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existence of operator on set which is nonassociative and invertible

Source: Putnam 1984 B3

September 2, 2021
Setsabstract algebra

Problem Statement

Prove or disprove the following statement: If FF is a finite set with two or more elements, then there exists a binary operation * on FF such that for all x,y,zx,y,z in FF,
(i)(\text i) xz=yzx*z=y*z implies x=yx=y (ii)(\text{ii}) x(yz)(xy)zx*(y*z)\ne(x*y)*z