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f (xy) = f (x) f (y) , f (x) =x^{-1} for more than 3/4 of members of finite gro

Source: 15th -b QEDMO problem 11 (19. - 22. 10. 2017) https://artofproblemsolving.com/community/c1512515_qedmo_2005

May 30, 2021
algebrafunctionalfunctional equation

Problem Statement

Let GG be a finite group and f:G→Gf: G \to G a map, such that f(xy)=f(x)f(y)f (xy) = f (x) f (y) for all x,y∈Gx, y \in G and f(x)=x−1f (x) = x^{-1} for more than 34\frac34 of all x∈Gx \in G is fulfilled. Show that f(x)=x−1f (x) =x^{-1} even holds for all x∈Gx \in G holds.
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