Abstract Algebra

My notes (not part of Algebra 1).

Group Axioms

A group $(G,\star )$ is a non-empty set $G$ together with a binary operation $\star$ that satisfy four properties:

Closure: $a\star b\in G\mathrm{\forall }a,b\in G$.

Associativity: $a\star b)\star c=a\star (b\star c)\mathrm{\forall }a,b,c\in G$.

Identity: $a\star e=e=e\star a\mathrm{\forall }a\in G$.

Inverse: $a\star a^{-1}=e=a^{-1}\star a\mathrm{\forall }a\in G$.

Citation: Rowland, Todd and Weisstein, Eric W. "Group." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Group.html https://mathworld.wolfram.com/Group.html