Group

Summary

A group is a set $G$ with a binary operation $\cdot$ that has these three properties. The operation is associative: $(a \cdot b) \cdot c = a \cdot (b \cdot c)$ for all $a, b, c$ in $G$. There is an identity element $e$ in $G$ such that $e \cdot a = a \cdot e = a$ for all $a$ in $G$. Each element $a$ of $G$ has an inverse $a^{-1}$ in $G$ such that $a \cdot a^{-1} = a^{-1} \cdot a = e$.

The order of a finite group is its number of elements.