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: \(a \star b \in G \quad \forall a, b \in G\).

Associativity: \(a \star b ) \star c = a \star (b \star c) \quad \forall a, b, c \in G \).

Identity: \(a \star e = e = e \star a \quad \forall a \in G\).

Inverse: \(a \star a^{-1} = e = a^{-1} \star a \quad \forall a \in G\).

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