A1 - definition of a group

let a set $G$ be equipped with a binary operation $*$
$(G, *)$ is said to be a group if the three conditions are satisfied:

associativity

for any $a, b, c \in G :$

(a * b) * c = a * (b * c)

identity

$e \in G$ such that for any $a \in G$ , there is $a * e = e * a = a$

inverse

for every element $a \in G \exists b \in G$ such that $a * b = b * a = e$