Cyclic Groups, Generators, Element Orders & Coset Partitions

The Order of an element a in G (o(a)) is the smallest positive integer k such that a^k = e (under multiplication) or k * a = 0 (under addition).

Corollary of Lagrange's Theorem: In any finite group G, o(a) divides |G|. Furthermore, a^{|G|} = e for all a in G.

Definition of a Cyclic Group:

A group G is cyclic if there exists an element g in G such that G = langle g rangle = { g^k : k in mathbb{Z} }. The element g is called a generator.

Fundamental Properties:

1. Every cyclic group is Abelian.

2. Any subgroup of a cyclic group is also cyclic.

3. In a finite cyclic group of order n, the element a^k has order n / gcd(k, n).

In the additive group mathbb{Z}_n = {0, 1, ..., n - 1} under addition modulo n:

An element k is a generator if and only if gcd(k, n) = 1.

Therefore, the total number of generators is phi(n) (Euler's Totient Function):

phi(n) = n * prod_{p | n} (1 - frac{1}{p})

For example, for n = 12 = 2^2 * 3:

phi(12) = 12 (1 - 1/2) (1 - 1/3) = 12 (1/2) (2/3) = 4 generators: {1, 5, 7, 11}.