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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 $$f(xy)+f(xy^{-1})=2f(x)$$, $$f(e)=0$$ are exactly the homomorphisms of $$G\to H$$ if, and only if, either $$H$$ has no element of order 2 or $$[G:G^ 2]\leq 2$$, where $$G^ 2:=\{x^ 2\mid\;x\in G\}$$. While this is not true in general for nonabelian groups, a partial result (in the “only if” direction) is presented in this case too.

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