Let-G-be-a-finite-group-f-be-an-automorphism-of-G-such-that-f-x-x-x-e-Then-prove-that-i-g-G-x-G-such-that-g-x-1-f-x-ii-If-x-G-f-f-x-x-G-is-Abelian-

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Question Number 193796 by Rajpurohith last updated on 20/Jun/23

Let G be a finite group,f be an automorphism of G such that f(x)=x ⇒x=e . Then prove that, (i)∀g∈G, ∃x∈G such that g=x^(−1) f(x). (ii)If ∀x∈G , f(f(x))=x ⇒ G is Abelian.

$${Let}\:{G}\:{be}\:{a}\:{finite}\:{group},{f}\:{be}\:{an}\:{automorphism}\:{of}\:{G} \\ $$$${such}\:{that}\:{f}\left({x}\right)={x}\:\Rightarrow{x}={e}\:. \\ $$$${Then}\:{prove}\:{that}, \\ $$$$\left(\boldsymbol{{i}}\right)\forall{g}\in{G},\:\exists{x}\in{G}\:{such}\:{that}\:{g}={x}^{−\mathrm{1}} {f}\left({x}\right). \\ $$$$\left(\boldsymbol{{ii}}\right){If}\:\forall{x}\in{G}\:,\:{f}\left({f}\left({x}\right)\right)={x}\:\Rightarrow\:{G}\:{is}\:{Abelian}. \\ $$$$ \\ $$

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Let-G-be-a-finite-group-f-be-an-automorphism-of-G-such-that-f-x-x-x-e-Then-prove-that-i-g-G-x-G-such-that-g-x-1-f-x-ii-If-x-G-f-f-x-x-G-is-Abelian-

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