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Endomorphism with exactly one fixed point

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December 14, 2019
functiongroup theoryendomorphismsmorphism

Problem Statement

Let be a finite group G G having an endomorphism η \eta that has exactly one fixed point.
a) Demonstrate that the function f:G⟶G f:G\longrightarrow G defined as f(x)=x−1⋅η(x) f(x)=x^{-1}\cdot\eta (x) is bijective. b) Show that G G is commutative if the composition of the function f f from a) with itself is the identity function.
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