▗▄▄▄▖▗▞▀▚▖▄▄▄▄  ▄    ■  ▗▞▀▜▌▄   ▄  ▄▄▄       ▗▄▖    ▗▖ ▄▄▄  
  █  ▐▛▀▀▘█ █ █ ▄ ▗▄▟▙▄▖▝▚▄▟▌█   █ █   █     ▐▌ ▐▌   ▗▖█   █ 
  █  ▝▚▄▄▖█   █ █   ▐▌        ▀▀▀█ ▀▄▄▄▀     ▐▌ ▐▌▄  ▐▌▀▄▄▄▀ 
  █             █   ▐▌       ▄   █           ▝▚▄▞▘▀▄▄▞▘      
                    ▐▌        ▀▀▀                            
                                                             

# 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 $$ a \star a^{-1} = a^{-1} \star a = e $$