用户:CaffeineP/笔记/Groups and Symmetry (M. A. Armstrong)

• group theory^w
• group^w
• symmetry^w

Chapter 2: Axioms

Definition of a group^w

$G$ is a set^w and $\cdot$ a binary operation^w, $ab:=a\cdot b$ , then

$(G,\cdot )\ is\ a\ group:\Leftrightarrow {\underset {x\in G,\,y\in G}{\forall }}xy\in G\land {\underset {x\in G,\,y\in G,\,z\in G}{\forall }}(xy)z=x(yz)\land {\underset {e\in G}{\exists }}\ {\underset {x\in G}{\forall }}\left(xe=x=ex\land {\underset {y\in G}{\exists }}yx=e=xy\right)$

Uniqueness of the identity element^w of a group

${\underset {e\in G,\,e'\in G}{\forall }}\left({\underset {x\in G}{\forall }}xe=x=ex\land {\underset {x\in G}{\forall }}xe'=x=e'x\Rightarrow e=ee'=e'\right)$

Uniqueness of the inverse^w of each element of a group

${\underset {x\in G}{\forall }}\ {\underset {y\in G,\,z\in G}{\forall }}(yx=e=xy\land zx=e=xz\Rightarrow y=ey=(zx)y=z(xy)=ze=z)$

Hence $(G,\cdot )\ is\ a\ group\Leftrightarrow$

${\underset {x\in G,\,y\in G}{\forall }}xy\in G\land {\underset {x\in G,\,y\in G,\,z\in G}{\forall }}(xy)z=x(yz)\land {\underset {e\in G}{\exists !}}\ {\underset {x\in G}{\forall }}\left(xe=x=ex\land {\underset {y\in G}{\exists !}}\left(yx=e=xy,\,y=:x^{-1}\right)\right)$