📖 Table of Contents

🔑 Key

Blue - Definitions

Red - Important Remark / Forumla

Green - Exercise for reader

✍️ Acronyms

WTS - Want To Show

RHS - Right Hand Side

LHS - Left Hand Side

St. - Such that

FTA - Fundamental Theorem of Algebra

📚 Review

Monoid

A monoid has two properties,

• ${\bf{Associative}}: x\cdot (y\cdot z) = (x\cdot y)\cdot z$
• ${\bf{Identity / Neutral\ Element}}: \exists\ e \in G: x\cdot e\ = e\cdot x \ =\ x$

Adding a third property

• ${\bf{Inverse}}: \forall\ x \in G\ \exists \ y \in G:\ x\cdot y = y\cdot x =\ e$
  • ($y = x^{-1}$)

Groups

A Group is a Monoid with Inverses

Abelian Group

A group with Communativity is an Abelian Group

• ${\bf{Commutativity}}: \forall x,y\in G: x\cdot y = y\cdot x$

Examples:

• $(\Z, +)$ - Abelian Group
• $(\Z, \cdot)$ - Monoid (Abelian Monoid because $n\cdot m = m \cdot n$)
  • 0 is not invertible
• $(\Z_n, +_{mod(n)})$ = {0, 1, ..., n - 1} - Abelian Group
• $(\Z_n, \cdot_{mod(n)})$ - Monoid
• $(\R, +)$ - Abelian Group