“
There is a somewhat surprising source of cyclic groups: if p is prime, the group ((Z/pZ) ! , ·) is cyclic. We will prove a more general statement when we have accumulated more machinery (Theorem IV.6.10), but the adventurous reader can already enjoy a proof by working out Exercise 4.11. This is a relatively deep fact; note that, for example, (Z/12Z) ! is not cyclic (cf. Exercise 2.19, and Exercise 4.10). The fact that (Z/pZ) ! is cyclic for p prime means that there must be integers a such that every non-multiple of p is congruent to a power of a; the usual proofs of this fact are not constructive, that is, they do not explicitly produce an integer with this property. There is a very pretty connection between the order of an element of the cyclic group (Z/pZ) ! and the so-called ‘cyclotomic polynomials’; but that will have to wait for a little field theory (cf. Exercise VII.5.15
”
”