Group and Abelian Group

Let $G$ be a set and let $\ast$ be a binary operation on $G$. The pair $(G,\ast)$ is called a group if the following three conditions hold.

1. (Associativity) The operation $\ast$ is associative in the sense of Binary Operation on a Set; that is, $(a\ast b)\ast c=a\ast(b\ast c)$ for all $a,b,c\in G$.
2. (Existence of an identity element) There exists $e\in G$ such that $e\ast a=a$ and $a\ast e=a$ for every $a\in G$. Any such element $e$ is called an identity element of $(G,\ast)$.
3. (Existence of inverses) There exists an identity element $e$ as in condition 2 such that for every $a\in G$ there is $b\in G$ with

Any such $b$ is called an inverse of $a$ with respect to $e$.

The group $(G,\ast)$ is called abelian (or commutative) if in addition $\ast$ is commutative in the sense of Binary Operation on a Set, that is, $a\ast b=b\ast a$ for all $a,b\in G$.

When the operation is clear from the context we write $ab$ instead of $a\ast b$ and speak of the group $G$.