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

39B52Functional equations for functions with more general domains and/or ranges
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