Exponent of a group

Definition

Let $G$ be a group. The exponent of $G$ is the least positive integer $n$ such that $g^n = e$ for each $g \in G$. If no such integer exists, then $G$ has infinite exponent.

Remarks

When the exponent is finite, it is the least common multiple of the orders of the elements of $G$. The exponent of a finite group divides the order of the group, by Lagrange's theorem. The exponent of the cyclic group of order $n$ is $n$, and the exponent of the Klein four-group is $2$. The exponent of $(\mathbb{Z}/n\mathbb{Z})^\times$ is the Carmichael function $\lambda(n)$.

Explanations

L1 human

The exponent is the smallest power that sends every element to the identity. It is a common multiple of all the element orders. A finite group always has a finite exponent, and the exponent divides the order.