\part{Cyclic Groups}

\chapter{Cyclic Group}
- cyclic group
- generator element

If the group operator is clear, one may write the set of elements generated by the element $g$ as the following
\[ \langle g \rangle = \{ g^{k} : k \in \mathbb{Z} \} \].

Finite cyclic groups of the same order are all isomorphic, hence we define a notation to represent cyclic groups of order n.

- Cyc(n)	Cyclic group of order n
- order
- cyclic subgroup


- cyclic subgroup

\chapter{Properties of Cyclic Groups}

- Cyclic group isomorphism to $\mathbb{Z}\setminus n\mathbb{Z}$
- subgroups of cyclic groups are cyclic
- d divides the order of cyclic group implies unique subgroup of order d
- sum of Euler totients of divisors of N equals N 
- Cauchy's theorem

