Exercise 4.3.1 Verify the claims in the proof of Theorem 4.3.2. Specifically, prove that (i) a -

a (91,..., 9p) =

(ga+1,..., ga+p) defines an action of Z, on GP and (ii) X = {(9₁,..., 9p) EGP | 91929p = e} is invariant by Zp

and hence the Z, action on GP restricts to an action on X. Hint: for the last part, you need to show that for all

a € Zp, 91 9p = e if and only if ga+1 ga+p = e.