Jensen's functional equation on groups, III

Summary.

A first extension of Jensen's functional equation on the real line is the equation f(xy) + f(xy ^-1) = 2f(x) where f maps a group G into an abelian group H. A second extension is f(xy) + f(y ^-1 x) = 2f(x). Results were reported on the first equation in two preceding articles. Here, we solve the second on free groups, and illustrate how it leads to solutions on more specific groups including linear groups $GL_n({\Bbb Z})$, $SL_2({\Bbb Z})$, symmetric groups S _n , alternating groups A _n , dihedral groups, and finite abelian groups.