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Solution of Whitehead equation on groups

Valeriĭ A. FaĭzievPrasanna K. Sahoo — 2013

Mathematica Bohemica

Let G be a group and H an abelian group. Let J * ( G , H ) be the set of solutions f : G H of the Jensen functional equation f ( x y ) + f ( x y - 1 ) = 2 f ( x ) satisfying the condition f ( x y z ) - f ( x z y ) = f ( y z ) - f ( z y ) for all x , y , z G . Let Q * ( G , H ) be the set of solutions f : G H of the quadratic equation f ( x y ) + f ( x y - 1 ) = 2 f ( x ) + 2 f ( y ) satisfying the Kannappan condition f ( x y z ) = f ( x z y ) for all x , y , z G . In this paper we determine solutions of the Whitehead equation on groups. We show that every solution f : G H of the Whitehead equation is of the form 4 f = 2 ϕ + 2 ψ , where 2 ϕ J * ( G , H ) and 2 ψ Q * ( G , H ) . Moreover, if H has the additional property that 2 h = 0 implies h = 0 for all h H , then every...

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