Definition:Order of Group Element/Infinite/Definition 3

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 the group $\gen x$ generated by $x$ is of infinite order.

$\order x = \infty \iff \order {\gen x} = \infty$

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

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 5.4$. Cyclic groups
• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $25$. Cyclic Groups and Lagrange's Theorem
• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 41$