GroupDefinition 1. A group $\langle G, \ast \rangle$ is a set $G$, closed under a binary operation $\ast$, such that the following axioms are satisfied:(1) $\forall a, b, c \in G$, $(a \ast b) \ast c = a \ast (b \ast c)$(2) $\exists e \in G$ such that $\forall x \in G$, $e \ast x = x \ast e = x$(3) $\forall a \in G$, $\exists a' \in G$ such that $a \ast a' = a' \ast a = e$.