Definition:Order of Group Element/Definition 2

Definition

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

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

The order of $x$ (in $G$), denoted $\order x$, is the order of the group generated by $x$:

$\order x := \order {\gen x}$

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 $\map {\operatorname {Ord} } x$.

Also see

• Equivalence of Definitions of Order of Group Element

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $25$. Cyclic Groups and Lagrange's Theorem
• 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.9$: Subgroups: Example $28$
• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 41$
• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.4$: Cyclic groups