Definition:Order of an Element

Definition

The order $\left|{x}\right|$ of an element $x$ of a group $G$ is the smallest $k \in \N: k > 1$ such that $x^k = e_G$, where $e_G$ is the identity of $G$.

If there is no such $k$, then $x$ is said to be of infinite order, or has infinite order.

Otherwise it is of finite order, or has finite order.

Also known as

Some sources call this the period of the element.

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 25$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 41$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 38$
• John F. Humphreys: A Course in Group Theory (1996): $\S 3$: Definition $3.9$