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☕ Description

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🔗 Links

📖 Textbook

W10 Discussion

103 Cheat Sheet

^ Not updated for final yet

📖 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

ord - order

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: