Ojo's Dojo

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# Group Theory

A binary operation on a set $G$ is function \[\star: G \times G \to G\] and, for convenience, for any $a,b \in G$ we write $\star(a,b)$ as $a \star b$

A group is a set $G$ equipped with a binary operation $\star$ on $G$ such that:

$\forall a,b \in G$ we write $\star (a,b)$ as $a \star b$
There exists an element $e \in G$ (which we call the identity) such that $\forall a \in G$ we have $$ e \star a = a \star e = a $$
For each element a $\in G$ there is an element $a^{-1} \in G$ (which we call the inverse of a) such that $$ a \star a^{-1} = a^{-1} \star a = e $$