a functional equation on groups with an involution related to quadratic polynomials in two variables

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An Erratum to this article was published on 27 October 2015

Abstract

Let G be a group and \({\mathbb{C}}\) the field of complex numbers. Suppose \({\sigma \colon G \to G}\) is an involution on G. In this paper, we determine the general solution \({f\colon G\times G \to \mathbb{C}}\) of the functional equation

$$\begin{aligned}f(x_1 \sigma y_1 , x_2 \sigma y_2) - f(x_1 \sigma y_1 , x_2) - f(x_1 , x_2 \sigma y_2)\\ \quad = f(x_1 y_1 , x_2 y_2) - f(x_1 y_1 , x_2) - f(x_1 , x_2 y_2)\end{aligned}$$

for all \({x_1, x_2 , y_1, y_2 \in G}\). In Chung et al. (J Korean Math Soc 38:37–47, 2001), the solution of the above equation was determined assuming (a) f is central in each variable, (b) \({\sigma (x) = x^{-1}}\) for all \({x \in G}\), and (c) x 2 = y has a solution on G for all \({x, y \in G}\). We do not require any such conditions to obtain its solution. No new solutions emerge on arbitrary groups.

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References

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Correspondence to Thomas Riedel.

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Hunt, H.B., Riedel, T. & Sahoo, P.K. On a functional equation on groups with an involution related to quadratic polynomials in two variables. Aequat. Math. 90, 87–96 (2016). https://doi.org/10.1007/s00010-015-0367-x

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  • DOI: https://doi.org/10.1007/s00010-015-0367-x

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