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Zbl 0755.39008
Parnami, J.C.; Vasudeva, H.L.
On Jensen's functional equation.
(English)
[J] Aequationes Math. 43, No.2-3, 211-218 (1992). ISSN 0001-9054; ISSN 1420-8903/e

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.
[J.Aczél (Waterloo / Ontario)]
MSC 2000:
*39B52 Functional equations for functions with more general domains

Keywords: Jensen's functional equation; 2-cancellativity; Zorn's lemma; abelian groups; homomorphisms

Cited in: Zbl 0938.39024

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