Solution of Whitehead equation on groups

Faĭziev, Valeriĭ A., and Sahoo, Prasanna K.. "Solution of Whitehead equation on groups." Mathematica Bohemica 138.2 (2013): 171-180. <http://eudml.org/doc/252553>.

@article{Faĭziev2013,
abstract = {Let $G$ be a group and $H$ an abelian group. Let $J^\{*\} (G, H)$ be the set of solutions $f \colon G \rightarrow H$ of the Jensen functional equation $f(xy)+f(xy^\{-1\}) = 2f(x)$ satisfying the condition $f(xyz) - f(xzy) = f(yz)-f(zy)$ for all $x, y , z \in G$. Let $Q^\{*\} (G, H)$ be the set of solutions $f \colon G \rightarrow H$ of the quadratic equation $f(xy)+f(xy^\{-1\}) = 2f(x) + 2f(y)$ satisfying the Kannappan condition $f(xyz) = f(xzy)$ for all $x, y, z \in G$. In this paper we determine solutions of the Whitehead equation on groups. We show that every solution $f \colon G \rightarrow H$ of the Whitehead equation is of the form $4f = 2 \varphi + 2 \psi $, where $2\varphi \in J^* (G, H)$ and $2\psi \in Q^* (G, H)$. Moreover, if $H$ has the additional property that $2h = 0$ implies $h = 0$ for all $h \in H$, then every solution $f \colon G \rightarrow H$ of the Whitehead equation is of the form $2f = \varphi + \psi $, where $\varphi \in J^*(G,H)$ and $2\psi (x) = B(x, x)$ for some symmetric bihomomorphism $B \colon G \times G \rightarrow H$.},
author = {Faĭziev, Valeriĭ A., Sahoo, Prasanna K.},
journal = {Mathematica Bohemica},
keywords = {homomorphism; Fréchet functional equation; Jensen functional equation; symmetric bihomomorphism; Whitehead functional equation; homomorphism; Fréchet functional equation; Jensen functional equation; symmetric bihomomorphism; Whitehead functional equation; quadratic equations},
language = {eng},
number = {2},
pages = {171-180},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Solution of Whitehead equation on groups},
url = {http://eudml.org/doc/252553},
volume = {138},
year = {2013},
}

TY - JOUR
AU - Faĭziev, Valeriĭ A.
AU - Sahoo, Prasanna K.
TI - Solution of Whitehead equation on groups
JO - Mathematica Bohemica
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 138
IS - 2
SP - 171
EP - 180
AB - Let $G$ be a group and $H$ an abelian group. Let $J^{*} (G, H)$ be the set of solutions $f \colon G \rightarrow H$ of the Jensen functional equation $f(xy)+f(xy^{-1}) = 2f(x)$ satisfying the condition $f(xyz) - f(xzy) = f(yz)-f(zy)$ for all $x, y , z \in G$. Let $Q^{*} (G, H)$ be the set of solutions $f \colon G \rightarrow H$ of the quadratic equation $f(xy)+f(xy^{-1}) = 2f(x) + 2f(y)$ satisfying the Kannappan condition $f(xyz) = f(xzy)$ for all $x, y, z \in G$. In this paper we determine solutions of the Whitehead equation on groups. We show that every solution $f \colon G \rightarrow H$ of the Whitehead equation is of the form $4f = 2 \varphi + 2 \psi $, where $2\varphi \in J^* (G, H)$ and $2\psi \in Q^* (G, H)$. Moreover, if $H$ has the additional property that $2h = 0$ implies $h = 0$ for all $h \in H$, then every solution $f \colon G \rightarrow H$ of the Whitehead equation is of the form $2f = \varphi + \psi $, where $\varphi \in J^*(G,H)$ and $2\psi (x) = B(x, x)$ for some symmetric bihomomorphism $B \colon G \times G \rightarrow H$.
LA - eng
KW - homomorphism; Fréchet functional equation; Jensen functional equation; symmetric bihomomorphism; Whitehead functional equation; homomorphism; Fréchet functional equation; Jensen functional equation; symmetric bihomomorphism; Whitehead functional equation; quadratic equations
UR - http://eudml.org/doc/252553
ER -