Cyclic group

(a) In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g (called a "generator" of the group) such that, when written multiplicatively, every element of the group is a power of g (a multiple of g when the notation is additive).

(b) Let |G| = p. Pick an element from G which is not the identity element
and call it x.

Compute:

x, x^2, x^3, ..., x^p = e

This list of p powers of x must include every element of G. To see this, suppose that this is not the case. Then <x> is some proper subgroup of G. Lagrange's Theorem says that the order of this proper subgroup must divide the order of G (which is the prime number p). Clearly this cannot happen since the only divisors of the prime p are 1 and p.Thus G = <x> and G is hence cyclic.