Group theory/Order of an element/Definition

Order of a group element

Let ${}G$ be a group and ${}g\in G$ an element. Then we call the smallest positive number ${}n$ with ${}g^{n}=e_{G}$ the order of ${}g$. For this, we write ${}\operatorname {ord} \,(g)$. If all positive powers of ${}g$ are different from the neutral element, then we set ${}\operatorname {ord} \,(g)=\infty$.