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Show that f is bijective on T

Source: IMO Shortlist 2017 A3

July 10, 2018
IMO Shortlistfunctionalgebra

Problem Statement

Let SS be a finite set, and let A\mathcal{A} be the set of all functions from SS to SS. Let ff be an element of A\mathcal{A}, and let T=f(S)T=f(S) be the image of SS under ff. Suppose that f∘g∘f≠g∘f∘gf\circ g\circ f\ne g\circ f\circ g for every gg in A\mathcal{A} with g≠fg\ne f. Show that f(T)=Tf(T)=T.
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