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On Jensen’s functional equation. (English) Zbl 0755.39008
The following is offered as main result. Let $\left(G,·\right)$ and $\left(H,+\right)$ be abelian groups, and $e$ the neutral element of $\left(G,·\right)$. The solutions $f:G\to H$ of $f\left(xy\right)+f\left(x{y}^{-1}\right)=2f\left(x\right)$, $f\left(e\right)=0$ are exactly the homomorphisms of $G\to H$ if, and only if, either $H$ has no element of order 2 or $\left[G:{G}^{2}\right]\le 2$, where ${G}^{2}:=\left\{{x}^{2}\mid \phantom{\rule{4pt}{0ex}}x\in G\right\}$. While this is not true in general for nonabelian groups, a partial result (in the “only if” direction) is presented in this case too.

MSC:
 39B52 Functional equations for functions with more general domains and/or ranges
References:
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