Questions regarding the group of symmetries of $\cos(2\pi x)$

Question

This problem introduces the group of symmetries of $\cos(2\pi x)$.

Let $H(x) = -x$ and $T(x) = x+1$

a) Show that $\cos(2\pi H(x))$ = $\cos(2\pi x)$ and that $\cos(2\pi T(x))$ = $\cos(2\pi x)$

b) Find the orders of $H$ and of $T$

I finished part a) already. I just don't understand the question of part b) and how to do part b). What is the identity here?

Thanks!

Answer 1

$H$ is an inversion, so of order two. $T$ is translation of order $\infty$. Call the identity operation $I$, then $I(x)=x$.