Definition:Order of Group Element/Infinite/Definition 1

Definition

Let $G$ be a group whose identity is $e_G$.

Let $x \in G$ be an element of $G$.

$x$ is of infinite order, or has infinite order, if and only if there exists no $k \in \Z_{> 0}$ such that $x^k = e_G$:

$\order x = \infty$

Also known as

Some sources refer to the order of an element of a group as its period.

Also denoted as

The order of an element $x$ in a group is sometimes seen as $\map o x$.

Some sources render it as $\map {\operatorname {Ord} } x$.

Hence, in the context of an element of infinite order, the notation $\map o x = \infty$ can sometimes be seen.

Also see

• Equivalence of Definitions of Infinite Order Element

Sources

• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Problem $\text{GG}$
• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 41$
• 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 5$: Groups $\text{I}$
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 38$. Period of an element
• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.4$: Cyclic groups
• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $3$: Elementary consequences of the definitions: Definition $3.9$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): order: 6. (of a group or element)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): order: 6. (of a group or element)