- #1

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## Main Question or Discussion Point

Problem: If ##H = \langle x \rangle## and ##|H| = n##, then ##x^n=1## and ##1,x,x^2,\dots, x^{n-1}## are all the distinct elements of ##H##.

This is just a proposition in my book with a proof following it. What I don't get is the very beginning of the proof: "Let ##|x| = n##. The elements ##1,x,x^2,\dots, x^{n-1}## are all distinct elements because..."

Isn't the fact that ##|x| = n## part of what we wanted to prove? Why does the proof just "let" this be the case?

This is just a proposition in my book with a proof following it. What I don't get is the very beginning of the proof: "Let ##|x| = n##. The elements ##1,x,x^2,\dots, x^{n-1}## are all distinct elements because..."

Isn't the fact that ##|x| = n## part of what we wanted to prove? Why does the proof just "let" this be the case?