Group (Mathematics)

In modern algebra, a group is a set of objects, together with a rule of composition, which obey certain rules.

Definition

A group is a set of objects, sometimes referred to as $G\$, and a rule of composition, $*\$, such that

1. . There is an element of $G\$, call it $e\$, such that for any $x\$ in $G\$, $e*x=x*e=x\$. This element is called the identity of the group. We write $x\in G\$ to mean that $x\$ is in $G\$.

1. . For any two $x,y\in G\$, their composition $x*y\in G\$.

1. . For any three $x,y,z\in G,x*(y*z)=(x*y)*z\$